ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra01RealNumbers due 1/1/10 at 2:00 AM
9.(1 pt) setAlgebra01RealNumbers/sw1 2 15a.pg
Add the fractions, and reduce your answer.
8
6
÷
57 36
The reduced answer is
/
10.(1 pt) setAlgebra01RealNumbers/sw1 2 15b.pg
Add the fractions, and reduce your answer.
8
8
−
5÷
4
4
1.(1 pt) setAlgebra01RealNumbers/srw1 1 8.pg
Evaluate the expression 3(−2)(2 − 1 − 2(1)).
(Your answer cannot be an algebraic expression. )
2.(1 pt) setAlgebra01RealNumbers/sw1 2 9.pg
Use the properties of real numbers to write the expression
12(5m)
in the form of
A · m.
The reduced answer is
/
11.(1 pt) setAlgebra01RealNumbers/order
Evaluate the expression
1
(−8 − 5/42)
7
The number A =
3.(1 pt) setAlgebra01RealNumbers/sw1 2 11.pg
Use properties of real numbers to write the expression
9
− (4x − 4y)
2
in the form of
A · x + B · y.
and the number B = .
The number A =
4.(1 pt) setAlgebra01RealNumbers/sw1 2 12.pg
Use properties of real numbers to write the expression
in the form of
of ops.pg
NOTE: Your answer cannot be an algebraic expression.
(7a)(8b + 9c − 9d)
K · ab + M · ac + N · ad.
,
,
.
The number K =
The number M =
The number N =
5.(1 pt) setAlgebra01RealNumbers/sw1 2 13a.pg
Add the fractions, and reduce your answer.
3 19
+
6
6
/
The reduced answer is
6.(1 pt) setAlgebra01RealNumbers/sw1 2 13b.pg
Add the fractions, and reduce your answer.
15 8
+
6
9
/
The reduced answer is
7.(1 pt) setAlgebra01RealNumbers/sw1 2 14a.pg
Add the fractions, and reduce your answer.
9
3
+
55 48
/
The reduced answer is
8.(1 pt) setAlgebra01RealNumbers/sw1 2 14b.pg
Add the fractions, and reduce your answer.
2
7
+
+3
8 17
/
The reduced answer is
12.(1 pt) setAlgebra01RealNumbers/sw1
Evaluate the expression |89|.
2 47a.pg
13.(1 pt) setAlgebra01RealNumbers/sw1
Evaluate the expression | − 70|.
2 47b.pg
14.(1 pt) setAlgebra01RealNumbers/srw1
Evaluate the expression |81|.
8 65.pg
15.(1 pt) setAlgebra01RealNumbers/srw1
Evaluate the expression | − 159|.
8 66.pg
16.(1 pt) setAlgebra01RealNumbers/srw1
Evaluate the expression |130 − 312|.
8 67.pg
17.(1 pt) setAlgebra01RealNumbers/ur ab 8 1.pg
Evaluate the expression | − (−43 − 192)|.
18.(1 pt) setAlgebra01RealNumbers/sw1
Evaluate the expression
2 49a.pg
|| − 41| − | − 3||.
19.(1 pt) setAlgebra01RealNumbers/sw1
Evaluate the expression
−37
.
| − 37|
Your answer is
20.(1 pt) setAlgebra01RealNumbers/sw1
Evaluate the expression
1
|4 − | − 47||.
2 49b.pg
2 50a.pg
Your answer is
21.(1 pt) setAlgebra01RealNumbers/sw1
Evaluate the expression
4. π ≥ 3.1416
2 50b.pg
28.(1 pt) setAlgebra01RealNumbers/lhp1 79-82.pg
Use absolute value and inequality notations to describe the following situations. You may use ”|” for absolute value sign; leq
for ≤ and geq for ≥. e.g. you may use ”|x| geq 5” for ”|x| ≥ 5”.
The distance between x and 3 is no more than 24
The distance between x and −11 is at least 28
x is at least 11 units from 0
x is at most 3 units from 0
29.(1 pt) setAlgebra01RealNumbers/srw1 1 21-28.pg
Match the statements defined below with the letters labeling
their equivalent expressions.
You must get all of the answers correct to receive credit.
1. x is any real number
2. x is greater than 10
3. x is greater than or equal to 10
4. The distance from x to 10 is more than 2
5. The distance from x to 10 is less than or equal to 2
A. 10 < x
B. |x − 10| > 2
C. −∞ < x < ∞
D. |x − 10| ≤ 2
E. x ≥ 10
−12 − |12 − | − 12||.
Your answer is
22.(1 pt) setAlgebra01RealNumbers/lhp1 49-58.pg
This exercise concerns the definition of absolute values. Evaluate the following expressions:
| − 18| =
−18 − | − 3| =
If 18 < x, simplify the expression by removing the absolute sign
|18 − x| =
If y < −3, evaluate the expression
|y + 3|
=
y+3
23.(1 pt) setAlgebra01RealNumbers/srw1 8 73.pg
|104 − 333|
Evaluate the expression
. Give you answer in deci| − 8|
mal notation correct to three decimal places or give your answer
as a fraction.
[NOTE: Your answer can be an algebraic expression.
Make sure to include all necessary (, ). ]
30.(1 pt) setAlgebra01RealNumbers/srw1 1 39-48.pg
Match the statements defined below with the letters labeling
their equivalent intervals.
You must get all of the answers correct to receive credit.
1. x ∈ [3, 8)
2. x ∈ (−∞, 3)
3. x ∈ (3, 8]
4. x ∈ [3, ∞)
5. x ∈ [3, 8]
A. 3 < x ≤ 8
B. 3 ≤ x ≤ 8
C. 3 ≤ x < 8
D. x < 3
E. 3 ≤ x
24.(1 pt) setAlgebra01RealNumbers/sw1 2 17.pg
Enter a T or an F in each answer space below to indicate whether
the corresponding statement is true or false.
You must get all of the answers correct to receive credit.
√
1. 2 < 9
2. 11 − 1 < 11
25.(1 pt) setAlgebra01RealNumbers/sw1 2 18.pg
Enter a T or an F in each answer space below to indicate whether
the corresponding statement is true or false.
You must get all of the answers correct to receive credit.
Your answer for the following statement is
22 24
<
23 25
Your answer for the following statement is
15
16
− <−
16
17
26.(1 pt) setAlgebra01RealNumbers/srw1 8
Find the distance betweem 355 and 372.
77.pg
[NOTE: Your answer can be an algebraic expression]
27.(1 pt) setAlgebra01RealNumbers/srw1 1 9-18.pg
Enter a T or an F in each answer space below to indicate whether
the corresponding statement is true or false.
You must get all of the answers correct to receive credit.
1. −1 < −10
2. 8 − 1 ≤ 8
3. −2 ≤ −2
2
31.(1 pt) setAlgebra01RealNumbers/lhp1 25-30.pg
Sketch the following sets on a piece of paper and write them in
interval notation. Enter the interval in the answer box. You may
use ”infinity” for ∞ and ”-infinity” for −∞. For example, you
may write (-infinity, 5] for the interval (−∞, 5].
x ≥ 18
x ≤ 26
x>1
x<5
32.(1 pt) setAlgebra01RealNumbers/lhp1 31-34.pg
Sketch the following sets on a piece of paper and write them in
interval notation. Enter the interval in the answer box. You may
use ”infinity” for ∞ and ”-infinity” for −∞. For example, you
may write (-infinity, 5] for the interval (−∞, 5].
1≤x≤4
3<x≤8
21 < x < 25
19 ≤ x < 24
1.
2.
3.
4.
A.
B.
C.
D.
33.(1 pt) setAlgebra01RealNumbers/lhp1 37-44.pg
This exercise concerns with operations with inequality and interval notations.
Match the sets and the inequalities by Placing the letter of the
inequality next to each set listed below:
1. All x in the interval [−17, 19]
2. x is no more than 19
3. All x in the interval (−17, 19]
4. All x in the interval (−17, 19)
5. x is greater than −17
A. −17 < x ≤ 19
B. −17 ≤ x ≤ 19
C. −17 < x
D. x ≤ 19
E. −17 < x < 19
S∪T
S∩T
S ∪W
T ∩W
Ø
(−∞, 3)
[−5, 7]
[0, 3)
37.(1 pt) setAlgebra01RealNumbers/ur ab 10 3.pg
Let S = (0, ∞), T = (−∞, 4], and W = [−6, 4).
For each intersection or union, choose the correct notation for
the resulting interval.
1. S ∪W
2. S ∩W
3. S ∩ T
4. T ∪W
A. (0, 4)
B. (0, 4]
C. (−∞, 4]
D. [−6, ∞)
34.(1 pt) setAlgebra01RealNumbers/ur ab 10 1.pg
Match each interval below with set-builder notation for the same
interval.
1. (2, 9]
2. (−∞, 2]
3. (2, ∞)
4. [2, 9)
5. (−∞, 2)
A. {x|2 < x ≤ 9}
B. {x|2 ≤ x < 9}
C. {x|x ≤ 2}
D. {x|x > 2}
E. {x|x < 2}
38.(1 pt) setAlgebra01RealNumbers/srw1 8 1.pg
Match the statements defined below with the letters labeling
their equivalent expressions.
You must get all of the answers correct to receive credit.
1. |x − 2| ≤ 7
2. |x − 2| ≥ 7
3. |x − 2| < ∞
4. |x − 2| > 7
5. |x − 2| < 7
A. x ∈ (−∞, ∞)
B. x ∈ [−5, 9]
C. x ∈ (−∞, −5) ∪ (9, ∞)
D. x ∈ (−∞, −5] ∪ [9, ∞)
E. x ∈ (−5, 9)
35.(1 pt) setAlgebra01RealNumbers/ur ab 8 2.pg
The interval described in set-builder notation by the inequality
|3x − 12| < 27 has interval notation (a, b)
for
a=
and
b=
36.(1 pt) setAlgebra01RealNumbers/ur ab 10 2.pg
Let S = [−5, 3), T = [0, 7], and W = (−∞, −1).
For each intersection or union, choose the correct notation for
the resulting interval.
39.(1 pt) setAlgebra01RealNumbers/ur ab 8 3.pg
Let S be the union of the two intervals (−∞, −3] and [17, ∞).
Then S can also be described in set-builder notation by the inequality |x − a| ≥ b
for
a=
and
b=
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
3
ARNOLD PIZER
rochester problib from CVS June 25, 2004
Rochester WeBWorK Problem Library
1.(1 pt) setAlgebra02ExponentsRadicals/srw1
Evaluate the expression −42 .
WeBWorK assignment Algebra02ExponentsRadicals due 1/2/10 at 2:00 AM
x=
2 2.pg
[NOTE: Your answer cannot be an algebraic expression. ]
2.(1 pt) setAlgebra02ExponentsRadicals/srw1
Evaluate the expression 63 + 64 .
2 33.pg
[NOTE: Your answer cannot be an algebraic expression. ]
3.(1 pt) setAlgebra02ExponentsRadicals/srw1
Evaluate the expression 2−3 64 .
2 3.pg
[NOTE: Your answer cannot be an algebraic expression. ]
4.(1 pt) setAlgebra02ExponentsRadicals/srw1
3
3
Evaluate the expression
.
−3
2 5.pg
5.(1 pt) setAlgebra02ExponentsRadicals/srw1
2−1
Evaluate the expression 2 .
2
2 7.pg
[NOTE: Your answer cannot be an algebraic expression. ]
[NOTE: Your answer cannot be an algebraic expression. ]
6.(1 pt) setAlgebra02ExponentsRadicals/srw1
42
Evaluate the expression −3 .
2
2 7a.pg
[NOTE: Your answer cannot be an algebraic expression. ]
7.(1 pt) setAlgebra02ExponentsRadicals/srw1 2 41.pg
24 25 2−3
The expression √
equals 2n where n is:
43 23 2−3
8.(1 pt) setAlgebra02ExponentsRadicals/srw1 2 53.pg
The expression (3a4 b4 c2 )2 (2a4 b2 c4 )3 equals nar bsct
where n, the leading coefficient, is:
and r, the exponent of a, is:
and s, the exponent of b, is:
and finally t, the exponent of c, is:
[NOTE: Your answers cannot be algebraic expressions.]
9.(1 pt) setAlgebra02ExponentsRadicals/srw1 2 60.pg
5 4 3 −2 −3
x y z x
The expression
equals xr ys zt
x 3 y 5 z4 y 3
where r, the exponent of x, is:
and s, the exponent of y, is:
and finally t, the exponent of z, is:
[NOTE: Your answers cannot be algebraic expressions.]
11.(1 pt) setAlgebra02ExponentsRadicals/sw1
Evaluate the expression (−2)6 .
Your answer is
3 1a.pg
12.(1 pt) setAlgebra02ExponentsRadicals/sw1
Evaluate the expression −26 .
Your answer is
3 1b.pg
13.(1 pt) setAlgebra02ExponentsRadicals/sw1
Evaluate the expression (−6)0 .
Your answer is
3 1c.pg
14.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression 64 7−2 equals n/d where
the numerator n is
the denorminator d is
.
3 3a.pg
15.(1 pt) setAlgebra02ExponentsRadicals/sw1
Evaluate the expression
106
104
Your answer is
3 3b.pg
16.(1 pt) setAlgebra02ExponentsRadicals/sw1
Evaluate the expression (22 · 22 )2 .
Your answer is
3 3c.pg
17.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
x9 (2x)5
x5
equals cxe where
the coefficient c is
, the exponent e is
3 19.pg
18.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
1 3
x (20x−11 )
x4
4
e
equals c/x where
the coefficient c is
, the exponent e is
3 21.pg
19.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
(rs)−2 (3s)5 (3r)5
3 23.pg
equals cr e sd where
the coefficient c is
nent d of s is
.
.
.
, the exponent e of r is
20.(1 pt) setAlgebra02ExponentsRadicals/sw1 3 25.pg
The expression
(6y3 )4
4y6
e
equals cy where
the coefficient c is
, the exponent e of y is
10.(1 pt) setAlgebra02ExponentsRadicals/nsc2s7p1.pg
Find x if
(5.4)x (5.4)6
= (5.4)3
(5.4)4
1
, the expo-
.
21.(1 pt) setAlgebra02ExponentsRadicals/sw1 3 27.pg
The expression
(x4 y2 )6 (xy6 )−2
x4 y3
equals xe /yd where
the exponent e of x is
, the exponent d of y is
29.(1 pt) setAlgebra02ExponentsRadicals/srw1
Evaluate the expression 125−4/3 .
[NOTE: Your answer cannot be an algebraic expression. ]
30.(1 pt) setAlgebra02ExponentsRadicals/srw1
2 88.pg
p
√
p
5
The expression x4 y4 3 x4 y4 x3 equals xr ys
where r, the exponent of x, is:
and s, the exponent of y, is:
.
22.(1 pt) setAlgebra02ExponentsRadicals/sw1 3 29.pg
The expression
(x3 y4 z6 )7
(x6 y4 z)6
r
s
t
equals y z /x where
r, the exponent of y, is:
s, the exponent of z, is:
t, the exponent of x, is:
[NOTE: Your answers cannot be algebraic expressions.]
31.(1 pt) setAlgebra02ExponentsRadicals/srw1
2 88-sol.pg
p
p
√
5
The expression x4 y4 3 x4 y4 x3 equals xr ys
where r, the exponent of x, is:
and s, the exponent of y, is:
32.(1 pt) setAlgebra02ExponentsRadicals/srw1
p
3 √
The expression
64x3 equals nxr
where n, the leading coefficient, is:
and r, the exponent of x, is:
23.(1 pt) setAlgebra02ExponentsRadicals/sw1 3 31.pg
The expression
−3 3 −2 −1
x y z
y−4 z4 x−6
equals zr /(xs yt ) where
r, the exponent of z, is:
s, the exponent of x, is:
t, the exponent of y, is:
[NOTE: Your answers cannot be algebraic expressions.]
24.(1 pt) setAlgebra02ExponentsRadicals/Test1 5.pg
The expression (4b−2 c−6 )0 (4b−3 a6 )1 equals nar bs ct
where n, the leading coefficient, is:
and r, the exponent of a, is:
and s, the exponent of b, is:
and finally t, the exponent of c, is:
[NOTE: Your answers cannot be algebraic expressions.]
25.(1 pt) setAlgebra02ExponentsRadicals/Test1 6.pg
−1 5 −6 −4
x y z
The expression
equals xr ys zt
(xy)3 z−6
where r, the exponent of x, is:
and s, the exponent of y, is:
and finally t, the exponent of z, is:
[NOTE: Your answers cannot be algebraic expressions.]
26.(1 pt) setAlgebra02ExponentsRadicals/srw1
√
Evaluate the expression 3 −8.
2 9.pg
[NOTE: Your answer cannot be an algebraic expression. ]
27.(1 pt) setAlgebra02ExponentsRadicals/srw1
√
Evaluate the expression 225 + 64.
2 80.pg
[NOTE: Your answer cannot be an algebraic expression.]
28.(1 pt) setAlgebra02ExponentsRadicals/srw1
q
Evaluate the expression
2 26.pg
2 17.pg
9·12
3 .
[NOTE: Your answer cannot be an algebraic expression. ]
2
2 91.pg
33.(1 pt) setAlgebra02ExponentsRadicals/sw1
Evaluate the expression
√
144
√
4
Your answer is
3 7a.pg
34.(1 pt) setAlgebra02ExponentsRadicals/sw1
Evaluate the expression
√
112
√
7
Your answer is
3 7b.pg
35.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
√
81
√
16
/
.
equals
3 7c.pg
36.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
−1/2
25
4
equals
/
.
3 9a.pg
37.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
2/3
8
−
27
equals
/
.
3 9b.pg
38.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
3/2
49
64
equals
/
.
3 9c.pg
39.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
√
√
125− 20
√
equals
× 5.
r, the exponent of c, is:
s, the exponent of d, is:
3 11.pg
40.(1 pt) setAlgebra02ExponentsRadicals/srw1 2 109.pg
If you rationalize the denominator of
1
√
√
11 5 − 2 3
then you will get
√
√
r 5+s 3
n
where r, s, and n are all positive integers (with no common factors).
r=
s=
n=
[NOTE: Your answers cannot be algebraic expressions.]
46.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
3/6
y1/5
3 39.pg
47.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
(5x6 y−4/5 )4 (4y2 )3/4
equals nxr /yt where
n, the coefficient, is:
r, the exponent of x, is:
t, the exponent of y, is:
3 41.pg
48.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
3 2/7
xy
y5
equals xr /yt where
r, the exponent of x, is:
t, the exponent of y, is:
3 43.pg
49.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
−1
2a−5
3b−1/6
r
t
equals na /b where
n, the coefficient, is:
r, the exponent of a, is:
t, the exponent of b, is:
3 45.pg
50.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
(16s−6t 3 )3/2
(64s5t −6 )2/3
a
b
equals nt /s where
n, the coefficient, is:
a, the exponent of t, is:
b, the exponent of s, is:
3 47.pg
3 49.pg
equals yr where
r, the exponent of y, is:
41.(1 pt) setAlgebra02ExponentsRadicals/rational denominator.pg
If you rationalize the denominator of
1
√
√
11x 5 − 4y 3
then you will get
A
B
where A =
and B =
42.(1 pt) setAlgebra02ExponentsRadicals/rational numerator.pg
If you rationalize the numerator of
√
√
3 2
x − 8 3 x + 64
√
x3 + 8
then you will get
A
B
where A =
and B =
43.(1 pt) setAlgebra02ExponentsRadicals/sw1 3 33.pg
The expression
x2/4 x3/4
equals xr
where r, the exponent of x, is:
44.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
(25b)1/2 (3b1/2 )
r
equals nb where
n, the coefficient, is:
r, the exponent of b, is:
3 35.pg
51.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
√
4
x4 = x r
where x is a non-negative real number.
r, the exponent of x, is:
45.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
(c5 d 2 )−1/2
equals 1/(cr d s ) where
3 37.pg
52.(1 pt) setAlgebra02ExponentsRadicals/sw1 3 51.pg
The expression
p
6 6 2
x y = x r ys
where x and y are non-negative real numbers.
r, the exponent of x, is:
s, the exponent of y, is:
3
53.(1 pt) setAlgebra02ExponentsRadicals/sw1
The expression
√
2
a5 b4
r
s
equals a b where
r, the exponent of x, is:
s, the exponent of y, is:
3 53.pg
54.(1 pt) setAlgebra02ExponentsRadicals/sw1
Rationalize the denominator of expression
1
√
7
i.e., write it in the form of
√
a
.
b
Your answer for a is :
Your answer for b is :
55.(1 pt) setAlgebra02ExponentsRadicals/sw1
Rationalize the denominator of expression
r
x
7y
3 57a.pg
i.e., write the expression in the form of
√
Ax
.
B
Your answer for A is :
Your answer for B is :
56.(1 pt) setAlgebra02ExponentsRadicals/sw1
Rationalize the denominator of expression
r
2
,
9
i.e., write it in the form of
√
a
.
b
Your answer for a is :
Your answer for b is :
57.(1 pt) setAlgebra02ExponentsRadicals/sw1
Rationalize the denominator of expression
1
√
2
x
i.e., write the expression in the form of
p
2
f (x)
.
g(x)
Your answer for the function f (x) is :
Your answer for the function g(x) is :
58.(1 pt) setAlgebra02ExponentsRadicals/sw1
Rationalize the denominator of expression
3 59b.pg
1
√
6 2
x
i.e., write the expression in the form of
p
6
f (x)
.
g(x)
Your answer for the function f (x) is :
Your answer for the function g(x) is :
3 57b.pg
59.(1 pt) setAlgebra02ExponentsRadicals/sw1
Rationalize the denominator of expression
3 59c.pg
1
√
8 3
x
i.e., write the expression in the form of
p
8
f (x)
.
g(x)
3 57c.pg
Your answer for the function f (x) is :
Your answer for the function g(x) is :
60.(1 pt) setAlgebra02ExponentsRadicals/Test1
7.pg
p
p
The expression 4 256g2y4 256g2y4 equals kxr ys
where r, the exponent of g, is:
and s, the exponent of y, is:
and k, the leading coefficient is:
61.(1 pt) setAlgebra02ExponentsRadicals/Test1
The expression
√
12
v108
3 59a.pg
equals
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
4
10.pg
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra03Expressions due 1/3/10 at 2:00 AM
x2 +
x+
10.(1 pt) setAlgebra03Expressions/sw1 4 3.pg
The expression 2(6x3 + 3x2 − 7x + 3) − (2x2 + 7x − 7) equals
x3 + x 2 +
x+
11.(1 pt) setAlgebra03Expressions/sw1 4 5.pg
The expression 4(7x + 3) − 5(5x − 2) equals
x+
12.(1 pt) setAlgebra03Expressions/sw1 4 9.pg
√
√
The expression x(2x − 5 x)equals Ax3/2 + Bx
where A equals:
and B equals:
1.(1 pt) setAlgebra03Expressions/srw1 3 1.pg
The expression 3(3 − 7x) + 6(x − 7) equals Ax + B
where A equals:
and B equals:
[NOTE: Your answers cannot be algebraic expressions.]
2.(1 pt) setAlgebra03Expressions/srw1 3 6.pg
The expression 6(7x2 + 3x + 3) − 6(7x2 + 4x + 6) equals
x2 +
x+
3.(1 pt) setAlgebra03Expressions/Test1 11-12.pg
Given
P = 3b3 + 9b − 7,
Q = b2 − 8b + 1,
R = b3 − 6
Then P + Q =
b3 +
b2 +
b+
6
5
and R(P + Q) =
b +
b +
b4 +
b3 +
b2 +
b+
4.(1 pt) setAlgebra03Expressions/srw1 3 13.pg
The expression (2x + 2)(6x − 7) equals Ax2 + Bx +C
where A equals:
and B equals:
and C equals:
13.(1 pt) setAlgebra03Expressions/sw1 4 13.pg
The expression (6t − 7)(2t − 4) equals At 2 + Bt +C
where A equals:
and B equals:
and C equals:
14.(1 pt) setAlgebra03Expressions/sw1 4 15.pg
The expression (7x + 6)(5x − 5) equals Ax2 + Bx +C
where A equals:
and B equals:
and C equals:
15.(1 pt) setAlgebra03Expressions/sw1 4 17.pg
The expression (5 − 4x)2equals Ax2 + Bx +C
where A equals:
and B equals:
and C equals:
5.(1 pt) setAlgebra03Expressions/srw1 3 16.pg
The expression (7t − 5)(6t + 4) + 6t − 3 equals At 2 + Bt +C
where A equals:
and B equals:
and C equals:
16.(1 pt) setAlgebra03Expressions/lhp3 26.pg
The expression 3(3x2 − 10x + 6) − (3x2 + 10x − 9) equals
x2 +
x+
17.(1 pt) setAlgebra03Expressions/lhp3 56.pg
The expression (3x − 6)(2x + 9) equals
x2 +
x+
6.(1 pt) setAlgebra03Expressions/srw1 3 21.pg
√
√
√
√
The expression (6 x + 6 y)(6 x − 6 y)equals Ax + By
where A equals:
and B equals:
7.(1 pt) setAlgebra03Expressions/srw1 3 22.pg
The expression (7x + 4)2equals Ax2 + Bx +C
where A equals:
and B equals:
and C equals:
18.(1 pt) setAlgebra03Expressions/lhp3 58.pg
The expression (4x + 3)2 equals
x2 +
x+
19.(1 pt) setAlgebra03Expressions/lhp3 62.pg
The expression (2x − 7)(2x + 7) equals
x2 −
20.(1 pt) setAlgebra03Expressions/Test1 13.pg
√
√
√
√
The expression (7 x + 3 y)(7 x − 3 y) equals
8.(1 pt) setAlgebra03Expressions/srw1 3 22a.pg
The expression (x − 6)(x2 + 4x + 7)equals Ax3 + Bx2 +Cx + D
where A equals:
and B equals:
and C equals:
and D equals:
21.(1 pt) setAlgebra03Expressions/Test1 19.pg
A square rug lies in the middle of a rectangular room. There are
6 feet of uncovered floor on 2 sides of the rug and 3 feet of uncovered floor on the other 2 sides. Find a polynomial expression
for the area of the room in terms of x, the side length of the rug.
.
The area of the room is
9.(1 pt) setAlgebra03Expressions/sw1 4 1.pg
The expression (7x2 + 3x + 2) + (7x2 − 2x − 2) equals
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
1
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra04ExpressionsFactoring due 1/4/10 at 2:00 AM
and A equals:
and B equals:
1.(1 pt) setAlgebra04ExpressionsFactoring/srw1 3 43.pg
Factor the polynomial x2 + 7x + 10. Your answer can be written
as (x + A)(x + B) where A < B
and A equals:
and B equals:
10.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 45.pg
Factor the polynomial x2 − 6x − 27. Your answer can be written
as (x + A)(x + B) where A < B
and A equals:
and B equals:
2.(1 pt) setAlgebra04ExpressionsFactoring/srw1 3 53.pg
Factor the polynomial x2 + 6x + 8. Your answer can be written
as (x + A)(x + B) where A < B
and A equals:
and B equals:
11.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 47.pg
Factor the polynomial x2 − 6x + 5. Your answer can be written
as (x + A)(x + B) where A < B
and A equals:
and B equals:
3.(1 pt) setAlgebra04ExpressionsFactoring/srw1 3 53a.pg
Factor the polynomial x2 + 3x − 4. Your answer can be written
as (x + A)(x + B) where A < B
and A equals:
and B equals:
12.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 49.pg
Factor the polynomial 20x2 + 37x + 15. Your answer can be
written as (5x + B)(Cx + D) with B, C, and D- integers
where B equals:
and C equals:
and D equals:
4.(1 pt) setAlgebra04ExpressionsFactoring/srw1 3 54.pg
Factor the polynomial 20x2 + 41x + 20. Your answer can be
written as (5x + B)(Cx + D) with B, C, and D- integers
where B equals:
and C equals:
and D equals:
13.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 51.pg
Factor the polynomial 2x2 − 32. Your answer can be written as
A(x + B)(x +C) with integerss A, B, C, and B < C
,B=
, and C =
where A =
5.(1 pt) setAlgebra04ExpressionsFactoring/srw1 3 55.pg
Factor the polynomial x4 +9x2 +18. Your answer can be written
as (x2 + A)(x2 + B) where A < B
and A equals:
and B equals:
14.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 53.pg
Factor the polynomial 20x2 + 27x − 14. Your answer can be
written as (5x − B)(Cx + D) with B, C, and D- integers
where B =
,C=
, and D =
15.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 55.pg
Factor the polynomial (x − 1)(x + 4)2 − (x − 1)2 (x + 4). Your
answer can be written as A(x + B)(x +C) with integers A, B, C
and B < C
where A =
,B=
, and C =
6.(1 pt) setAlgebra04ExpressionsFactoring/srw1 3 56.pg
Factor the polynomial t 7 + 3t 6 − 18t 5 . Your answer can be written as t N (t + A)(t + B) where A < B.
N equals:
and A equals:
and B equals:
16.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 57.pg
Factor the polynomial y3 (y + 4)3 + y4 (y + 4)4 . Your answer can
be written as yr (y + A)s (y2 + By +C) with r, s A, B, and C- integers
where r equals:
and s equals:
and A equals:
and B equals:
and C equals:
7.(1 pt) setAlgebra04ExpressionsFactoring/srw1 3 57.pg
Factor the polynomial x3 − 64. Your answer can be written as
(x − A)(x2 + Bx +C)
where A equals:
and B equals:
and C equals:
8.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 41.pg
Factor the polynomial 5y5 − 4y4. Your answer can be written as
ya (Ayb + B) where
a equals:
and A equals:
and b equals:
and B equals:
17.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 63.pg
Factor the polynomial 16x2 − 40x + 25. Your answer can be
written as (Ax − B)r with A, B, and r being integers.
A=
,B=
, and r =
18.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 69.pg
Factor the polynomial t 6 +6t 5 −7t 4 . Your answer can be written
as t N (t + A)(t + B) where A < B.
N=
,A=
, and B =
9.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 43.pg
Factor the polynomial x2 + 5x + 4. Your answer can be written
as (x + A)(x + B) where A < B
1
19.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 73.pg
Factor the polynomial x4 + 5x2 − 6. Your answer can be written
as (x + A)(x + B)(x2 +C) with integers A, B, C, and A < B
where A =
,B=
, and C =
into the product of two polynomials, A · B where the coefficient
of c in A is less than the coefficient of c in B. Find A and B.
A=
B=
29.(1 pt) setAlgebra04ExpressionsFactoring/factoring hard.pg
The polynomial 15x3 + 10x2 + 18x + 12 can be factored into
the product of two polynomials, A · B where the degree of A
is greater than the degree of B. Find A and B.
A=
B=
30.(1 pt) setAlgebra04ExpressionsFactoring/factor hard2.pg
The polynomial 25x3 + 175x2 − 1x − 7 can be factored into the
product of three polynomials, A · B ·C where the constant term
of A is less than or equal to the constant term of B which is less
than or equal to the constant term of C. Find A, B and C.
A=
B=
C=
31.(1 pt) setAlgebra04ExpressionsFactoring/factor hard3.pg
The polynomial 81x3 − 49xy2 + 729x2 − 441y2 can be factored
into the product of three polynomials, A · B ·C where the coefficient of y in A is less than the coefficient of y in B which is less
than the coefficient of y in C. Find A, B and C.
A=
B=
C=
32.(1 pt) setAlgebra04ExpressionsFactoring/factor hard4.pg
The polynomial 36a2 + 60ab + 25b2 − 36 can be factored into
the product of two polynomials, A · B where the constant term in
A is less than the constant term in B. Find A and B.
A=
B=
33.(1 pt) setAlgebra04ExpressionsFactoring/factor hard5.pg
The polynomial 25a8 + 80a4 b + 64b2 − 9c−8 can be factored
into the product of two polynomials, A · B where the coefficient
of c in A is less than the coefficient of c in B. Find A and B.
A=
B=
34.(1 pt) setAlgebra04ExpressionsFactoring/factor1.pg
The polynomial 729x3 − 64y3 can be factored into the product
of two polynomials, A · B where the degree of A is greater than
the degree of B. Find A and B.
A=
B=
35.(1 pt) setAlgebra04ExpressionsFactoring/factor2.pg
The polynomial 24x3 + 30x2 + 28x + 35 can be factored into
the product of two polynomials, A · B where the degree of A
is greater than the degree of B. Find A and B.
A=
B=
20.(1 pt) setAlgebra04ExpressionsFactoring/sw1 4 75.pg
Factor the polynomial x3 − 125. Your answer can be written as
(x − A)(x2 + Bx +C)
where A =
, and B =
, and C =
21.(1 pt) setAlgebra04ExpressionsFactoring/lhp4
Factor 16 − 4x2
16 − 4x2 = (A − Bx)(C + Dx)
where A= , B= , C= , and D= .
19-24.pg
22.(1 pt) setAlgebra04ExpressionsFactoring/lhp4
Factor the trinomial 25x2 − 30x + 9
25x2 − 30x + 9 = (Ax − B)2
where A is
and B is
.
23.(1 pt) setAlgebra04ExpressionsFactoring/lhp4
Factor the trinomial x2 − 9x + 18
29-34.pg
where A =
43-50.pg
x2 − 9x + 18 = (x − A)(x − B)
and B =
with A < B.
24.(1 pt) setAlgebra04ExpressionsFactoring/lhp4
Factor the trinomial 18x2 − 51x + 35
51-54.pg
18x2 − 51x + 35 = (Ax − B)(Cx − D})
where A = , B = , C = , and D = , with A < C and
B < D.
25.(1 pt) setAlgebra04ExpressionsFactoring/Test1 15.pg
The polynomial 8x6 − 1y15 can be factored into the product of
two polynomials, A · B where the degree of A is greater than the
degree of B. Find A and B.
A=
B=
26.(1 pt) setAlgebra04ExpressionsFactoring/Test1 16.pg
The polynomial 49x2 − 4y10 can be factored into the product of
two polynomials, A · B where the coefficient of y in A is greater
than the coefficient of y in B. Find A and B.
A=
B=
27.(1 pt) setAlgebra04ExpressionsFactoring/Test1 17.pg
The polynomial 25x18 + 40x9 y8 + 16y16 can be factored into the
product of two polynomials, A · B where the coefficient of y in
A is greater than or equal to the coefficient of y in B. Find A and
B.
A=
B=
28.(1 pt) setAlgebra04ExpressionsFactoring/Test1 18.pg
The polynomial 81a10k − 90a5k b + 25b2 − 64c6 can be factored
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
2
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra05RationalExpressions due 1/5/10 at 2:00 AM
x3
x+8
C. x3 (x + 8)
x+8
D.
x3
B.
1.(1 pt) setAlgebra05RationalExpressions/srw1 4 1.pg
Match the expressions below with the letters labeling their
equivalent expressions.
You must get all of the answers correct to receive credit.
x2 + 0x − 49
1. 2
x + 13x + 42
x2 + 3x + 2
2. 2
x + 8x + 12
x2 + 8x + 7
3. 2
x + 4x − 21
x+1
A.
x+6
x+1
B.
x−3
x−7
C.
x+6
4.(1 pt) setAlgebra05RationalExpressions/srw1 4 20.pg
Match the expressions below with the letters labeling their
equivalent expressions.
You must get all of the answers correct to receive credit.
1
1
1.
−
x+6 x−3
1
1
2.
−
x+3 x+6
1
1
+
3.
x−6 x−3
3
A.
(x + 3)(x + 6)
−9
B.
(x − 3)(x + 6)
2x − 9
C.
(x − 3)(x − 6)
2.(1 pt) setAlgebra05RationalExpressions/srw1 4 7.pg
Match the expressions below with the letters labeling their
equivalent expressions.
You must get all of the answers correct to receive credit.
x2 − 9
1. 2
x + 6x + 9
x−3 x+3
2. 2
·
x + 9 x2 − 9
x2 − 9
3. 3
x − 27
1
A. 2
x +9
x−3
B.
x+3
x+3
C. 2
x + 3x + 9
5.(1 pt) setAlgebra05RationalExpressions/srw1 4 29.pg
Match the expressions below with the letters labeling their
equivalent expressions.
You must get all of the answers correct to receive credit.
1
1
1.
− 2
x+3 x −9
1
1
2.
−
x − 3 x2 − 9
1
1
3.
+ 2
x+3 x +9
x−4
A. 2
x −9
x2 + x + 12
B.
(x + 3)(x2 + 9)
x+2
C. 2
x −9
3.(1 pt) setAlgebra05RationalExpressions/srw1 4 13.pg
Match the expressions below with the letters labeling their
equivalent expressions.
You must get all of the answers correct to receive credit.
1.
2.
3.
4.
A.
x3
x+3
x6
x2 +11x+24
x6
x2 +11x+24
x3
x+3
x3
x2 +11x+24
x6
x+3
x6
x+3
x3
x2 +11x+24
1
x3 (x + 8)
1
6.(1 pt) setAlgebra05RationalExpressions/srw1 4 35-37.pg
Match the expressions below with the letters labeling their
equivalent expressions.
You must get all of the answers correct to receive credit.
b
b + a−1
1.
b
b − a−1
b
−a
2. 1a b1
− a2
b2
a
3. b − b a
a+b
a
A.
a−2
B. −ba
b3
C. 2
b + a2
Your answer for the function f (x) is :
Your answer for the function g(x) is :
11.(1 pt) setAlgebra05RationalExpressions/lhp5
7.(1 pt) setAlgebra05RationalExpressions/srw1 4 50.pg
Match the expressions below with the letters labeling their
equivalent expressions.
You must get all of the answers correct to receive credit.
√
√
y + h+ y
1.
√ h √
y + h− y
2.
h
1
A. √
√
y + h− y
1
B. √
√
y + h+ y
Simplify the expression
x2 − 8x + 12
x2 − 7x + 10
and give your answer in the form of
f (x)
.
g(x)
Your answer for the function f (x) is :
Your answer for the function g(x) is :
12.(1 pt) setAlgebra05RationalExpressions/lhp5
55-56.pg
Simplify the expression
8.(1 pt) setAlgebra05RationalExpressions/srw1 4 55-59.pg
Enter a T or an F in each answer space below to indicate whether
the corresponding equation is true or false. An equation is true
ony if it is true for all values of the variables. Disregard values
that make denominators 0.
You must get all of the answers correct to receive credit.
40
40
1.
= 1−
40 − c
c
99 99
99
=
+
2.
40 + x 40
x
x + 99 x
=
3.
y + 99 y
99 + a
a
4.
= 1+
99
99
2x − 2 2 − x
+
x−1 x−1
and give your answer in the form of
f (x)
.
g(x)
Your answer for the function f (x) is :
Your answer for the function g(x) is :
13.(1 pt) setAlgebra05RationalExpressions/lhp5
57-58.pg
Simplify the expression
2
x−3
and give your answer in the form of
2−
9.(1 pt) setAlgebra05RationalExpressions/srw1 4 60-64.pg
Enter a T or an F in each answer space below to indicate whether
the corresponding equation is true or false. An equation is true
ony if it is true for all values of the variables. Disregard values
that make denominators 0.
You must get all of the answers correct to receive credit.
−16a
16a
1.
=−
b b
a
97a
2. 97
=
b
97b
2
x +1
3.
= x−1
x+1
2
x −1
= x+1
4.
x−1
10.(1 pt) setAlgebra05RationalExpressions/lhp5
25-32.pg
f (x)
.
g(x)
Your answer for the function f (x) is :
Your answer for the function g(x) is :
14.(1 pt) setAlgebra05RationalExpressions/lhp5
60.pg
Simplify the expression
1
1
−
x+2 x+4
and give your answer in the form of
f (x)
.
g(x)
15-24.pg
Simplify the expression
Your answer for the function f (x) is :
Your answer for the function g(x) is :
6y4
xy − 6y
and give your answer in the form of
15.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
f (x)
.
g(x)
2
x2 − 8x + 12
x2 − 3x + 2
5 3.pg
and give your answer in the form of
and give your answer in the form of
f (x)
.
g(x)
f (x)
.
g(x)
Your answer for the function f (x) is :
Your answer for the function g(x) is :
Your answer for the function f (x) is :
Your answer for the function g(x) is :
16.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
21.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
5 5.pg
2
3x
+
(x + 2)2 x + 2
y2 + 4y
y2 − 16
and give your answer in the form of
and give your answer in the form of
f (x)
.
g(x)
f (y)
.
g(y)
Your answer for the function f (x) is :
Your answer for the function g(x) is :
Your answer for the function f (y) is :
Your answer for the function g(y) is :
17.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
22.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
1
x+1+
x+1
and give your answer in the form of
5 7.pg
7x3 − 12x2 − 4x
4x2 − 11x + 6
and give your answer in the form of
Your answer for the function f (x) is :
Your answer for the function g(x) is :
Your answer for the function f (x) is :
Your answer for the function g(x) is :
23.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
5 11.pg
f (x)
.
g(x)
f (x)
.
g(x)
Your answer for the function f (x) is :
Your answer for the function g(x) is :
Your answer for the function f (x) is :
Your answer for the function g(x) is :
24.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
5 13.pg
1
2x2 + 9x + 4 x2 + 7x + 12
÷
x2 + 2x − 8 2x2 − 5x + 2
and give your answer in the form of
x2 + 5x + 4
1
x2 − 4x − 5
f (x)
.
g(x)
Your answer for the function f (x) is :
Your answer for the function g(x) is :
Your answer for the function f (x) is :
Your answer for the function g(x) is :
1
1
−
x+3 x+5
−
5 35.pg
and give your answer in the form of
f (x)
.
g(x)
20.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
5 27.pg
5
3
+
x2 x2 + x
and give your answer in the form of
x2 + 5x + 6 x2 + 4x + 3
·
x2 + 7x + 6 x2 + 8x + 12
and give your answer in the form of
19.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
5 25.pg
f (x)
.
g(x)
f (x)
.
g(x)
18.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
5 23.pg
25.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
5 21.pg
5
1 + c−5
3
5
1 − c−5
5 39.pg
and give your answer in the form of
(−6)−4 + (−6)−2
−6−3
f (c)
.
g(c)
Your answer for the function f (c) is :
Your answer for the function g(c) is :
26.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
5
x−1
x
x−1
33.(1 pt) setAlgebra05RationalExpressions/Test2
In lowest terms, the fraction
5 41.pg
(a − 1)4 − (b + 1)4
(a − 1)3 − (b + 1)3
1
− x+1
A
can be written as where
B
A=
and B =
34.(1 pt) setAlgebra05RationalExpressions/Test2
1
+ x+1
and give your answer in the form of
f (x)
.
g(x)
Your answer for the function f (x) is :
Your answer for the function g(x) is :
27.(1 pt) setAlgebra05RationalExpressions/sw1
Simplify the expression
4
a+h
where A =
and B =
5 47.pg
31.(1 pt) setAlgebra05RationalExpressions/Test1
The expression
7 8
4 3 2
t − y
12
17
equals:
14.pg
4.pg
x2 − (−1x − 4) 5x2 + 7 + 1x
A
=
+
x2 + 4x + 3
−3 − 4x − x2 B
h
and give your answer in the form of
A
.
B
Your answer for A is :
Your answer for B is :
28.(1 pt) setAlgebra05RationalExpressions/sw1 5 59.pg
Rationalize the denominator of expression
2
√
6+ 5
=
/
29.(1 pt) setAlgebra05RationalExpressions/Test1 8.pg
The expression
√
3
64h−5 s7
√
3 8 −3
hs
equals khr st
where r, the exponent of h, is:
and t, the exponent of s, is:
and k, the leading coefficient is:
9.pg
3.pg
−7
7x + 8
A
−7
−
+
=
x − 9 x − 1 x2 − 10x + 9 B
35.(1 pt) setAlgebra05RationalExpressions/Test2
− 4a
30.(1 pt) setAlgebra05RationalExpressions/Test1
The expression
√
9
v81
equals
2.pg
where A =
and B =
36.(1 pt) setAlgebra05RationalExpressions/Test2 5.pg
When written as a simple fraction, without negative exponents,
the fraction
A
1x−12 + 1y−12
=
(x4 + y4 )y−4
B
where A =
and B =
37.(1 pt) setAlgebra05RationalExpressions/Test2 12.pg
The total resistance, R, of a particular group is given by the formula:
!
1
R = S+ 1 1
T +U
This formula can be simplified to the form AB where A and B
contain no fractions.
A=
B=
Suppose that S = 22Ω, T = 39ΩandU = 89Ω.
Then R =
Note: Your answer must be a decimal.
38.(1 pt) setAlgebra05RationalExpressions/rational
If you rationalize the denominator of
1
√
√
5x 5 − 2y 3
then you will get
where A =
and B =
32.(1 pt) setAlgebra05RationalExpressions/Test2 1.pg
Write the following as a simple fraction in lowest terms.
4
A
B
denominator.pg
39.(1 pt) setAlgebra05RationalExpressions/rational
If you rationalize the numerator of
√
√
3 2
x + 7 3 x + 49
√
x3 − 7
then you will get
numerator.pg
where A =
and B =
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
5
A
B
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra06EqnGraphs due 1/6/10 at 2:00 AM
Is the graph symmetric with respect to the origin? Input yes or
no here :
7.(1 pt) setAlgebra06EqnGraphs/srw1 8 63.pg
For the graph of the equation x = y2 − 4, answer the following
questions:
The x- intercept(s) is x =
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
The y - intercept(s) is y=
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
Is the graph symmetric with respect to the x-axis? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
Is the graph symmetric with respect to the origin? Input yes or
no here :
8.(1 pt) setAlgebra06EqnGraphs/srw1 8 65.pg
For the graph of the equation x2 y2 + xy = 2, answer the following questions:
Is the graph symmetric with respect to the x-axis? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
Is the graph symmetric with respect to the origin? Input yes or
no here :
9.(1 pt) setAlgebra06EqnGraphs/srw1 8 67.pg
For the graph of the equation y = x3 + 16x, answer the following
questions:
Is the graph symmetric with respect to the x-axis? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
Is the graph symmetric with respect to the origin? Input yes or
no here :
10.(1 pt) setAlgebra06EqnGraphs/sw2 2 11.pg
Find the x- and y-intercepts of the graph of the equation x2 +
y2 = 64.
The x-intercepts are : x1 =
, x2 =
with x1 ≤ x2 ;
The y-intercepts are : y1 =
, y2 =
with y1 ≤ y2 .
1.(1 pt) setAlgebra06EqnGraphs/srw1 8 39.pg
Determine whether the given points are on the graph of y =
2x + 3. Enter Yes or No for your answers:
Is (2,7) on the graph?
Is (1,5) on the graph?
Is (-6,-10) on the graph?
Is (3,4) on the graph?
2.(1 pt) setAlgebra06EqnGraphs/srw1 8 43.pg
Find the x- and y-intercepts of the graph of the equation y =
x − 2.
The x-intercept(s) have x =
Note: If there is more than one, give a comma separated list. If
there are none, type none .
The y-intercept(s) have y =
Note: If there is more than one, give a comma separated list. If
there are none, type none .
3.(1 pt) setAlgebra06EqnGraphs/srw1 8 45.pg
Find the x- and y-intercepts of the graph of the equation y =
x2 + 3x − 10.
The x-intercept(s) have x =
Note: If there is more than one, give a comma separated list. If
there are none, type none .
The y-intercept(s) have y =
Note: If there is more than one, give a comma separated list. If
there are none, type none .
4.(1 pt) setAlgebra06EqnGraphs/srw1 8 46.pg
For the graph of the equation y = 18x + 4, draw a sketch of the
graph on a piece of paper. Then answer the following questions:
The x-intercept is :
, The y-intercept is :
Is the graph symmetric with respect to the x-axis? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
Is the graph symmetric with respect to the origin? Input yes or
no here :
5.(1 pt) setAlgebra06EqnGraphs/srw1 8 47.pg
Find the x- and y-intercepts of the graph of the equation x2 +
y2 = 4.
The x-intercepts are : x1 =
, x2 =
with x1 ≤ x2 ;
The y-intercepts are : y1 =
, y2 =
with y1 ≤ y2 .
11.(1 pt) setAlgebra06EqnGraphs/sw2 2 18.pg
For the graph of the equation y = 10x + 7, draw a sketch of the
graph on a piece of paper. Then answer the following questions:
, The y-intercept is :
The x-intercept is :
Is the graph symmetric with respect to the x-axis? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
Is the graph symmetric with respect to the origin? Input yes or
no here :
6.(1 pt) setAlgebra06EqnGraphs/srw1 8 53.pg
For the graph of the equation y = x2 − 36, draw a sketch of the
graph on a piece of paper. Then answer the following questions:
The x-intercepts are : x1 =
, x2 =
with x1 ≤ x2 .
The y-intercept is :
Is the graph symmetric with respect to the x-axis? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
1
12.(1 pt) setAlgebra06EqnGraphs/sw2 2 25.pg
For the graph of the equation y = x2 − 25, draw a sketch of the
graph on a piece of paper. Then answer the following questions:
The x-intercepts are : x1 =
, x2 =
with x1 ≤ x2 .
The y-intercept is :
Is the graph symmetric with respect to the x-axis? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
Is the graph symmetric with respect to the origin? Input yes or
no here :
13.(1 pt) setAlgebra06EqnGraphs/sw2 2 41.pg
For the graph of the equation y = x4 + x2 , answer the following
questions:
Is the graph symmetric with respect to the x-axis? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
Is the graph symmetric with respect to the origin? Input yes or
no here :
14.(1 pt) setAlgebra06EqnGraphs/symm1.pg
For the equation 10x − 2y2 = 4 answer the following questions.
Is the equation symmetric with respect to the y-axis? (yes or no
)
Is the equation symmetric with respect to the x-axis? (yes or no
)
Is the equation symmetric with respect to the origin? (yes or no
)
Is the graph symmetric with respect to the x-axis? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
Is the graph symmetric with respect to the origin? Input yes or
no here :
17.(1 pt) setAlgebra06EqnGraphs/p1.pg
For the graph of the equation x = y2 − 16, answer the following
questions:
the x- intercepts are x =
Note: If there is more than one answer enter them separated by
commas.
the y - intercepts are y=
Note: if there is more than one answer enter them separated by
commas.
Is the graph symmetric with respect to the x-axis? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
Is the graph symmetric with respect to the origin? Input yes or
no here :
18.(1 pt) setAlgebra06EqnGraphs/p2.pg
For the graph of the equation y = −x3 + 6, answer the following
questions.
The x-intercepts have x =
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
The y-intercepts have y =
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
Is the graph symmetric with respect to the x-axis? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
Is the graph symmetric with respect to the origin? Input yes or
no here :
15.(1 pt) setAlgebra06EqnGraphs/beth1.pg
For the equation
y = −5|x| + 9
answer the following questions:
What are the x-intercept(s) written as ordered pair(s)?
Note: If there is more than one write them separated by a
comma (i.e.: (1,2),(3,4)). If there are none, type none in the
answer blank.
x-intercept(s):
What is the y-intercept written as an ordered pair?
Note: If there is more than one write them separated by a
comma (i.e.: (1,2),(3,4)). If there are none, type none in the
answer blank.
y-intercept:
Is the graph symmetric with respect to the x-axis? (yes or no )
19.(1 pt) setAlgebra06EqnGraphs/p3.pg
√
For the graph of the equation y = x + 1, answer the following
questions:
The x-intercepts have x =
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
The y-intercepts have y =
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
Is the graph symmetric with respect to the x-axis? Input yes
or no here :
Is the graph symmetric with respect to the y-axis? Input yes or
no here :
Is the graph symmetric with respect to the origin? Input yes or
no here :
Is the graph symmetric with respect to the y-axis? (yes or no )
Is the graph symmetric with respect to the origin? (yes or no )
16.(1 pt) setAlgebra06EqnGraphs/beth1grapheq.pg
For the graph of the equation x2 y4 + x3 y3 = 14, answer the following questions:
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
2
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra07PointsCircles due 1/7/10 at 2:00 AM
9.(1 pt) setAlgebra07PointsCircles/srw1 9 4-sol.pg
Consider the two points (2, −5) and (−4, −2). The distance between them is:
The x co-ordinate of the midpoint of the line segment that joins
them is:
The y co-ordinate of the midpoint of the line segment that joins
them is:
1.(1 pt) setAlgebra07PointsCircles/sw2 1 7.pg
Find the midpoint of the segment that joins the points (−4, 0)
and (−1, 1).
Input your answer here: (
,
)
2.(1 pt) setAlgebra07PointsCircles/sw2 1 11.pg
Plot the points A(5,0), B(9,0), C(8,6) and D(6,6) on a coordinate
plane on a piece of paper. Draw the segments AB, BC, CD and
DA.
The quadrilateral ABCD is commonly called
;
Its area equals
;.
10.(1 pt) setAlgebra07PointsCircles/sApB 1-6.pg
Find the distance between (3, 8) and (-1, -2).
11.(1 pt) setAlgebra07PointsCircles/sApB x.pg
Find the perimeter of the triangle with the vertices at
(3, -1), (-6, 3), and (-6, -5).
3.(1 pt) setAlgebra07PointsCircles/sw2 1 17.pg
Sketch the region given by the set {(x, y)|2 ≤ x ≤ 4, 4 ≤ y ≤ 10}
on a piece of paper.
.
The area of the region is
12.(1 pt) setAlgebra07PointsCircles/sApB x-sol.pg
Find the perimeter of the triangle with the vertices at
(2, 1), (−3, 3), and (−4, −5).
4.(1 pt) setAlgebra07PointsCircles/sw2 1 19.pg
Sketch the region given by the set {(x, y)|xy < 0} on a piece of
paper. Which quadrants of the plane are included in the set?
Input Yes or No at the corresponding space below:
,
the first quadrant is included:
the second quadrant is included:
,
the third quadrant is included:
,
the fourth quadrant is included:
.
Be careful, you only have one chance to enter your answer!!!
13.(1 pt) setAlgebra07PointsCircles/midpoint hard.pg
The midpoint of AB is at (−4, −5). If A = (3, 5), find B.
B is:( , )
14.(1 pt) setAlgebra07PointsCircles/perimeter of triangle.pg
Find the perimeter of the triangle with the vertices at
(5, -2), (-3, 3), and (-6, -5).
5.(1 pt) setAlgebra07PointsCircles/sw2 1 25.pg
Which of the points A(5, 6) or B(−4, 7) is closer to the origin?
Input the corresponding letter A or B here:
;
Be careful, you only have one chance to enter your answer!!!
15.(1 pt) setAlgebra07PointsCircles/ur ab 9 1.pg
Find the point (0, b) on the y-axis that is equidistant from the
points (2, 2) and (5, −4).
b=
6.(1 pt) setAlgebra07PointsCircles/srw1 9 2.pg
Consider the two points (2, −3) and (6, 9). The distance between them is:
The x co-ordinate of the midpoint of the line segment that joins
them is:
The y co-ordinate of the midpoint of the line segment that joins
them is:
7.(1 pt) setAlgebra07PointsCircles/srw1 9 2-sol.pg
Consider the two points (5, −5) and (9, 8). The distance between them is:
The x co-ordinate of the midpoint of the line segment that joins
them is:
The y co-ordinate of the midpoint of the line segment that joins
them is:
8.(1 pt) setAlgebra07PointsCircles/srw1 9 4.pg
Consider the two points (4, −3) and (−9, −9). The distance between them is:
The x co-ordinate of the midpoint of the line segment that joins
them is:
The y co-ordinate of the midpoint of the line segment that joins
them is:
16.(1 pt) setAlgebra07PointsCircles/dist between 2
Find the distance between (2, 6) and (-4, 2).
pts.pg
17.(1 pt) setAlgebra07PointsCircles/Manhattan.pg
Find the Manhattan distance between (272, -114) and (-260, 36).
The Manhattan distance between the given points is
.
18.(1 pt) setAlgebra07PointsCircles/dist midpoint.pg
Consider the two points (0, 5) and (−8, 9). The distance between them is:
The x co-ordinate of the midpoint of the line segment that joins
them is:
The y co-ordinate of the midpoint of the line segment that joins
them is:
19.(1 pt) setAlgebra07PointsCircles/equidist off axis.pg
Find the point (x, y) on the line y = x that is equidistant from the
points (8, 0) and (7, −3).
x=
y=
1
20.(1 pt) setAlgebra07PointsCircles/equidist off axis hard.pg
Find the point (x, y) on the line y = 5x − 2 that is equidistant
from the points (2, 6) and (−5, −2).
x=
y=
The radius is :
29.(1 pt) setAlgebra07PointsCircles/srw1 8 86.pg
Find the center and radius of the circle given by the equation
x2 + y2 + 8x + 6y + 24 = 0
21.(1 pt) setAlgebra07PointsCircles/equidist on axis.pg
Find the point (x, y) on the x-axis that is equidistant from the
points (6, 10) and (4, 0).
x=
y=
,
)
The center is : (
The radius is :
30.(1 pt) setAlgebra07PointsCircles/srw1 8 89.pg
Find the area of the region that lies outside the circle
22.(1 pt) setAlgebra07PointsCircles/sw2 2 51.pg
Find an equation of the circle with center (11, −2) and radius 6
in the form of (x−A)2 +(y−B)2 = C where A, B,C are constant.
Then
A is :
B is :
C is :
23.(1 pt) setAlgebra07PointsCircles/sw2 2 53.pg
Find an equation of the circle with center at the origin and passing through (−5, 6) in the form of (x−A)2 +(y−B)2 = C where
A, B,C are constant. Then
A is :
B is :
C is :
24.(1 pt) setAlgebra07PointsCircles/sw2 2 61.pg
Find the center and radius of the circle given by the equation
x2 + y2 − 6x − 8y + 16 = 0
The center is : (
,
)
The radius is :
25.(1 pt) setAlgebra07PointsCircles/sw2 2 67.pg
Find the center and radius of the circle given by the equation
x2 + y2 + 4x + 12y + 36 = 0
The center is : (
,
)
The radius is :
26.(1 pt) setAlgebra07PointsCircles/srw1 8 73.pg
Find an equation of the circle with center (19, −19) and radius 1
in the form of (x−A)2 +(y−B)2 = C where A, B,C are constant.
Then
A is :
B is :
C is :
27.(1 pt) setAlgebra07PointsCircles/srw1 8 74.pg
Find an equation of the circle with center at the origin and passing through (−1, 5) in the form of (x−A)2 +(y−B)2 = C where
A, B,C are constant. Then
A is :
B is :
C is :
28.(1 pt) setAlgebra07PointsCircles/srw1 8 81.pg
Find the center and radius of the circle given by the equation
but inside the circle
The center is : (
x2 + y 2 = 4
x2 + y2 − 4y − 60 = 0.
Your answer is
31.(1 pt) setAlgebra07PointsCircles/circle2.pg
Find the standard form for the equation of a circle
(x − h)2 + (y − k)2 = r 2
with a diameter that has endpoints (−5, −1) and (6, 3).
h=
k=
r=
32.(1 pt) setAlgebra07PointsCircles/circle3.pg
Find the center (h, k) and the radius r of the circle
5x2 + 2x + 5y2 − 5y − 10 = 0.
h=
k=
r=
33.(1 pt) setAlgebra07PointsCircles/circlenot1.pg
Find the standard form for the equation of a circle (x − h)2 +
(y − k)2 = r 2 with a diameter that has endpoints of (−2, 9) and
(9, −3).
h=
k=
r2 =
34.(1 pt) setAlgebra07PointsCircles/p4.pg
Find an equation of the circle with center at (−8, 2) and passing
through (3, −4) in the form of (x − A)2 + (y − B)2 = C where
A, B,C are constant. Then
A is :
B is :
C is :
35.(1 pt) setAlgebra07PointsCircles/p5.pg
Find an equation of the circle with center at (−4, 3) that is tangent to the y-axis in the form of (x − A)2 + (y − B)2 = C where
A, B,C are constant. Then
A is :
B is :
C is :
36.(1 pt) setAlgebra07PointsCircles/center radius from gen.pg
Find the center and radius of the circle whose equation is
x2 + 10x + y2 + 10y + 15 = 0.
x2 + y2 − 8x − 8y + 31 = 0
,
)
2
The center of the circle is (
,
).
The radius of the circle is
.
Note: Your answers must be decimals.
37.(1 pt) setAlgebra07PointsCircles/center radius from gen hard.pg
Find the center and radius of the circle whose equation is
3x2 − 5x + 3y2 − 8y − 18 = 0.
,
).
The center of the circle is (
The radius of the circle is
.
Note: Your answers must be decimals.
38.(1 pt) setAlgebra07PointsCircles/eqn from center pt.pg
Complete the equation of the circle centered at (1, 10) that
passes through (6, 10).
= 0.
39.(1 pt) setAlgebra07PointsCircles/eqn from center radius.pg
Complete the equation of the circle centered at (−10, 4) with
radius 18.
= 0.
40.(1 pt) setAlgebra07PointsCircles/ur geo 1 5.pg
Plot the points A = (0, −2), B = (2, 1), and C = (−6, 2). Notice
that these points are vertices of a right triangle (the angle A is
90 degrees).
Find the distance between A and B:
Find the distance between A and C:
Find the area of the triangle ABC:
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
3
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra08LinearEqns due 1/8/10 at 2:00 AM
x=
Please also use your calculator to verify your answer by finding
the x-intercept of the graph of y = 7x + 9 − (−7).
1.(1 pt) setAlgebra08LinearEqns/sw1 6 11.pg
Solve the equation −6x − 1 = 6x − 8.
x=
2.(1 pt) setAlgebra08LinearEqns/sw1 6 13.pg
1
1
Solve the equation y + 7 = y.
2
8
y=
9.(1 pt) setAlgebra08LinearEqns/sw3 1 3.pg
Solve the equation 7x + 1 = 3x − 6 algebraically.
x=
Please also use your calculator to verify your answer by finding
the x-intercept of the graph of y = 7x + 1 − (3x − 6).
3.(1 pt) setAlgebra08LinearEqns/sw1 6 19.pg
−5 −2
=
+ 4.
Solve the equation
x
3x
x=
4.(1 pt) setAlgebra08LinearEqns/sw1 6 29.pg
8
5
2
Solve the equation
− =
.
x + 1 2 3x + 3
x=
5.(1 pt) setAlgebra08LinearEqns/sw1 6 39.pg
x
1
Solve the equation
−1 =
.
4x − 16
x−4
Does the equation have a solution?
Input Yes or No here:
If your answer is Yes, input your solution here: x =
10.(1 pt) setAlgebra08LinearEqns/sw3 1 7.pg
5
4
Solve the equation +
= 4 algebraically.
x 2x
x=
Please also use your calculator to verify your answer.
11.(1 pt) setAlgebra08LinearEqns/srw1
Solve the equation 6x + 5 = 3x − 9.
x=
12.(1 pt) setAlgebra08LinearEqns/srw1
Solve the equation for x
5 5.pg
5 14.pg
8(x + 8) + 3 = −11(x − 9) − 10
6.(1 pt) setAlgebra08LinearEqns/sw1 6 71.pg
Solve the equation PV = nRT for R.
Your answer is :
Note: The answer is case sensitive. P, V and T are capital
letters!
7.(1 pt) setAlgebra08LinearEqns/sw1 6 74.pg
Solve the equation P = 2l + 2w for w.
Your answer is :
Note: The answer is case sensitive!
8.(1 pt) setAlgebra08LinearEqns/sw3 1 1.pg
Solve the equation 7x + 9 = −7 algebraically.
x=
13.(1 pt) setAlgebra08LinearEqns/Test2
Solve for y.
y=
10.pg
−3y − 6(−5y − 4) = 6 − 5(6 − y)
14.(1 pt) setAlgebra08LinearEqns/Test2 11.pg
Solve for k.
6
−7
6
k+ = 1−
k
5
5
9
k=
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
1
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra09LinearEqnsModeling due 1/9/10 at 2:00 AM
Your answer is :
1.(1 pt) setAlgebra09LinearEqnsModeling/sw3 2 9.pg
The distance (in miles) traveled when driving at a certain speed
s for 16 hours, then driving 8 miles/hour faster for another hour.
Express the distance in terms of s.
Your answer is :
10.(1 pt) setAlgebra09LinearEqnsModeling/srw1 6 11.pg
A cash register contains only five dollar and ten dollar bills. It
contains twice as many five’s as ten’s and the total amount of
money in the cash register is 540 dollars. How many ten’s are
in the cash register?
2.(1 pt) setAlgebra09LinearEqnsModeling/sw3 2 11.pg
Express the average age of three sisters in the terms of the age
a of the firstborn (in years) if the second was born 2 years after
the first and the third was born 4 years after the second.
Your answer is :
11.(1 pt) setAlgebra09LinearEqnsModeling/srw1 6 20.pg
After robbing a bank in Dodge City, a robber gallops off at 14
mi/h. 10 minutes later, the marshall leaves to pursue the robber at 16 mi/h. How long (in hours) does it take the marshall to
catch up to the robber?
3.(1 pt) setAlgebra09LinearEqnsModeling/sw3 2 13.pg
An executive in an engineering firm earns a monthly salary plus
a Christmas bonus of 6900 dollars. If she earns a total of 97600
dollars per year, what is her monthly salary in dollars?
Your answer is :
12.(1 pt) setAlgebra09LinearEqnsModeling/srw1 6 25.pg
Two cyclists, 108 miles apart, start riding toward each other at
the same time. One cycles 2 times as fast as the other. If they
meet 4 hours later, what is the speed (in mi/h) of the faster cyclist?
4.(1 pt) setAlgebra09LinearEqnsModeling/sw3 2 17.pg
The oldest child in a family of four children is twice as old as
the yougest. The two middle children are 12 and 15 years old. If
the average age of the children is 13.5, how old is the youngest
child?
Your answer is :
13.(1 pt) setAlgebra09LinearEqnsModeling/srw1 6 29.pg
What quantity of 60 per cent acid solution must be mixed with
a 30 solution to produce 180 mL of a 50 per cent solution?
5.(1 pt) setAlgebra09LinearEqnsModeling/sw3 2 23.pg
A rectangular garden is 20 ft wide. If its area is 1600 ft2 , what
is the length of the garden?
Your answer is :
14.(1 pt) setAlgebra09LinearEqnsModeling/srw1 6 31.pg
The radiator in a car is filled with a solution of 75 per cent
antifreeze and 25 per cent water. The manufacturer of the antifreeze suggests that for summer driving, optimal cooling of the
engine is obtained with only 50 per cent antifreeze. If the capacity of the raditor is 3.4 liters, how much coolant (in liters) must
be drained and replaced with pure water to reduce the antifreeze
concentration to 50 per cent?
6.(1 pt) setAlgebra09LinearEqnsModeling/sw3 2 25.pg
Phyllis invested 70000 dollars, a portion earning a simple interest rate of 4 percent per year and the rest earning a rate of
6 percent per year. After one year the total interest earned on
these investments was 3500 dollars. How much money did she
invest at each rate?
At rate 4 percent :
At rate 6 percent :
15.(1 pt) setAlgebra09LinearEqnsModeling/tax.pg
Taxylvania has a tax code that rewards charitable giving. If
a person gives p% of his income to charity, that person pays
(32 − 1.2p) % tax on the remaining money. For example, if a
person gives 10% of his income to charity, he pays 20 % tax on
the remaining money. If a person gives 26.6666666666667 % of
his income to charity, he pays no tax on the remaining money.
A person does not receive a tax refund if he gives more than
26.6666666666667 % of his income to charity. Count Taxula
earns $ 59000. What percentage of his income should he give to
charity to maximize the money he has after taxes and charitable
giving?
% to charity.
The count should give
If the count did receive a tax refund for giving more than
26.6666666666667 % of his income to charity, how much
should he give to charity?
The count should give
% to charity.
7.(1 pt) setAlgebra09LinearEqnsModeling/sw3 2 28.pg
A change purse contains an equal number of pennies, nickels,
and dimes. The total value of the coins is 416 cents. How many
coins of each type does the purse contain?
Number of pennies :
8.(1 pt) setAlgebra09LinearEqnsModeling/sw3 2 52.pg
Stan and Hilda can mow the lawn in 40 min if they work together. If Hilda works twice as fast as Stan, how long would it
take Stan to mow the lawn alone?
Give your answer in munites here:
9.(1 pt) setAlgebra09LinearEqnsModeling/sw3 2 57.pg
Wilma drove at an average speed of 35 mi/h from her home in
City A to visit her sister in City B. She stayed in City B 15 hours,
and on the trip back averaged 60 mi/h. She returned home 49
hours after leaving. How many miles is City A from City B
1
NOTE: Your answers must be numbers. No arithmetic operations are allowed.
The three numbers in increasing order are
16.(1 pt) setAlgebra09LinearEqnsModeling/c0s1p5.pg
A Norman window has the shape of a rectangle surmounted by
a semicircle. If the perimeter of the window is 56.400 ft. give
the area A of the window in square feet when the width is 9.600
ft. Give the answer to two decimal places.
20.(1 pt) setAlgebra09LinearEqnsModeling/lh1-3 31.pg
This exercise concerns with modeling with linear equations.
One positive number is 5 times another number. The difference
between the two numbers is 1412, find the numbers.
and
The two numbers in increasing order are
, and
21.(1 pt) setAlgebra09LinearEqnsModeling/lh1-3 32.pg
This exercise concerns with modeling with linear equations.
One positive number is one-fifth of another number. The difference between the two numbers is 272, find the numbers.
The two numbers in increasing order are
and
17.(1 pt) setAlgebra09LinearEqnsModeling/c0s1p6.pg
A Norman window has the shape of a rectangle surmounted by
a semicircle. The perimeter is 47.000 ft. Order the widths listed
below according to the area of the corresponding Norman window from the lowest area (1) to highest area (5).
You will need to enter the numbers 1 through 5 in the entry
blanks below.
1.
2.
3.
4.
5.
,
22.(1 pt) setAlgebra09LinearEqnsModeling/lh1-3
What is 45 percent of 330?.
Your answers is :
23.(1 pt) setAlgebra09LinearEqnsModeling/lh1-3
125 is what percent of 620?.
percent.
Your answers is :
Width = 5.600 ft.
Width = 12.200 ft.
Width = 5.800 ft.
Width = 8.300 ft.
Width = 12.800 ft.
36.pg
38.pg
24.(1 pt) setAlgebra09LinearEqnsModeling/lh1-3 39.pg
290 is 20 percent of what number?
Your answers is :
25.(1 pt) setAlgebra09LinearEqnsModeling/lh1-3 46.pg
Your weekly paycheck is 10 percent less than your coworker’s.
Your two paychecks total 815. Find the amount of each paycheck.
and yours is
.
Your coworker’s is :
Remark: To be able to order the sizes of the windows you
are going to have to calculate the area for all five windows from
knowing their widths. Since there are several calculations it will
save time to figure out and simplify a formula which calculates
the area from the width and the perimeter. This is in contrast to
the previous problem where, with only one calculation to make,
it wasn’t necessarily worth the effort to find a general formula.
You can use that example to check your formula however.
I do this very frequently when I am doing research and solving problems. Work out a special case first. THEN work out a
formula for the general case and use the solution to the special
case to check the formula.
26.(1 pt) setAlgebra09LinearEqnsModeling/lh1-3 51.pg
A rectangular room is 2 times as long as it is wide, and its
perimeter is 28 meters. Find the dimension of the room.
The length is :
meters and the width is
meters.
27.(1 pt) setAlgebra09LinearEqnsModeling/lh1-3 54.pg
Suppose that you are taking a course that has 4 tests. The first
three tests are for 100 points each and the fourth test is for 200
points. To get an B in the course, you must have an average of
at least 80 percent on the 4 tests. Your scores on the first 3 tests
were 73, 80, and 79. What is the minimum score you need on
the fourth test to get an B for the course?.
Your answers is :
.
28.(1 pt) setAlgebra09LinearEqnsModeling/ur ab 6 1.pg
A student has scores of 77.25, 79, and 83.25 on his first three
tests. He needs an average of at least 80 to earn a grade of B.
What is the minimum score that the student needs on the fourth
test to ensure a B?
Note: The answer need not be an integer.
18.(1 pt) setAlgebra09LinearEqnsModeling/paintingpartners.pg
Mutt and Jeff need to paint a fence. Mutt can do the job alone
3 hours faster than Jeff. If together they work for 17 hours and
finish only 12 of the job, how long would Jeff need to do the job
alone?
Your answer must be a number. No arithmetic operations are
allowed.
It would take Jeff
hours to do the job alone.
19.(1 pt) setAlgebra09LinearEqnsModeling/lh1-3 30.pg
This exercise concerns with modeling with linear equations.
The sum of three consecutive natural numbers is 423, find the
numbers.
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
2
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra10QuadraticEqns due 1/10/10 at 2:00 AM
1.(1 pt) setAlgebra10QuadraticEqns/sw1 6 43.pg
Find all real solutions of the equation x2 = 64.
x1 =
and x2 =
with x1 < x2 !!!
14.(1 pt) setAlgebra10QuadraticEqns/sw3 3 37.pg
Find all real solutions of equation 3x2 + 4x + 3 = 0.
Does the equation have real solutions? Input Yes or No:
If your answer is Yes, input the solutions:
x1 =
and x2 =
with x1 ≤ x2 .
2.(1 pt) setAlgebra10QuadraticEqns/sw1 6 47.pg
Find all real solutions of the equation x2 − 343 = 0.
x1 =
and x2 =
with x1 < x2 !!!
15.(1 pt) setAlgebra10QuadraticEqns/sw3 3 63.pg
A rectangular garden is 10 ft longer than it is wide. Its area is
3000 ft2 . What are its dimensions?
Its width equals
and its length equals
3.(1 pt) setAlgebra10QuadraticEqns/sw1 6 51.pg
Find all real solutions of the equation (x − 2)2 = 36.
x1 =
and x2 =
with x1 < x2 !!!
16.(1 pt) setAlgebra10QuadraticEqns/sw3 3 69.pg
A box with a square base and no top is to be made from a square
piece of carboard by cutting 4 in. squares from each corner and
folding up the sides. The box is to hold 2916 in3 . How big a
piece of cardboard is needed?
Your answer is:
in. by
in.
4.(1 pt) setAlgebra10QuadraticEqns/sw1 6 53.pg
How many real solutions does the equation x3 = 343 have?
Input your answer here:
How many real solutions does the equation x3 = −343 have?
Input your answer here:
5.(1 pt) setAlgebra10QuadraticEqns/sw3 1 9.pg
Solve the equation x2 − 25 = 0 algebraically.
x1 =
and x2 =
with x1 ≤ x2 .
Please also use your calculator to verify your answer.
17.(1 pt) setAlgebra10QuadraticEqns/sw3 4 49.pg
Find all solutions of the equation x2 − 4x + 6 = 0 and express
them in the form a + bi:
First input the solution with b < 0 here:
the real number a equals
and the real number b equals
6.(1 pt) setAlgebra10QuadraticEqns/sw3 3 1.pg
Solve the equation x2 − 4x − 12 = 0 by factoring.
The solutions are x1 =
and x2 =
with x1 ≤ x2 .
Then input the solution with b > 0 here:
the real number a equals
and the real number b equals
7.(1 pt) setAlgebra10QuadraticEqns/sw3 3 3.pg
Solve the equation x2 − 10x + 24 = 0 by factoring.
The solutions are x1 =
and x2 =
with x1 ≤ x2 .
18.(1 pt) setAlgebra10QuadraticEqns/sw3
8.(1 pt) setAlgebra10QuadraticEqns/sw3 3 5.pg
Solve the equation 3x2 + 22x + 35 = 0 by factoring.
The solutions are x1 =
and x2 =
with x1 ≤ x2 .
Find all solutions of the equation t +4+
7
= 0 and express them
t
in the form a + bi:
First input the solution with b < 0 here:
and the real number b
the real number a equals
equals
Then input the solution with b > 0 here:
the real number a equals
and the real number b
equals
9.(1 pt) setAlgebra10QuadraticEqns/sw3 3 13.pg
Solve the equation x2 − 2x − 15 = 0 by completing the square.
The solutions are x1 =
and x2 =
with x1 ≤ x2 .
10.(1 pt) setAlgebra10QuadraticEqns/sw3 3 17.pg
Solve the equation 9x2 + 12x + 3 = 0 by completing the square.
The solutions are x1 =
and x2 =
with x1 ≤ x2 .
19.(1 pt) setAlgebra10QuadraticEqns/srw1 5 35.pg
By completing the square, the expression x2 + 2x + 99 equals
(x + A)2 + B
where A =
and B =
20.(1 pt) setAlgebra10QuadraticEqns/srw1 5 36.pg
By completing the square, the expression x2 − 12x + 71 equals
(x + A)2 + B
where A =
and B =
21.(1 pt) setAlgebra10QuadraticEqns/srw1 5 41.pg
The equation x2 + 7x − 10 = 0 has two solutions A and B where
A<B
and A =
and B =
22.(1 pt) setAlgebra10QuadraticEqns/srw1 5 47.pg
The equation 3x2 +14x +3 = 0 has two solutions A and B where
A<B
11.(1 pt) setAlgebra10QuadraticEqns/sw3 3 19.pg
Solve the equation 5x2 − 6x = 0 by completing the square.
The solutions are x1 =
and x2 =
with x1 ≤ x2 .
12.(1 pt) setAlgebra10QuadraticEqns/sw3 3 25.pg
Find all real solutions of equation 2x2 + 5x − 2 = 0.
Does the equation have real solutions? Input Yes or No:
If your answer is Yes, input the solutions:
x1 =
and x2 =
with√
x 1 ≤ x2 .
Note: Use sqrt(10) or 10**(1/2) for 10, etc.
13.(1 pt) setAlgebra10QuadraticEqns/sw3 3 31.pg
Find all real solutions of equation 3 + 8z + z2 = 0.
Does the equation have real solutions? Input Yes or No:
If your answer is Yes, input the solutions:
z1 =
and z2 =
with z1 ≤ z2 .
4 55.pg
1
and A =
and B =
23.(1 pt) setAlgebra10QuadraticEqns/srw1 6 9.pg
The length of a rectangular garden is 8 feet longer than its width.
If the garden’s perimeter is 196 feet, what is the area of the garden in square feet?
Note: Your answer must be a number. It may not contain any
arithmetic operations.
The volume of the cube is
cm3 .
31.(1 pt) setAlgebra10QuadraticEqns/maxprofit.pg
The Acme Widget Company has found that if widgets are priced
at $ 383, then 5000 will be sold. They have also found that for
every increase of $ 18, there will be 200 fewer widgets sold.
The marginal cost of widgets is $ 172.35. The fixed costs for
the Acme Widget Company are $ 44000.
If x represents the price of a widget find the following in terms
of x:
The number of widgets that will be sold:
The revenue generated by the sale of widgets:
The cost of producing just enough widgets to meet demand:
24.(1 pt) setAlgebra10QuadraticEqns/ur ab 6 4.pg
A factory is to be built on a lot measuring 240 ft by 320 ft. A
local building code specifies that a lawn of uniform width and
equal in area to the factory must surround the factory.
What must the width of the lawn be?
If the dimensions of the factory are A ft by B ft with A < B, then
A=
and B =
25.(1 pt) setAlgebra10QuadraticEqns/ur ab 6 5.pg
The difference of two positive numbers is 6 and the sum of their
squares is 50. Find the numbers.
The bigger number is
, and the smaller number is
.
The profit from selling widgets:
Find the price that will maximize profits from the sale of widgets:
26.(1 pt) setAlgebra10QuadraticEqns/ur ab 6 6.pg
The area of a rectangle is 40, and its perimeter is 28. Find its
dimensions and diagonal.
Longer side:
Shorter side:
Diagonal:
32.(1 pt) setAlgebra10QuadraticEqns/rocket.pg
NASA launches a rocket at t = 0 seconds. Its height, in meters above sea-level, as a function of time is given by h(t) =
−4.9t 2 + 250t + 122.
Assuming that the rocket will splash down into the ocean, at
what time does splashdown occur?
The rocket splashes down after
seconds.
How high above sea-level does the rocket get at its peak?
The rocket peaks at
meters above sea-level.
27.(1 pt) setAlgebra10QuadraticEqns/findequation.pg
Find b and c so that y = −10x2 + bx + c has vertex (5, 2).
b=
.
c=
.
28.(1 pt) setAlgebra10QuadraticEqns/area2diagonal.pg
The width of a rectangle is 3 less than twice its length. If the area
of the rectangle is 51 cm2 , what is the length of the diagonal?
Note: Your answer must be a number. It may not contain any
arithmetic operations.
The length of the diagonal is
cm.
33.(1 pt) setAlgebra10QuadraticEqns/wireproblem.pg
You have a wire that is 29 cm long. You wish to cut it into two
pieces. One piece will be bent into the shape of a square. The
other piece will be bent into the shape of a circle. Let A represent the total area of the square and the circle. What is the
circumference of the circle when A is a minium?
The circumference of the circle is
cm.
29.(1 pt) setAlgebra10QuadraticEqns/area2perimeter.pg
Given that the area of an equilateral triangle is 248 cm2 , find its
perimeter.
Note: Your answer must be a number. No arithmetic operations
are allowed.
The perimeter of the triangle is
cm.
34.(1 pt) setAlgebra10QuadraticEqns/SA2volume.pg
Given that the surface area of a sphere is 143 π cm2 , find its
volume.
Note: Your answer must be a number. No arithmetic operations
are allowed.
The volume of the sphere is
cm3 .
30.(1 pt) setAlgebra10QuadraticEqns/area2volume.pg
The surface area of a cube is 177 cm2 . What is the volume of
the cube?
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
2
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra11ComplexNumbers due 1/11/10 at 2:00 AM
6.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 2.pg
More on complex numbers. (For additional help check out the
appendix in Stewart’s Calculus book. There is an entire appendix of hints for working with complex numbers.)
An apology: The exponents don’t print very well on the
screen version of this problem. You can get a better idea of
what the notation looks like from the hard copy and/or you can
use the ”typeset” mode to get a better printing. Unfortunately in
typset mode you won’t be able to enter the answers which are
within equations.
1.(1 pt) setAlgebra11ComplexNumbers/adding.pg
Evaluate the expression (−2 + 2i) + (3 +4i) and write the result
in the form a + bi.
The sum is
.
2.(1 pt) setAlgebra11ComplexNumbers/Multiply.pg
Evaluate the expression (4 − 9i)(2 + 5i) and write the result in
the form a + bi.
The product is
.
3.(1 pt) setAlgebra11ComplexNumbers/Subtract.pg
Evaluate the expression (−2 + 6i) − (−3 + 3i) and write the result in the form a + bi.
.
The difference is
4.(1 pt) setAlgebra11ComplexNumbers/Divide.pg
Evaluate the expression
2 − 3i
7 − 3i
and write the result in the form a + bi.
.
The quotient is
5.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 1.pg
For some practice working with complex numbers:
Calculate
(3 + 4i) + (1 − 2i) =
,
(3 + 4i) − (1 − 2i) =
,
(3 + 4i)(1 − 2i) =
.
The complex conjugate of (1 + i) is (1 − i). In general to obtain
the complex conjugate reverse the sign of the imaginary part.
(Geometrically this corresponds to finding the ”mirror image”
point in the complex plane by reflecting through the x-axis. The
complex conjugate of a complex number z is written with a bar
over it: z and read as ”z bar”.
Notice that if z = a + ib, then
(z) (z) = |z|2 = a2 + b2
which is also the square of the distance of the point z from the
origin. (Plot z as a point in the ”complex” plane in order to see
this.)
If z = 3 + 4i then z =
and |z| =
.
You can use this to simplify complex fractions. Multiply the
numerator and denominator by the complex conjugate of the denominator to make the denominator real.
3 + 4i
=
+i
.
1 − 2i
Two convenient functions to know about pick out the real and
imaginary parts of a complex number.
Re(a + ib) = a (the real part (coordinate) of the complex
number), and
Im(a + ib) = b (the imaginary part (coordinate) of the complex
number. Re and Im are linear functions – now that you know
about linear behavior you may start noticing it often.
The red point represents the complex number z1 =
,
and the blue point represents the complex number z2 =
.
|z1 | =
.
We can also write these complex numbers in polar coordinates (r, θ). The angle is sometimes called the ”argument” of
the complex number and r is called the ”modulus” or the absolute value of the number.
By comparing Taylor series we find that
eiθ = cos(θ) + i sin(θ).
This is a very important and very useful formula. One use is
to relate the polar coordinate and cartesian coordinate formulas
for the complex number. If z can be represented by both coordinates x + iy and by polar coordinates r, θ then
reiθ = r cos(θ) + ir sin(θ) = x + iy = z.
Represent z1 (the red point) in polar coordinates (use an angle
between −π and π):
ei
.
Represent z2 (the blue point) in polar coordinates:
ei
1
.
Using the law of exponents it is really easy to multiply complex numbers represented in polar coordinates – the angles just
add!
B:
C:
9.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 7.pg
Write the
numbers in a + bi form:
following
i
=
+
i,
(a) −3
2
(b) (−2 − 5i) − (−5 + 5i) =
+
i,
3
(c) =
+
i,
i
10.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 8.pg
Write the following numbers in a + bi form:
(a) (−4 + i)2 =
+
i,
1 + 4i
(b) i
=
+
i,
(r1 eiθ1 )(r2 eiθ2 ) = r1 r2 ei(θ1+θ2 ) .
Find z1 · z2 using polar coordinates and your answer above:
z1 · z2 =
ei
,
.
.
Check your answer by doing the standard multiplication and
then converting to polar coordinates. Can you plot this number
on the graph?
7.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 5.pg
Enter the complex coordinates of the following points:
1
(c)
1 + 4i
1
1
=
+
i.
11.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 9.pg
Write the following numbers in a + bi form:
(a) (1 − 5i)2 =
+
i,
(b) i(π − 2i) =
+
i,
−3 + 5i
(c)
+
i.
=
i
12.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 10.pg
Write the following numbers in a + bi form:
4 + 3i
=
+
i,
(a)
2 + 5i
−5
5
(b)
+
=
+
i,
2i
4i
3
+
i.
(c) (i) =
A:
B:
C:
+
+
+
13.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 11.pg
Write the following numbers in a + bi form:
2
2+i
(a)
=
+
i,
3i − (−2 − 2i)
(b) (i)2 (−4 + i)2 =
+
i.
i,
i,
i.
8.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 6.pg
Enter the complex coordinates of the following points:
14.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 12.pg
Write the following numbers in a + bi form:
(a) (4 + 4i)(−1 − 5i)(5 + 2i) =
+
i,
(b) ((2 + 3i)2 − 1)i =
+
i.
15.(1 pt) setAlgebra11ComplexNumbers/ur
Calculate the following:
(a) i2 =
,
,
(b) i3 =
(c) i4 =
,
(d) i5 =
,
(e) i67 =
,
(f) i0 =
,
(g) i−1 =
,
(h) i−2 =
,
(i) i−3 =
,
(j) i−47 =
.
A:
,
2
cn 1 13.pg
16.(1 pt) setAlgebra11ComplexNumbers/ur cn 1
Let z = −1 − 4i. Calculate the following:
14.pg
(a) z2 + 2z + 1 =
+
i,
(b) z2 + iz − (2 + i) =
+
i,
(z − 4)2
(c)
=
+
i.
z+i
17.(1 pt) setAlgebra11ComplexNumbers/sw3 4 7.pg
Evaluate the expression (5 + 4i) + (−3 −7i) and write the result
in the form a + bi.
The real number a equals
The real number b equals
25.(1 pt) setAlgebra11ComplexNumbers/srw3 4 13.pg
Evaluate the expression (−4 + 6i) + (−6 + 3i) and write the result in the form a + bi.
The real number a equals
The real number b equals
26.(1 pt) setAlgebra11ComplexNumbers/srw3 4 17.pg
Evaluate the expression (5 + 7i) − (6 − 2i) and write the result
in the form a + bi.
The real number a equals
The real number b equals
18.(1 pt) setAlgebra11ComplexNumbers/sw3 4 11.pg
Evaluate the expression (2 − 8i) − (−4 −2i) and write the result
in the form a + bi.
The real number a equals
The real number b equals
27.(1 pt) setAlgebra11ComplexNumbers/srw3 4 23.pg
Evaluate the expression (−1 − 4i)(−1 + 2i) and write the result
in the form a + bi.
The real number a equals
The real number b equals
19.(1 pt) setAlgebra11ComplexNumbers/sw3 4 15.pg
Evaluate the expression (4 + 1i)(−1 − 4i) and write the result in
the form a + bi.
The real number a equals
The real number b equals
20.(1 pt) setAlgebra11ComplexNumbers/sw3
Evaluate the expression
4 + 2i
3 + 1i
and write the result in the form a + bi.
The real number a equals
The real number b equals
28.(1 pt) setAlgebra11ComplexNumbers/srw3
Evaluate the expression
−4 − 2i
−2 + 2i
and write the result in the form a + bi.
The real number a equals
The real number b equals
4 23.pg
4 29.pg
29.(1 pt) setAlgebra11ComplexNumbers/srw3 4 35.pg
Evaluate the expression −1−4i
and write the result in the form
8i
a + bi.
The real number a equals
The real number b equals
21.(1 pt) setAlgebra11ComplexNumbers/sw3 4 29.pg
Evaluate the expression −3−3i
and write the result in the form
4i
a + bi.
The real number a equals
The real number b equals
30.(1 pt) setAlgebra11ComplexNumbers/srw3 4 41.pg
Evaluate the expression i102 and write the result in the form
a + bi.
The real number a equals
The real number b equals
22.(1 pt) setAlgebra11ComplexNumbers/sw3 4 35.pg
Evaluate the expression i88 and write the result in the form
a + bi.
The real number a equals
The real number b equals
31.(1 pt) setAlgebra11ComplexNumbers/srw3
4 43.pg
√
Evaluate the expression −9 and write the result in the form
a + bi.
The real number a equals
The real number b equals
23.(1 pt) setAlgebra11ComplexNumbers/beth1complex.pg
Evaluate the expression
(3 − 3i)(−4i)
1 + 1i
and write the result in the form a + bi.
Then a =
and b =
24.(1 pt) setAlgebra11ComplexNumbers/jj1.pg
If we write the following complex number in standard form
√
√
√
√
( 8 + 10i)( 8 − 10i) = a + bi
32.(1 pt) setAlgebra11ComplexNumbers/srw3
4 45.pg
√ √
Evaluate the expression −3 −192 and write the result in the
form a + bi.
The real number a equals
The real number b equals
33.(1 pt) setAlgebra11ComplexNumbers/srw3
√
√ 4 47.pg
Evaluate the expression (4 − −4)(2 + −1) and write the result in the form a + bi.
The real number a equals
The real number b equals
then
a=
b=
Your answers here have to be simplified so that they are just
numbers.
34.(1 pt) setAlgebra11ComplexNumbers/srw3
4 49.pg
√
−2+√ −4
Evaluate the expression −4+ −25 and write the result in the
form a + bi.
3
41.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 20.pg
Write the following numbers in the polar form reiθ , −π < θ ≤ π:
(a) πi
r= √ ,θ=
,
(b) −2 3 − 4i
r=
, θ√=
,
(c) (1 − i)(− 3 + i)
r= √
,θ=
,
(d) ( 3 − 4i)2
r=
,
√, θ =
−5 + 3i
(e)
3 + 3i
,θ=
,
r= √
− 7(1 + i)
(f) √
2+i
r=
,θ=
,
The real number a equals
The real number b equals
35.(1 pt) setAlgebra11ComplexNumbers/srw3
4 51.pg
√
√
Evaluate the expression √−6−5
and
write
the
result in the form
−5
a + bi.
The real number a equals
The real number b equals
36.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 15.pg
Solve the following equations for z:
(a) iz = 4 − zi
z =
+
i,
z
= 1 − 5i
(b)
1−z
z=
+
i,
(c) (2 − i)z + 8z2 = 0
(This question has two solutions, one of which is 0, find the
other)
+
i.
z=
37.(1 pt) setAlgebra11ComplexNumbers/ur cn 1
Calculate:
4 + 4i (a) ,
=
−4
−
i
,
(b) (1 + i)(3 − 2i)(3 − i) =
i(3 + 2i)3 (c) ,
=
(3 − 2i)2
(π + i)100 .
(d) =
(π − i)100
42.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 21.pg
Write each of the given numbers in the form a + bi :
iπ
(a) e− 3
+
i,
e(1+i4π)
(b)
iπ
e(−1+ 2 )
+
i,
i
(c) ee
+
i.
16.pg
43.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 22.pg
Write each of the given numbers in the form a + bi :
e6i − e−6i
(a)
2i
+
i,
iπ
(b) 8e(6+ 6 )
+
i,
iπ
5e( 3 )
(c) e
+
i.
38.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 17.pg
Answer the following questions (T or F):
1. arg z = Arg z + 2πk, (k = 0, ±1, ±2, ±3...) and if
z 6= 0.
2. Arg z = −Arg z, if z is not real.
3. Arg zz12 = Arg z1 − Arg z2 , if z1 6= 0 , z2 6= 0.
4. Arg(0) is undefined.
5. Arg z1 z2 = Arg z1 + Arg z2 , if z1 6= 0 , z2 6= 0.
39.(1 pt) setAlgebra11ComplexNumbers/ur
Place the following in order:
(a) |z2 | − |z1 | ,
(b) |z1 + z2 | ,
(c) |z
1 | + |z2 | ,
(d) |z2 | − |z1 | .
≤
≤
≤
.
44.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 23.pg
Write each of the given numbers in the polar form reiθ, −π <
θ ≤ π.
8−i
(a)
7
r=
,√
θ=
,
(b) −6π(5 + i 3)
r=
,θ=
,
(c) (1 + i)3
r=
,θ=
.
cn 1 18.pg
40.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 19.pg
Write the following numbers in the polar form reiθ , 0 ≤ θ < 2π:
1
(a)
7
r=
,θ=
,
(b) 5 + 5i
r=
,θ=
,
(c) 5 − 5i
r=
,θ=
.
45.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 24.pg
Write each of the given numbers in the polar form reiθ, −π <
θ ≤π.
3
2π
2π
(a) cos
+ i sin
9
9
r=
,θ=
,
6 − 6i
(b) √
− 3+i
r=
,θ=
,
4
(c)
r=
4i
3e(4+i)
,θ=
• E. |z| ≥ 2
• F. −1 < Im z ≤ 1
.
46.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 25.pg
Determine which of the following properties of the real exponential function remain true for the complex exponential ( i.e.,
for x replaced by z ).
Answer T or F:
1. ex is defined for all x.
2. ex is never zero.
3. e−x = e1x .
4. ex is a one-to-one function.
52.(1 pt) setAlgebra11ComplexNumbers/Sqrt.pg
Find the square root of 1+2i so that the real part of your answer
is positive.
The square root is
.
47.(1 pt) setAlgebra11ComplexNumbers/ur
Which of the following sets are open?
• A. (Re z)2 > 1
• B. |Arg z| < π4
• C. |z| ≥ 2
• D. 0 < |z − 2| < 3
• E. −1 < Im z ≤ 1
• F. |z − 1 + i| ≤ 3
cn 1 26.pg
(2) 1 5
Place all answers in the following blank, separated by commas:
48.(1 pt) setAlgebra11ComplexNumbers/ur
Which of the given sets are bounded?
• A. |Arg z| < π4
• B. −1 < Im z ≤ 1
• C. 0 < |z − 2| < 3
• D. |z| ≥ 2
• E. (Re z)2 > 1
• F. |z − 1 + i| ≤ 3
cn 1 27.pg
49.(1 pt) setAlgebra11ComplexNumbers/ur
Which of the given sets are regions?
• A. −1 < Im z ≤ 1
• B. (Re z)2 > 1
• C. |z| ≥ 2
• D. |z − 1 + i| ≤ 3
• E. 0 < |z − 2| < 3
• F. |Arg z| < π4
cn 1 28.pg
1
1
(3) i 4
Place all answers in the following blank, separated by commas:
54.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 32.pg
Find all the values of the following :
√ 1
(1) (1 + 3i) 3
Place all answers in the following blank, separated by commas:
1
(2) (i + 1) 2
Place all answers in the following blank, separated by commas:
1
6
4i
(3)
1+i
Place all answers in the following blank, separated by commas:
55.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 33.pg
Solve the following equations for z, find all solutions :
(1) 5z2 + z + 4 = 0
Place all answers in the following blank, separated by commas:
(2) z2 − (3 − 2i)z + 1 − 3i = 0
Place all answers in the following blank,separated by commas:
50.(1 pt) setAlgebra11ComplexNumbers/ur cn 1
Which of the given sets are closed regions?
• A. −1 < Im z ≤ 1
• B. |z| ≥ 2
• C. |z − 1 + i| ≤ 3
• D. 0 < |z − 2| < 3
• E. |Arg z| < π4
• F. (Re z)2 > 1
51.(1 pt) setAlgebra11ComplexNumbers/ur
Which of the given sets are domains?
• A. |z − 1 + i| ≤ 3
• B. 0 < |z − 2| < 3
• C. (Re z)2 > 1
• D. |Arg z| < π4
53.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 31.pg
Find all the values of the following.
1
(1) (−16) 4
Place all answers in the following blank, separated by commas:
(3) z2 − 2z + i = 0
Place all answers in the following blank, separated by commas:
29.pg
56.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 34.pg
Let z = −3 + 6i. Write the following numbers in a + bi form:
(a) −7z =
+
i,
(b) z̄ =
+
i,
1
(c) =
+
i.
z
cn 1 30.pg
57.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 35.pg
Write the following numbers in the polar form r(cosφ + i sin φ),
0 ≤ φ < 2π.
(a) 8
r=
,φ=
,
(b) 3i
5
r=
,φ=
(c) 5 + 3i
r=
,φ=
61.(1 pt) setAlgebra11ComplexNumbers/sw3 4 55.pg
Find all solutions of the equation t + 2 + 5t = 0 and express them
in the form a + bi:
First input the solution with b < 0 here:
the real number a equals
and the real number b
equals
Then input the solution with b > 0 here:
the real number a equals
and the real number b
equals
,
.
58.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 36.pg
Let z = 9(cos0.4 + i sin0.4).
Write the following numbers in the polar form r(cosφ + i sin φ),
0 ≤ φ < 2π.
(a) 5z
r=
,φ=
,
(b) z̄
r=
,φ=
,
1
(c)
z
r=
,φ=
.
62.(1 pt) setAlgebra11ComplexNumbers/beth2complex.pg
Find all solutions of the equation x2 + 3x + 6 = 0 and express
them in the form a + bi:
solutions:
(Note: If there is more than one solution, enter a comma separated list (i.e.: 1+2i,3+4i).)
59.(1 pt) setAlgebra11ComplexNumbers/ur cn 1 37.pg
Let z = 7ei2.8 .
Write the following numbers in the polar form reiφ , 0 ≤ φ < 2π.
(a) 4z
r=
,φ=
,
(b) z̄
,φ=
,
r=
1
(c)
z
,φ=
.
r=
Then input the solution with b > 0 here:
the real part a equals
and the imaginary part b equals
60.(1 pt) setAlgebra11ComplexNumbers/sw3 4 49.pg
Find all solutions of the equation x2 + 1x + 6 = 0 and express
them in the form a + bi:
First input the solution with b < 0 here:
the real number a equals
and the real number b equals
64.(1 pt) setAlgebra11ComplexNumbers/srw3 4 61.pg
Find all solutions of the equation t + 2 + 5t = 0 and express them
in the form a + bi:
First input the solution with b < 0 here:
the real part a equals
and the imaginary part b equals
Then input the solution with b > 0 here:
the real number a equals
and the real number b equals
Then input the solution with b > 0 here:
the real part a equals
and the imaginary part b equals
63.(1 pt) setAlgebra11ComplexNumbers/srw3 4 55.pg
Find all solutions of the equation x2 − 2x + 8 = 0 and express
them in the form a + bi:
First input the solution with b < 0 here:
the real part a equals
and the imaginary part b equals
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
6
ARNOLD PIZER
rochester problib from CVS June 25, 2004
Rochester WeBWorK Problem Library
WeBWorK assignment Algebra12EqnsOtherTypes due 1/12/10 at 2:00 AM
10.(1 pt) setAlgebra12EqnsOtherTypes/SolveSqrt.pg
Solve for the only possible solution. Give your answer to the
nearest thousandth.
1.(1 pt) setAlgebra12EqnsOtherTypes/srw1 5 57.pg
The real solution of the equation x3 = 64 is:
√
−7 1x − 6 = −5
2.(1 pt) setAlgebra12EqnsOtherTypes/srw1 5 59.pg
The equation x4 − 4 = 0 has two real solutions A and B where
A < B.
A=
and B =
3.(1 pt) setAlgebra12EqnsOtherTypes/srw1 5 61.pg
The equation 4x4 − 6x3 − 3x2 = 0 has three real solutions A, B,
and C where A < B < C.
A=
,
B=
,
C=
.
4.(1 pt) setAlgebra12EqnsOtherTypes/lh1-6 21.pg
√
Solve the equation x − 4 x − 12 = 0 by factoring.
The only solution is x =
.
.
x=
Does your solution satisfy the equation? (yes or no)
11.(1 pt) setAlgebra12EqnsOtherTypes/equationW2radicals.pg
Solve for t:
√
√
t − 33− t + 41 = 15
The only possible root ist =
. It is a(n)
root. (Fill in
the second blank with REAL or EXTRANEOUS)
12.(1 pt) setAlgebra12EqnsOtherTypes/srw1
Solve the equation for t
8
5
8
+
+
=0
9 − t 9 + t 81 − t 2
5.(1 pt) setAlgebra12EqnsOtherTypes/quadWfractions.pg
Solve for x:
t=
13.(1 pt) setAlgebra12EqnsOtherTypes/ur ab
Solve the equation
2
x + 66
x + 66
− 59
+ 114 = 0
x − 68
x − 68
The smaller solution is
.
The larger solution is
.
6.(1 pt) setAlgebra12EqnsOtherTypes/lh1-6
√
Solve the equation 10 − x+ x = 4.
The only solution is x =
.
1
5 1.pg
−5
8
x+1
=
+ 2
x − 1 x + 3 x + 2x − 3
Hint: There is only one non-extraneous root.
x=
36.pg
7.(1 pt) setAlgebra12EqnsOtherTypes/ur ab 5
Solve the equation
5 22.pg
14.(1 pt) setAlgebra12EqnsOtherTypes/lh1-6
1
1
1
−
= .
Solve the equation
x + 4 x + 5 12
The solutions are x1 =
and x2 =
where x1 ≤ x2 .
4.pg
1
(x − 1)− 2 (x − 7) + 1(x − 1) 2 = 0
15.(1 pt) setAlgebra12EqnsOtherTypes/lh1-6
Solve the equation |2x − 3| = 18.
The solutions are x1 =
and x2 =
where x1 ≤ x2 .
.
x=
8.(1 pt) setAlgebra12EqnsOtherTypes/SolveDualSqrt.pg
Solve for the only possible solution. Give your answer to the
nearest thousandth.
59.pg
66.pg
16.(1 pt) setAlgebra12EqnsOtherTypes/ur ab 5 2.pg
The equation |8x − 24| = 24 has two solutions.
The sum of those two solutions is
.
17.(1 pt) setAlgebra12EqnsOtherTypes/ur ab 5 3.pg
The equation |5x + 10| = 5 has two solutions.
The distance between those two solutions is
18.(1 pt) setAlgebra12EqnsOtherTypes/pn4.pg
Solve the following equation.
|5x − 10| = 10
√
√
2x − 7 = 7x − 5
x=
.
Does your solution satisfy the equation? (yes or no)
9.(1 pt) setAlgebra12EqnsOtherTypes/SolveHardSqrt.pg
Find all possible solutions. Give your answers in increasing order. Give your answers to the nearest thousandth.
√
3x + 3+ 3 = 7x
. Is it a solution? (yes or
The smaller possible solution is
no)
The larger possible solution is
. Is it a solution? (yes or
no)
.
Answer:
Note: If there is more than one answer, write them separated
by commas (e.g., 1, 2).
19.(1 pt) setAlgebra12EqnsOtherTypes/pn5.pg
Solve the following equation.
1
1
=3
|11 − 10x|
List the four possible roots in increasing order. Below each possible root, enter ROOT if it is a root or EXTRANEOUS if it is
an extraneous root.
<
<
<
,
,
,
Answer:
Note: If there is more than one answer, write them separated
by commas (e.g., 1, 2).
21.(1 pt) setAlgebra12EqnsOtherTypes/volume2SA.pg
Given that the volume of a cylinder is 97, and the radius of the
cylinder is twice the height, find the surface Area of the cylinder.
Note: Your answer must be a number. No arithmetic operations
are allowed.
The surface area of the cylinder is
cm3 .
20.(1 pt) setAlgebra12EqnsOtherTypes/absolutevalue.pg
Solve the following equation for x:
x | x − 8| = 28x + 8
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
2
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra13Inequalities due 1/13/10 at 2:00 AM
For each interval, answer YES or NO to whether the interval
is included in the solution.
(−∞, A)
(A, ∞)
1.(1 pt) setAlgebra13Inequalities/pn2.pg
Express the inequality using interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
x < −1
Answer:
2.(1 pt) setAlgebra13Inequalities/pn3.pg
Express the inequality using interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
x ≥ −8
Answer:
3.(1 pt) setAlgebra13Inequalities/srw1 7 3.pg
The inequality 3x + 9 > 3 means that x is greater than A where
A is
4.(1 pt) setAlgebra13Inequalities/srw1 7 9.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
8.(1 pt) setAlgebra13Inequalities/p1.pg
Solve the following inequality. Write the answer in interval notation. If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as ”infinity”.
−4 − x < 1
Answer:
9.(1 pt) setAlgebra13Inequalities/pn6.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
−1 − x < 4
Answer:
10.(1 pt) setAlgebra13Inequalities/p13.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
−2x + 4 < 5(1x + 3) − 5
Answer:
11.(1 pt) setAlgebra13Inequalities/pn1.pg
Express the inequality using interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
−3 < x ≤ 10
Answer:
12.(1 pt) setAlgebra13Inequalities/srw1 7 19.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
8−x≥ 7
Answer:
5.(1 pt) setAlgebra13Inequalities/srw1 7 13.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
3x + 16 ≤ 6x + 8
Answer:
6.(1 pt) setAlgebra13Inequalities/srw1 7 15.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
1
x − 16 > 12
2
.
Answer:
7.(1 pt) setAlgebra13Inequalities/ur ab 7 1.pg
Consider the inequality
5 ≤ x + 4 < 10
4 + 7x < 4x + 8
The solution of this inequality consists one or more of the
following intervals: (−∞, A) and (A, ∞)
Find A
1
Answer:
13.(1 pt) setAlgebra13Inequalities/srw1 7 67.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
5
20 ≤ (F − 32) ≤ 33
9
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
|x − 6| < 3
Answer:
23.(1 pt) setAlgebra13Inequalities/p9.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
|x + 6| ≥ 1
Answer:
24.(1 pt) setAlgebra13Inequalities/p16.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
Note that an absolute value
the left hand side.
encloses
5x + 3 8 ≤7
Answer:
25.(1 pt) setAlgebra13Inequalities/p19.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞
as -infinity .
4 − 1 x ≤ 6
2 Answer:
14.(1 pt) setAlgebra13Inequalities/p2.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
−5 ≤ 4x + 1 < −1
Answer:
15.(1 pt) setAlgebra13Inequalities/srw1 7 16.pg
The inequality 2x + 6 ≤ x + 11 ≤ 3x + 3 means that x is in the
closed interval [A, B] where A is:
and B is:
16.(1 pt) setAlgebra13Inequalities/srw1 7 61.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
|2x − 5| ≤ 11
Answer:
17.(1 pt) setAlgebra13Inequalities/srw1 7 65.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
4|x + 2| − 4 < 5
Answer:
18.(1 pt) setAlgebra13Inequalities/srw1 8 23.pg
To say that |x − 8| ≤ 4 is the same as saying x is in the closed
interval [A, B]
where A is:
and where B is:
19.(1 pt) setAlgebra13Inequalities/srw1 8 26.pg
To say that |x + 8| ≤ 2 is the same as saying x is in the closed
interval [A, B] where A is:
and where B is:
20.(1 pt) setAlgebra13Inequalities/srw1
8 29.pg
x − 8
≤ 4 is the same as saying x is in the closed
To say that 4 interval [A, B]
where A is:
and where B is:
21.(1 pt) setAlgebra13Inequalities/srw1
8 29-sol.pg
x − 5
To say that ≤ 2 is the same as saying x is in the closed
5 interval [A, B]
where A is:
and where B is:
22.(1 pt) setAlgebra13Inequalities/p8.pg
Solve the following inequality. Write the answer in interval notation.
Answer:
26.(1 pt) setAlgebra13Inequalities/srw1 7 23.pg
Solve the inequality x2 + 4x − 60 < 0. The solution is x is in the
open interval (A, B) where A is:
and B is:
27.(1 pt) setAlgebra13Inequalities/srw1 7 25.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
(x − 8)(x − 17) > 0
Answer:
28.(1 pt) setAlgebra13Inequalities/srw1 7 29.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
2x2 + x ≥ 8
Answer:
29.(1 pt) setAlgebra13Inequalities/srw1 7 37.pg
Solve the following inequality. Write the answer in interval
2
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
x3 − 81x ≤ 0
−2x2 ≤ 30
Answer:
Answer:
30.(1 pt) setAlgebra13Inequalities/p3.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
(x − 3)(x − 2) ≤ 0
Answer:
31.(1 pt) setAlgebra13Inequalities/ur ab 7 2.pg
Consider the inequality
36.(1 pt) setAlgebra13Inequalities/srw1 7 53.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
x4 > 25x2
Answer:
37.(1 pt) setAlgebra13Inequalities/srw1 7 39.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
x2 < 5x + 36
The solution of this inequality consists one or more of the following intervals: (−∞, A), (A, B), and (B, ∞) where A < B.
Find A
Find B
For each interval, answer YES or NO to whether the interval is
included in the solution.
(−∞, A)
(A, B)
(B, ∞)
x−9
≥0
x + 12
Answer:
38.(1 pt) setAlgebra13Inequalities/p5.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
2−x
≥0
x−7
Answer:
32.(1 pt) setAlgebra13Inequalities/p4.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
x2 − 7x + 6 > 0
Answer:
33.(1 pt) setAlgebra13Inequalities/p11.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
x2 − 1x > 0
Answer:
34.(1 pt) setAlgebra13Inequalities/p12.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
−x2 + 3x ≥ 0
Answer:
35.(1 pt) setAlgebra13Inequalities/p6.pg
Solve the following inequality. Write the answer in interval
39.(1 pt) setAlgebra13Inequalities/p14.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
1
>6
x−6
Answer:
40.(1 pt) setAlgebra13Inequalities/p15.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
x−6
≤ −5
x−4
Answer:
41.(1 pt) setAlgebra13Inequalities/p17.pg
Solve the following inequality. Write the answer in interval
3
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
x
> −4
x−5
Answer:
42.(1 pt) setAlgebra13Inequalities/p18.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
1
1
≤
x−8 x−4
Answer:
43.(1 pt) setAlgebra13Inequalities/ur ab 7 3.pg
Consider the inequality
x−4
>0
2
x (x + 4)
The solution of this inequality consists of one or more of the
following intervals: (−∞, A), (A, B), (B,C),and (C, ∞) where
A < B < C.
Find A
Find B
Find C
For each interval, answer YES or NO to whether the interval is
included in the solution.
(−∞, A)
(A, B)
(B,C)
(C, ∞)
Find C
For each interval, answer YES or NO to whether the interval is
included in the solution.
(−∞, A)
(A, B)
(B,C)
(C, ∞)
46.(1 pt) setAlgebra13Inequalities/srw1 7 47.pg
Solve the following inequality. Write the answer in interval
notation. Note: If the answer includes more than one interval write the intervals separated by the ”union” symbol, U. If
needed enter ∞ as infinity and −∞ as -infinity .
2
2
1+
≤
x+1 x
Answer:
47.(1 pt) setAlgebra13Inequalities/p7.pg
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
x(x − 1)
≤0
x2 − 1x − 20
Answer:
48.(1 pt) setAlgebra13Inequalities/Inequality.pg
Solve the inequality
(x − 4)4 (x − 14)13
≥0
x − 24
Give your answer in interval notation.
x∈
Note: Enter your answer without spaces. If you need − inf, type
-inf. If you need inf, type inf. Remember that punctuation is
important.
44.(1 pt) setAlgebra13Inequalities/ur ab 7 4.pg
Consider the inequality
x+7
> −1
x+9
The solution of this inequality consists one or more of the following intervals: (−∞, A), (A, B),and (B, ∞) where A < B.
Find A
Find B
For each interval, answer YES or NO to whether the interval is
included in the solution.
(−∞, A)
(A, B)
(B, ∞)
49.(1 pt) setAlgebra13Inequalities/p10.pg
Solve the following inequality. Write the answer in interval notation. If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as ”infinity”.
1
≥4
|x + 2|
Answer:
50.(1 pt) setAlgebra13Inequalities/srw1 7 49.pg
You arrive in Paris and the forcast is for a low of 18 and a high
of 24 degrees Celsius. What is the forcasted low temperature in
Fahrenheit?
What is the forcasted high temperature in Fahrenheit?
45.(1 pt) setAlgebra13Inequalities/ur ab 7 5.pg
Consider the inequality
x
x
<
x−1 7
The solution of this inequality consists one or more of the
following intervals: (−∞, A), (A, B), (B,C),and (C, ∞) where
A < B < C.
Find A
Find B
51.(1 pt) setAlgebra13Inequalities/srw1 7 50.pg
Your friend from Paris arrives in New York and the forcast is for
a low of 50 and a high of 77 degrees Fahrenheit.
What is the forcasted low temperature in Celsius?
What is the focasted high temperature in Celsius?
4
52.(1 pt) setAlgebra13Inequalities/srw1 7 73.pg
A car rental company offers two plans for renting a car.
Plan A: 30 dollars per day and 15 cents per mile
Plan B: 50 dollars per day with free unlimited mileage
For what range of miles will plan B save you money?
Your answer is that the mileage must be greater than
is nonnegative (greater than or equal to zero) so that the radical
√
3x + 2
defines a real number. Write your answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
For example, you may write (-infinity, 5] for the interval (−∞, 5]
and (-infinity, 5]U(7,9) for (−∞, 5] ∪ (7, 9).
Your answer:
.
53.(1 pt) setAlgebra13Inequalities/lh1-7 75.pg
Find the interval on the real number line for which the radicand
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
5
ARNOLD PIZER
rochester problib from CVS June 25, 2004
Rochester WeBWorK Problem Library
1.(1 pt) setAlgebra14Lines/lh2-1
WeBWorK assignment Algebra14Lines due 1/14/10 at 2:00 AM
3.(1 pt) setAlgebra14Lines/lh2-1 7.pg
Find an equation y = mx+b for the line whose graph is sketched
1.pg
The slope m equals
.
The y-intercept b equals
Match the Lines L1 (blue), L2 ( red) and L3 (green) with the
slopes by placing the letter of the slopes next to each set listed
below:
1. The slope of line L1
2. The slope of line L2
3. The slope of line L3
A. m = 1.8
B. m = 0
C. m = −1.9
.
4.(1 pt) setAlgebra14Lines/lh2-1 9.pg
Find an equation y = mx+b for the line whose graph is sketched
2.(1 pt) setAlgebra14Lines/lh2-1 5.pg
Find an equation y = mx+b for the line whose graph is sketched
The slope m equals
.
The y-intercept b equals
The slope m equals
.
The y-intercept b equals
.
5.(1 pt) setAlgebra14Lines/sw2 4 11.pg
Find an equation y = mx+b for the line whose graph is sketched
.
1
The number m equals
The number b equals
14.(1 pt) setAlgebra14Lines/srw1 10 10.pg
The equation of the line with slope −5 that goes through the
point (−6, −6) can be written in the form y = mx + b where m
is:
and where b is:
15.(1 pt) setAlgebra14Lines/pt slope to slope int.pg
The equation of the line with slope −1 that goes through the
point (−8, −6) can be written in the form y = mx + b where m
is:
and where b is:
16.(1 pt) setAlgebra14Lines/sw2 4 21.pg
The equation of the line with slope 3 and y-intercept −5 can be
written in the form y = mx + b where
the number m is:
the number b is:
17.(1 pt) setAlgebra14Lines/sw2 4 3.pg
The equation of the line that goes through the points (2, 1) and
(9, 7) can be written in the form y = mx + b where
the slope m is:
.
.
6.(1 pt) setAlgebra14Lines/sApB 7-10.pg
Find the slope of the line through (0, −3) and (−4, 10).
18.(1 pt) setAlgebra14Lines/sw2 4 5.pg
The equation of the line that goes through the points (−3, −10)
and (8, 9) can be written in the form y = mx + b where its slope
m is:
19.(1 pt) setAlgebra14Lines/sw2 4 19.pg
The equation of the line that goes through the points (2, 1) and
(8, 6) can be written in the form y = mx +b where m is:
and b is:
20.(1 pt) setAlgebra14Lines/srw1 10 11.pg
The equation of the line that goes through the points (5, 8) and
(8, 3) can be written in the form y = mx +b where m is:
and where b is:
21.(1 pt) setAlgebra14Lines/srw1 10 12.pg
The equation of the line that goes through the points (−2, −10)
and (3, 5) can be written in the form y = mx + b where m is:
7.(1 pt) setAlgebra14Lines/slope from pts num.pg
Find the slope of the line through (7, 1) and (−2, 7).
8.(1 pt) setAlgebra14Lines/slope from pts var.pg
Find the slope of the line passing through the points (a, 5a + 9)
and (a + h, 5(a + 3h) + 9).
The slope is
9.(1 pt) setAlgebra14Lines/sApB 21-26.pg
A line through (0, −9) with a slope of 4 has a y-intercept at
10.(1 pt) setAlgebra14Lines/sw2 4 17.pg
The equation of the line with slope −2 that goes through the
point (8, −3) can be written in the form y = mx + b where m is:
and where b is:
22.(1 pt) setAlgebra14Lines/srw1 10 13.pg
The equation of the line that goes through the points (−4, 10)
and (3, −10) can be written in the form y = mx + b where m is:
and b is:
11.(1 pt) setAlgebra14Lines/srw1 10 7.pg
The equation of the line with slope 2 that goes through the
point (6, 3) can be written in the form y = mx + b where m is:
and where b is:
23.(1 pt) setAlgebra14Lines/pts to slope int.pg
The equation of the line that goes through the points (−2, 6) and
(4, 3) can be written in the form y = mx +b where m is:
and where b is:
24.(1 pt) setAlgebra14Lines/pts to gen.pg
The equation of the line that goes through the points (−5, −6)
and (−9, 8) can be written in general form Ax + By + C = 0
where
A=
B=
C=
and where b is:
12.(1 pt) setAlgebra14Lines/srw1 10 8.pg
The equation of the line with slope 2 that goes through the
point (−5, 5) can be written in the form y = mx + b where m
is:
and where b is:
13.(1 pt) setAlgebra14Lines/srw1 10 9.pg
The equation of the line with slope −5 that goes through the
point (7, −5) can be written in the form y = mx + b where m is:
and where b is:
2
25.(1 pt) setAlgebra14Lines/sw2 4 23.pg
The equation of the line with x-intercept 2 and y-intercept −3
can be written in the form y = mx + b where
the number m is:
the number b is:
26.(1 pt) setAlgebra14Lines/sw2 4 39.pg
Find the slope and y-intercept of the line x + y = 4.
the slope of the line is:
the y-intercept of the line is:
is parallel to the line going through the points (−6, 2) and (5, 2)
can be written in the form y = mx + b where
m is:
and b is:
36.(1 pt) setAlgebra14Lines/sApB 31-36.pg
An equation of a line through (3, 6) which is perpendicular to
the line y = 4x + 2 has slope:
and y-intercept at:
27.(1 pt) setAlgebra14Lines/sw2 4 41.pg
Find the slope and y-intercept of the line 17x + 17y = 0.
the slope of the line is:
the y-intercept of the line is:
37.(1 pt) setAlgebra14Lines/sw2 4 31.pg
The equation of the line that goes through the point (5, 10) and
is perpendicular to the line 2x + 3y = 4 can be written in the
form y = mx + b where m is:
and b is:
28.(1 pt) setAlgebra14Lines/sw2 4 43.pg
Find the slope and y-intercept of the line 20x − 5y = 3.
the slope of the line is:
the y-intercept of the line is:
38.(1 pt) setAlgebra14Lines/srw1 10 19a.pg
The equation of the line that goes through the point (9, 4) and is
perpendicular to the line 4x + 5y = 2 can be written in the form
y = mx + b where m is:
and where b is:
29.(1 pt) setAlgebra14Lines/slope int from gen.pg
Find the slope, x-intercept, and y-intercept for the line −1x −
2y + 20 = 0.
The slope is
.
The x-intercept is
.
The y-intercept is
.
Note: Your answers must be decimals.
30.(1 pt) setAlgebra14Lines/sApB 31-36a.pg
An equation of a line through (1, 1) which is parallel to the line
y = 2x + 2 has slope:
39.(1 pt) setAlgebra14Lines/srw1 10 19a-sol.pg
The equation of the line that goes through the point (4, 3) and is
perpendicular to the line 5x + 2y = 4 can be written in the form
y = mx + b where m is:
and where b is:
40.(1 pt) setAlgebra14Lines/given pt and line.pg
An equation of a line through (-2, 6) which is perpendicular to
the line y = 4x + 1 has slope:
and y-intercept at:
and y-intercept at:
31.(1 pt) setAlgebra14Lines/srw1 10 17.pg
The equation of the line that goes through the point (17, 36) and
is parallel to the x-axis can be written in the form y = mx + b
where m is:
and where b is:
32.(1 pt) setAlgebra14Lines/srw1 10 19.pg
The equation of the line that goes through the point (2, 4) and
is parallel to the line 2x + 4y = 5 can be written in the form
y = mx + b where m is:
and where b is:
33.(1 pt) setAlgebra14Lines/sw2 4 27.pg
The equation of the line that goes through the point (5, 3) and
is parallel to the line 2x + 4y = 3 can be written in the form
y = mx + b where m is:
and where b is:
34.(1 pt) setAlgebra14Lines/srw1 10 20.pg
The equation of the line that goes through the point (−3, 4) and
is parallel to the line 3x + 2y = 2 can be written in the form
y = mx + b where m is:
and where b is:
35.(1 pt) setAlgebra14Lines/sw2 4 33.pg
The equation of the line that goes through the point (3, 3) and
41.(1 pt) setAlgebra14Lines/given pt and line hard.pg
The equation, in general form, of the line that passes through
the point (−3, 10) and is parallel to the line 7x + 7y + 2 = 0 is
Ax + By +C = 0, where
A=
B=
C=
42.(1 pt) setAlgebra14Lines/faris1.pg
The demand equation for a certain product is given by p =
116 − 0.035x , where p is the unit price (in dollars) of the product and x is the number of units produced. The total revenue
obtained by producing and selling x units is given by R = xp.
Determine prices p that would yield a revenue of 6020 dollars.
Lowest such price =
Highest such price =
43.(1 pt) setAlgebra14Lines/ur geo 2 1.pg
The line whose equation is 3x − 3y = −5 goes through the point
(−6,t) for
t=
3
44.(1 pt) setAlgebra14Lines/ur geo 2 2.pg
The line through (−6, 4) and (12, −7) also goes through the
point (t, 2) for
t=
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
4
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra15Functions due 1/15/10 at 2:00 AM
9.(1 pt) setAlgebra15Functions/srw2 1 23.pg
Given the function
(
x ≤ −1
x2 + 2x, if
f (x) =
x + 8,
if
x > −1
1.(1 pt) setAlgebra15Functions/srw2 1 1.pg
Express the rule ”Multiply by 18, then add 16” as the function
f (x) =
.
2.(1 pt) setAlgebra15Functions/srw2 1 3.pg
Express the rule ”Subtract 23, then square” as the function
f (x) =
.
Calculate the following values:
f (−4) =
f (−1) =
f (1) =
3.(1 pt) setAlgebra15Functions/srw2 1 5.pg
x
f (x) = + 22 can be expressed in words as ”Add 22, then di7
vide by 7”. Is this statement true?
Your answer is (input Yes or No):
.
10.(1 pt) setAlgebra15Functions/sw4 1 11.pg
Given the function f (x) = 4x2 − 4x + 8. Calculate the following
values:
f (−2) =
f (−1) =
f (0) =
f (1) =
f (2) =
4.(1 pt) setAlgebra15Functions/srw2 1 7.pg
f (x) = 10x2 − 9 can be expressed in words as ”Square, multiply
by 10, then subtract 9”. Is this statement true?
Your answer is (input Yes or No):
.
5.(1 pt) setAlgebra15Functions/srw2 1 11.pg
Given the function f (x) = 3x2 − 5x + 2. Calculate the following
values:
f (−2) =
f (−1) =
f (0) =
f (1) =
f (2) =
11.(1 pt) setAlgebra15Functions/sw4 1 17.pg
Given the function f (x) = 8x2 + 7x − 8. Calculate the following
values:
f (0) =
f (2) =
f (−2) =
f (x + 1) =
f (−x) =
6.(1 pt) setAlgebra15Functions/ur fn 1 5.pg
x
Let f (x) be the function
− 1. Find the following:
x+1
f (9) =
f (−8) =
1
f( )
8
1
f (− )
7
7.(1 pt) setAlgebra15Functions/srw2 1 17.pg
Given the function f (x) = 4x2 + 2x − 8. Calculate the following
values:
f (0) =
f (2) =
f (−2) =
f (x + 1) =
f (−x) =
12.(1 pt) setAlgebra15Functions/sw4 1 19.pg
Given the function f (x) = 6|x − 4|. Calculate the following values:
f (0) =
f (2) =
f (−2) =
f (x + 1) =
f (x2 + 2) =
In your answer, use abs(g(x)) for |g(x)|.
13.(1 pt) setAlgebra15Functions/ns1
Let f (x) = 3x2 + x − 12.
Find
f (0)
f (−2)
f (6)
√
f ( 5)√
f (1 + 5)
8.(1 pt) setAlgebra15Functions/srw2 1 19.pg
Given the function f (x) = 5|x − 6|, calculate the following values:
f (0) =
f (2) =
f (−2) =
f (x + 1) =
f (x2 + 2) =
Note: In your answer, you may use abs(g(x)) for |g(x)|.
1 19.pg
14.(1 pt) setAlgebra15Functions/s0 1 1.pg √
Let f (x) = 5x3 + 5x2 + 3x + 5. Find f (2 + 2).
1
15.(1 pt) setAlgebra15Functions/lh2-2 35.pg
Given the function
4x + 4 if x < 0
f (x) =
4x + 8 if x ≥ 0
Calculate the following values:
f (−1) =
f (0) =
f (2) =
f (x + 1) =
f (x) + f (8) =
23.(1 pt) setAlgebra15Functions/srw2 1 29.pg
Given the function f (x) = 2x−8, calculate the following values:
f (a) =
f (a + h) =
f (a + h) − f (a)
=
h
24.(1 pt) setAlgebra15Functions/srw2 1 33.pg
Given the function f (x) = −7+3x2 , calculate the following values:
f (a) =
f (a + h) =
f (a + h) − f (a)
=
h
25.(1 pt) setAlgebra15Functions/nc1s1p1.pg
Find the domain of this function:
√
3
−4 + 4x
16.(1 pt) setAlgebra15Functions/lh2-2 36.pg
Given the function
2
4x + 3 if x < 1
f (x) =
8x2 + 3 if x ≥ 1
Calculate the following values:
f (−2) =
f (1) =
f (2) =
17.(1 pt) setAlgebra15Functions/faris1.pg
Let
x+4
f (x) =
.
3x − 2
Compute the following values. If one is not defined, type Undefined .
f (0) =
f (5) =
f (2/3) =
(which reads the 3th root of −4 + 4x ).
The function is defined on the interval from
.
Use INF for infinity or -INF for minus infinity.
18.(1 pt) setAlgebra15Functions/sw4 1 31.pg
Let f (x) = 14. Calculate the following values:
f (a) =
f (a + h) =
f (a + h) − f (a)
=
for h 6= 0
h
Now find the domain of this function:
√
4
−4 + 4x
(which reads the 4th root of −4 + 4x ).
The function is defined on the interval from
to
.
26.(1 pt) setAlgebra15Functions/s0 1 10.pg
29
The domain of the function f (x) =
is all real numbers
11x − 34
x except for x where x equals
19.(1 pt) setAlgebra15Functions/sw4 1 33.pg
Let f (x) = 4 − 2x + 12x2. Calculate the following values:
f (a) =
f (a + h) =
f (a + h) − f (a)
for h 6= 0
=
h
20.(1 pt) setAlgebra15Functions/s0
27.(1 pt) setAlgebra15Functions/s0 1 11.pg
√
The domain of the function f (x) = −3x − 34 consists of one
or more of the following intervals: (−∞, A] and [A, ∞).
Find A
For each interval, answer YES or NO to whether the interval is
included in the solution.
(−∞, A)
(A, ∞)
1 2.pg
f (3 + h) − f (3)
Let f (x) = 3x2 + 5x + 2 and let g(h) =
.
h
Determine each of the following:
(a) g(1) =
(b) g(0.1) =
(c) g(0.01) =
You will notice that the values that you entered are getting closer
and closer to a number L. This number is called the limit of g(h)
as h approaches 0 and is also called the derivative of f (x) at the
point when x = 3. We will see more of this when we get to the
calculus textbook.
Enter the value of L:
21.(1 pt) setAlgebra15Functions/s0
Let f (x) =
q(0.01) =
3x2 + 3x + 3
1 2a.pg
and let q(h) =
to
28.(1 pt) setAlgebra15Functions/s0 1 √
11a.pg
The domain of the function f (x) = 2x − 32 is all real numbers
in the interval [A, ∞) where A equals
29.(1 pt) setAlgebra15Functions/s0 1 18.pg
√
The domain of the function f (x) = −x2 + 10x − 9 consists of
one or more of the following intervals: (−∞, A], [A, B] and [B, ∞)
where A < B.
Find A
Find B
For each interval, answer YES or NO to whether the interval is
included in the solution.
f (3 + h) − f (3)
. Then
h
22.(1 pt) setAlgebra15Functions/srw2 1 25.pg
Given the function f (x) = 5+x2 , calculate the following values:
2
(−∞, A]
[A, B]
[B, ∞)
Greatest Value=
38.(1 pt) setAlgebra15Functions/ur
Find the domain of the function f (x) =
30.(1 pt) setAlgebra15Functions/s0 1 18a.pg
√
The domain of the function f (x) = 10 + 3x − x2 is the closed
interval [A, B]
where A =
and B =
31.(1 pt) setAlgebra15Functions/s0 1 18a-sol.pg
√
The domain of the function f (x) = 10 + 3x − x2 is the closed
interval [A, B] where A equals
and where B equals
greatest value of x in the domain?
Greatest Value=
40.(1 pt) setAlgebra15Functions/ur fn 1 7.pg
1
For x <
, the function f (x) = |11x − 1| + 1 is equivalent to
11
the function g(x) = mx + b for:
m=
and
b=
Now for fun, try graphing f (x) . . .
33.(1 pt) setAlgebra15Functions/p2.pg
The domain of the function
r
3x
x2 − 4
is
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
41.(1 pt) setAlgebra15Functions/p4.pg
Find domain and range of the function
5x2 − 3
Domain:
Range:
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
34.(1 pt) setAlgebra15Functions/p3.pg
The domain of the function
p
x(x − 2)
is
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
42.(1 pt) setAlgebra15Functions/p5.pg
Find domain and range of the function
13x2 + 13
Domain:
Range:
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
35.(1 pt) setAlgebra15Functions/ur fn 1 1.pg
√
Find the domain of the function f (x) = x3 − 81x. What is the
least value of x in the domain?
Least Value=
36.(1 pt) setAlgebra15Functions/ur fn 1 2.pg
1
Find the domain of the function f (x) =
. What is the only
8x + 3
value of x not in the domain?
Only Value=
fn 1 3.pg
r
Find the domain of the function f (x) =
is the greatest value of x not in the domain?
1
x2 + 7x − 30
11 − 2x
. What is the
4 + 2x
39.(1 pt) setAlgebra15Functions/ur fn 1 6.pg
Define a(function f (x) by:
12 − 6x, if x ≥ 6
f (x) =
36 − x2, if x < 6
f (7) =
f (−2) =
Looking only at values of x to the left of 6, what would you
expect f (6) to be?
Looking only at values of x to the right of 6, what would you
expect f (6) to be?
Now for fun, try graphing f (x) . . .
32.(1 pt) setAlgebra15Functions/p1.pg
The domain of the function
1
√
20x + 7
is
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
37.(1 pt) setAlgebra15Functions/ur
fn 1 4.pg
r
43.(1 pt) setAlgebra15Functions/p6.pg
Find domain and range of the function
√
x + 10
Domain:
Range:
Write the answer in interval notation.
. What
3
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
is
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
44.(1 pt) setAlgebra15Functions/p7.pg
The domain of the function
x + 13
x2 − 400
is
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
Domain:
Range:
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
55.pg
52.(1 pt) setAlgebra15Functions/s0 1 77-82.pg
For each of the following functions, decide whether it is even,
odd, or neither. Enter E for an EVEN function, O for an ODD
function and N for a function which is NEITHER even nor odd.
Note: You will only have four attempts to get this problem
right!
1. f (x) = x2 + 3x8 + 2x9
2. f (x) = x2 − 6x8 + 3x2
3. f (x) = −5x2 − 3x8 − 2
4. f (x) = x3 + x3 + x9
(x + 4)(x − 9)
Domain:
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
1 44.pg
11x2 + 10
is
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
1 45.pg
20x + 15,−20 ≤ x ≤ 12
53.(1 pt) setAlgebra15Functions/srw2
The function f (x) = x−2 is
ther).
2 51.pg
54.(1 pt) setAlgebra15Functions/srw2
The function f (x) = x2 + x is
ther).
2 53.pg
55.(1 pt) setAlgebra15Functions/srw2
The function f (x) = x3 − x is
ther).
2 55.pg
(enter even, odd, or nei-
(enter even, odd, or nei-
(enter even, odd, or nei-
56.(1 pt) setAlgebra15Functions/srw2 8 9.pg
The domain of the function h(x) = (x + 6)2(2x − 4)1/4 is
.
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
is
.
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
49.(1 pt) setAlgebra15Functions/srw2 1
The domain of the function
x+4
2
x − 121
51.(1 pt) setAlgebra15Functions/srw2 1
The domain of the function
√
2x − 48
.
is
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
46.(1 pt) setAlgebra15Functions/p9.pg
Find the domain of the function
48.(1 pt) setAlgebra15Functions/srw2
The domain of the function
53.pg
.
is
Write the answer in interval notation.
Note: If the answer includes more than one interval write the
intervals separated by the union symbol, U. If needed enter −∞
as - infinity and ∞ as infinity .
45.(1 pt) setAlgebra15Functions/p8.pg
Find domain and range of the function
√
15 x − 7
47.(1 pt) setAlgebra15Functions/srw2
The domain of the function
50.(1 pt) setAlgebra15Functions/srw2 1
The domain of the function
√
3
t − 29
49.pg
57.(1 pt) setAlgebra15Functions/srw2 2 41 51.pg
Enter Yes or No in each answer space below to indicate whether
the corresponding equation defines y as a function of x.
4
Note:
right.
1.
2.
3.
4.
Be careful, You only have TWO chances to get them
48450. It costs the company $ 1.65 to make each thing-a-mabob and the company charges $ 4.56 for each thing-a-ma-bob.
Let x represent the number of thing-a-ma-bobs made.
Write the cost function for this company.
C(x) =
x2 + 4y = 10
x + 10 = y2
10x = y2
1 + x = y3
Write the revenue function for this company.
58.(1 pt) setAlgebra15Functions/beth2.pg
List all real values of x such that f (x) = 0. If there are no such
real x, type DNE in the answer blank. If there is more that one
real x, give a comma separated list (e.g. 1,2).
f (x) =
x=
Write the profit function for this company. P(x) = R(x) −C(x)
=
What is the minimum number of thing-a-ma-bobs that the
company must pruduce and sell to make a profit? answer =
19x2 + 171x − 0
13x2 + 286x + 1365
65.(1 pt) setAlgebra15Functions/8.pg
0n a remote tropical island, the average life expectancy of
women is given by the model
p
y = 6000 + 67x − 2.2x2 + .04x3,
59.(1 pt) setAlgebra15Functions/beth3.pg
List all real values of x such that f (x) = 0. If there are no such
real x, type DNE in the answer blank. If there is more that one
real x, give a comma separated list (e.g. 1,2).
f (x) =
x=
where y is the ”average life expectancy for a woman” since
1980. Thus, x is the number of years 1980. What is the predicted average life expectancy of a woman in the year 2007?
Average Life Expectancy =
−16
8
+
x − 20 x + 10
66.(1 pt) setAlgebra15Functions/box.pg
An open box is to be made from a flat piece of material 15 inches
long and 3 inches wide by cutting equal squares of length xfrom
the corners and folding up the sides.
Write the volume V of the box as a function of x. Leave it as a
product of factors, do not multiply out the factors.
V=
If we write the domain of the box as an open interval in the form
(a,b), then what is a =?
a=
and what is b =?
b=
60.(1 pt) setAlgebra15Functions/beth4.pg
List all real values of x such that f (x) = 0. If there are no such
real x, type DNE in the answer blank. If there is more that one
real x, give a comma separated list (e.g. 1,2).
x=
f (x) = −20 +
−3
x+5
61.(1 pt) setAlgebra15Functions/beth5.pg
List all real values of x such that f (x) = 0. If there are no such
real x, type DNE in the answer blank. If there is more that one
real x, give a comma separated list (e.g. 1,2).
f (x) =
x=
4x − 18
3
67.(1 pt) setAlgebra15Functions/jay4.pg
An open box is to be made from a flat square piece of material
10 inches in length and width by cutting equal squares of length
x from the corners and folding up the sides.
Write the volume V of the box as a function of x. Leave it as a
product of factors; you do not have to multiply out the factors.
V=
If we write the domain of the box as an open interval in the form
(a, b), then what is a?
a=
and what is b?
b=
62.(1 pt) setAlgebra15Functions/beth6.pg
List all real values of x such that f (x) = 0. If there are no such
real x, type DNE in the answer blank. If there is more that one
real x, give a comma separated list (e.g. 1,2).
x=
R(x) =
f (x) = −19x + 15
63.(1 pt) setAlgebra15Functions/beth7.pg
List all real values of x such that f (x) = 0. If there are no such
real x, type none in the answer blank. If there is more that one
real x, give a comma separated list (e.g. 1,2).
68.(1 pt) setAlgebra15Functions/jay5.pg
A company produces very unusual CD’s for which the variable
cost is $12 per CD and the fixed costs are $30000. They will sell
the CD’s for $62 each. Let x be the number of CD’s produced.
Write the total cost C as a function of the number of CD’s produced.
C =$
f (x) = 1x2 + 11x + 3
x=
64.(1 pt) setAlgebra15Functions/beth8.pg
A company that makes thing-a-ma-bobs has a start up cost of $
5
Write the total revenue R as a function of the number of CD’s
produced.
R =$
Write the total profit P as a function of the number of CD’s produced.
P =$
Find the number of CD’s which must be produced to break even.
The number of CD’s which must be produced to break even is
h(p) =
70.(1 pt) setAlgebra15Functions/ns1 2 11.pg
At the surface of the ocean, the water pressure is the same as
the air pressure above the water, about 15 lb/in2 , Below the
surface the water pressure increases by about 3.64 lb/in2 for
every 10 ft of descent.
Write a function f (x) which expresses the water pressure in
pounds per square inch as a function of the depth in inches below the ocean surface.
f (x) =
At what depth is the pressure 80 lb/in2 ? Include the units in
your answer:
69.(1 pt) setAlgebra15Functions/ur fn 20.pg
The altitude of a right triangle is 5 cm. Let h be the length of the
hypotenuse and let p be the perimeter of the triangle. Express h
as a function of p.
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
6
ARNOLD PIZER
rochester problib from CVS June 25, 2004
Rochester WeBWorK Problem Library
1.(1 pt) setAlgebra16FunctionGraphs/sw4 2
For the function h(x) given in the graph
WeBWorK assignment Algebra16FunctionGraphs due 1/16/10 at 2:00 AM
1.pg
2.
its domain is
;
its range is
;
Write the answer in interval notation.
and then enter the corresponding function value in each answer
space below:
1.
2.
3.
4.
h(0)
h(−2)
h(2)
h(−3)
3.
2.(1 pt) setAlgebra16FunctionGraphs/c4s2p5 7/c4s2p5 7.pg
Enter Yes or No in each answer space below to indicate whether
the corresponding curve defines y as a function of x.
4.
1.
1
4. 3
A
B
C
D
5.(1 pt) setAlgebra16FunctionGraphs/sw4 2 41 51.pg
Enter Yes or No in each answer space below to indicate whether
the corresponding equation defines y as a function of x.
Note: Be careful, You only have TWO chances to get them
right.
5.
1.
2.
3.
4.
8 + x = y3
x2 + 4y = 7
x2 y + y = 6
2|x| + y = 7
6.(1 pt) setAlgebra16FunctionGraphs/c4s2p59 72/c4s2p59 72.pg
Match the functons with their graphs. Enter the letter of the
graph below which corresponds to the function.
1. Piecewise fucntion: f (x) = x if x ≤ 0, and f (x) = x + 1
if x > 0
2. Piecewise fucntion: f (x) = 1 − x if x < −2, and f (x) =
4 if x ≥ −2
3. Piecewise fucntion: f (x) = 2x+3 if x < −1, and f (x) =
3 − x if x ≥ −1
4. Piecewise fucntion: f (x) = 1 if x ≤ 1, and f (x) = x + 1
if x > 1
6.
3.(1 pt) setAlgebra16FunctionGraphs/sw4 2 7.pg
Consider the function given in the following graph.
A.
B.
C.
D.
7.(1 pt) setAlgebra16FunctionGraphs/c0s1p1/c0s1p1.pg
The simplest functions are the linear (or affine) functions — the
functions whose graphs are a straight line. They are important
because many functions (the so-called differentiable functions)
“locally” look like straight lines. (“locally” means that if we
zoom in and look at the function at very powerful magnification
it will look like a straight line.)
Enter the letter of the graph of the function which corresponds to each statement.
1. The graph of the line is increasing
2. The graph of the line is decreasing
3. The graph of the line is constant
4. The graph of the line is not the graph of a function
What is its domain?
What is its range?
Note: Write the answer in interval notation.
4.(1 pt) setAlgebra16FunctionGraphs/c4s2p19 40/c4s2p19 40.pg
Match the functons with their graphs. Enter the letter of the
graph below which corresponds to the function. (Click on image for a larger view.)
1. |2x|
2. x2 − 4
3. −x2
2
A
B
C
8.(1 pt) setAlgebra16FunctionGraphs/lh2-3
D
30a.pg
Find the intervals over which the function is increasing or decreasing. If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. You may use ”infinity” for ∞ and ”-infinity” for −∞. For example, you may write
(-infinity, 5] for the interval (−∞, 5] and (-infinity, 5]U(7,9) for
(−∞, 5] ∪ (7, 9).
The interval over which the function is increasing:
Consider the function whose graph is sketched:
The interval over which the function is decreasing:
10.(1 pt) setAlgebra16FunctionGraphs/c0s1p2/c0s1p2.pg
Enter the letter of the graph of the function which corresponds
to each statement.
1. The graph of the line is not the graph of a function
2. The graph of the line is decreasing
3. The graph of the line is constant
4. The graph of the line is increasing
Find the intervals over which the function is increasing or decreasing. If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. You may use ”infinity” for ∞ and ”-infinity” for −∞. For example, you may write
(-infinity, 5] for the interval (−∞, 5] and (-infinity, 5]U(7,9) for
(−∞, 5] ∪ (7, 9).
The interval over which the function is increasing:
A
C
D
11.(1 pt) setAlgebra16FunctionGraphs/c0s1p3/c0s1p3.pg
11.(1 pt) setAlgebra16FunctionGraphs/c0s1p3/c0s1p3.pg
Almost any kind of quantitative data can be represented by a
graph and most of these graphs represent functions. This is why
functions and graphs are the objects analyzed by calculus. The
next two problems illustrate data which can be represented by
a graph. Match the following descriptions with their graphs below:
1. The graph of the velocity of a car as it drives along a
city street vs. time.
2. The graph of the distance traveled by a car as it enters a
superhighway vs. time.
The interval over which the function is decreasing:
9.(1 pt) setAlgebra16FunctionGraphs/lh2-3
B
48a.pg
Consider the function whose graph is sketched:
3
3. The graph of the distance traveled by a car as it drives
along a city street vs. time.
4. The graph of the velocity of a car entering a superhighway vs. time.
A
B
C
B.
C.
D.
E.
The firm registers a loss on this interval.
The profit of the firm increases on this interval.
The profit of the firm decreases on this interval.
Assuming the profits are reinvested in the firm the networth of the company is increasing on this interval.
F. Assuming the profits are reinvested in the firm the networth of the company is decreasing on this interval.
D
12.(1 pt) setAlgebra16FunctionGraphs/c0s1p4/c0s1p4.pg
Match the following descriptions with their graphs below:
1. The graph of the number of days until next Friday vs.
time.
2. The graph of the amount of time until midnight next
Friday as a function of time.
3. The graph of the number of days to the nearest Friday
(in the future or in the past) as a function of time.
4. The graph of the amount of time to the nearest Friday
at midnight vs. time.
A
B
C
14.(1 pt) setAlgebra16FunctionGraphs/c0s1p8/c0s1p8.pg
D
13.(1 pt) setAlgebra16FunctionGraphs/c0s1p7/c0s1p7.pg
The function above represents the velocity of a race car as it
travels a linear track. Negative velocities mean the car is backing up.
For each interval, enter all letters whose corresponding statements are true for that interval.
1.
2.
3.
4.
5.
The interval from a to b
The interval from b to c
The interval from c to d
The interval from d to e
The interval from e to f
A. The car is moving forward on this interval
B. The car is backing up on this interval.
C. The forward velocity of the car is increasing on this interval.
D. The forward velocity of the car is decreasing on this
interval.
E. The distance from the starting point is increasing on this
interval.
F. The distance from the starting point is decreasing on
this interval.
The following questions concern the profits of firm N. The graph
of the profits vs. time is given above. For each of the intervals enter the letters corresponding to the descriptions which
describe the behavior of the graph on that interval. (The letters in each answer must be in alphabetical order with no spaces
between the letters.)
1. The interval from a to b
2. The interval from b to c
3. The interval from c to d
4. The interval from d to e
5. The interval from e to f
A. The firm makes a profit on this interval.
4
15.(1 pt) setAlgebra16FunctionGraphs/c0s1p8a/c0s1p8a.pg
A 5 gram weight is suspended from a string next to a ruler held
vertically. The string is jiggled up and down and the graph of
the POSITION of the weight vs. time in seconds is given above.
The ruler is calibrated in inches and 0 is in the center of the ruler.
Enter the letters for the intervals which correspond to the statements below.
1. The interval from a to b
2. The interval from b to c
3. The interval from c to d
4. The interval from d to e
5. The interval from e to f
A. The weight is moving upward on this interval.
B. The weight is moving downward on this interval.
C. The upward velocity of the weight is increasing on this
interval.
D. The upward velocity of the weight is decreasing on this
interval.
E. The (signed) distance from the starting point is increasing on this interval.
F. The (signed) distance from the starting point is decreasing on this interval.
The function above represents the displacement of a toy race
car as it travels a linear track. Negative numbers mean the car
is behind the starting line, positive numbers mean it is in front.
Postive velocities mean it is moving forward, while negative velocities mean it is moving backwards.
Remember that a value which changes from -2 to -1 to 0 is increasing!
For each interval, enter all letters whose corresponding statements are true for that interval.
1.
2.
3.
4.
5.
17.(1 pt) setAlgebra16FunctionGraphs/c0s1p12/c0s1p12.pg
The interval from a to b
The interval from b to c
The interval from c to d
The interval from d to e
The interval from e to f
A.
B.
C.
D.
E.
The car is in front of the starting line on this interval
The car is behind the starting line on this inteval.
The velocity of the car is positive on this interval.
The velocity of the car is negative on this interval.
The displacement of the car from the starting line is increasing on this interval.
F. The displacement of the car from the starting line is decreasing on this interval.
The graph indicates the RATE of absorbtion of carbon dioxide
into a body of water. The rate varies with time. Positive quantities mean that the carbon dioxide is being abosrbed into solution, while negative quantities mean the carbon dioxide is being
released to the air.
For each interval, enter all letters whose corresponding statements are true for that interval.)
1. The interval from a to b
2. The interval from b to c
3. The interval from c to d
4. The interval from d to e
5. The interval from e to f
A. Carbon dioxide is being absorbed by the water on this
interval.
B. Carbon dioxide is being released from the water on this
interval.
C. The rate at which the carbon dioxide is being absorbed
is increasing on this interval.
D. The rate at which the carbon dioxide is being absorbed
is decreasing on this interval.
E. The total amount of carbon dioxide in the water is increasing on this interval.
16.(1 pt) setAlgebra16FunctionGraphs/c0s1p11/c0s1p11.pg
5
F. The total amount of carbon dioxide in the water is decreasing on this interval.
The graph shown is the graph of the SLOPE of the tangent line
of the original function. (This slope is also called the derivative
of f.)
For each interval, enter all letters whose corresponding statements are true for that interval.
18.(1 pt) setAlgebra16FunctionGraphs/c0s1p13/c0s1p13.pg
1.
2.
3.
4.
5.
A.
B.
C.
D.
E.
F.
G.
Answer the questions about the function whose graph is shown
above.
Enter the letters for the intervals which correspond to the statements below. The letters for each entry should be in alphabetical
order with no spaces.
1.
2.
3.
4.
5.
H.
The interval from a to b
The interval from b to c
The interval from c to d
The interval from d to e
The interval from e to f
The interval from a to b
The interval from b to c
The interval from c to d
The interval from d to e
The interval from e to f
The slope of the original function is positive on this interval
The slope of the original function is negative on this
interval.
The slope of the original function is increasing on this
interval.
The slope of the original function is decreasing on this
interval.
The original function is increasing on this interval.
The original function is decreasing on this interval.
The shape of the original function is concave up on this
interval.
The shape of the original function is concave down on
this interval.
20.(1 pt) setAlgebra16FunctionGraphs/c0s5p3.pg
Determine which of the following statements are true and which
are false. Enter the T or F in front of each statement.
Remember that x ∈ (−1, 1) is the same as −1 < x < 1
and x ∈ [−1, 1] means −1 ≤ x ≤ 1.
1. The function sin(x) on the domain x ∈ (−π/2, π/2) has
at least one input which produces a largest output value.
2. The function f (x) = x2 with domain x ∈ [−3, 3] has at
least one input which produces a largest output value.
3. The function f (x) = x2 with domain x ∈ [−3, 3] has at
least one input which produces a smallest output value.
4. The function f (x) = x2 with domain x ∈ (−3, 3) has at
least one input which produces a smallest output value.
5. The function sin(x) on the domain x ∈ [−π/2, π/2] has
at least one input which produces a largest output value.
A.
B.
C.
D.
E.
The function is increasing on this interval.
The function is decreasing on this interval.
The slope of the function is increasing on this interval.
The slope of the function is decreasing on this interval.
The total (signed) area between the graph of the function and the x axis is increasing on this interval.
F. The total (signed) area between the graph of the function and the x axis is decreasing on this interval.
19.(1 pt) setAlgebra16FunctionGraphs/c0s1p14/c0s1p14.pg
21.(1 pt) setAlgebra16FunctionGraphs/c0s5p4.pg
Determine which of the following statements are true and which
are false. Enter the T or F in front of each statement.
Remember that x ∈ (−1, 1) is the same as −1 < x < 1
and x ∈ [−1, 1] means −1 ≤ x ≤ 1.
1. The function sin(x) on the domain x ∈ [−π, π] has at
least one input which produces a smallest output value.
2. The function f (x) = x3 with domain x ∈ (−3, 3) has at
least one input which produces a largest output value.
3. The function sin(x) on the domain x ∈ (−π, π) has at
least one input which produces a smallest output value.
4. The function sin(x) on the domain x ∈ (−π, π) has at
least one input which produces a largest output value.
6
5. The function sin(x) on the domain x ∈ [−π, π] has at
least one input which produces a largest output value.
22.(1 pt) setAlgebra16FunctionGraphs/ns1
1 45.pg
1.
Write the equation describing the graph above:
for x in the interval [
to
]
for x in the interval [
to
]
f (x) =
23.(1 pt) setAlgebra16FunctionGraphs/ns1
1 2.pg
2.
Given the graphs of f (in blue) and g (in red) to the left answer
these questions:
3.
1. What is the value of f at -4?
2. For what values of x is f (x) = g(x): Separate answers
by spaces (e.g “ 5 7”)
3. Estimate the solution of the equation g(x) = −4
4. On what interval is the function f decreasing? (Separate answers by a space: e.g. “-2 4”)
24.(1 pt) setAlgebra16FunctionGraphs/sc c1s3p2.pg
Use a graphing calculator to find the largest value of x which
satisfies x4 − 4.000x − 3.000 = 5.000x3 + 3.000x2 . Give the answer to 2 decimal places.
Remember to calculate the trig functions in radian mode.
4.
25.(1 pt) setAlgebra16FunctionGraphs/c2s2p5 7/c2s2p5 7.pg
Enter Yes or No in each answer space below to indicate whether
the corresponding curve defines y as a function of x.
7
A
B
C
D
28.(1 pt) setAlgebra16FunctionGraphs/jj1.pg
Consider the function shown in the following graph.
5.
Where is the function decreasing?
6.
Note: use interval notation to enter your answer.
26.(1 pt) setAlgebra16FunctionGraphs/c2s2p19 40/c2s2p19 40.pg
Match the functons with their graphs. Enter the letter of the
graph below which corresponds to the function. (Click on image for a larger view )
29.(1 pt) setAlgebra16FunctionGraphs/2.pg
Enter Yes or No in each answer space below to indicate whether
the corresponding equation defines y as a function of x.
Note: Be careful, You only have TWO chances to get them
right.
1. x2 y + y = 8
2. 2|x| + y = 4
3. 5x = y2
4. x2 + 4y = 8
1. 2x + 3
2. 1x
3. |2x|
4. |x| + x + 1
A
B
C
30.(1 pt) setAlgebra16FunctionGraphs/3.pg
Enter Yes or No in each answer space below to indicate whether
the corresponding equation defines y as a function of x.
Note: Be careful, You only have TWO chances to get them
right.
1. 2|x| + y = 1
2. x + 5 = y2
3. x2 + 4y = 9
4. x2 y + y = 2
D
27.(1 pt) setAlgebra16FunctionGraphs/c2s2p59 72/c2s2p59 72.pg
Match the functons with their graphs. Enter the letter of the
graph below which corresponds to the function. (Click on image for a larger view )
1. Piecewise fucntion : f (x) = x, if x ≤ 0 and f (x) = x +
1, if x > 0
2. Piecewise fucntion : f (x) = 2x + 3, if x < −1 and f (x) =
3 − x, if x ≥ −1
3. Piecewise fucntion : f (x) = 3, if x < 2 andf (x) = x −
1, if x ≥ 2
4. Piecewise fucntion : f (x) = 1 − x, if x < −2 andf (x) =
4, if x ≥ −2
31.(1 pt) setAlgebra16FunctionGraphs/4.pg
Enter Yes or No in each answer space below to indicate whether
the corresponding equation defines y as a function of x.
Note: Be careful, You only have TWO chances to get them
right.
1. 7 + x = y3
2. x + 5 = y2
8
3. x2 y + y = 5
4. x2 + 5y = 6
3. 2x = y2
4. 8 + x = y3
34.(1 pt) setAlgebra16FunctionGraphs/c1s1p2.pg
Let p(x) = 1.5x2 . Use a calculator or a graphing program to find
the slope of the tangent line to the point (x, p(x)) when x = 2.2.
Give the answer to 3 places.
32.(1 pt) setAlgebra16FunctionGraphs/5.pg
Enter Yes or No in each answer space below to indicate whether
the corresponding equation defines y as a function of x.
Note: Be careful, You only have TWO chances to get them
right.
1. x2 y + y = 5
2. x2 + 2y = 10
3. 4x = y2
4. 9 + x = y3
35.(1 pt) setAlgebra16FunctionGraphs/sc c1s3p1.pg
Use a graphing calculator to find the positive value of x
which satisfies x = 4.5 cos(x). Give the answer to 2 decimal
places.
Remember to calculate the trig functions in radian mode.
If you don’t have a graphing calculator you can use the program Xfunctions which is installed on most of the Macintoshes
in CLARC (except for the ones in the Mac classrooms). The
program is free and you can download it for your own computer
– see Mac Software – if you have a Mac. If you have a PC
try the CD that came with the textbook – see if that will graph
equations for you.
33.(1 pt) setAlgebra16FunctionGraphs/6.pg
Enter Yes or No in each answer space below to indicate whether
the corresponding equation defines y as a function of x.
Note: Be careful, You only have TWO chances to get them
right.
1. 2|x| + y = 7
2. x2 + 3y = 10
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
9
ARNOLD PIZER
rochester problib from CVS June 25, 2004
Rochester WeBWorK Problem Library
WeBWorK assignment Algebra17FunComposition due 1/17/10 at 2:00 AM
1.(1 pt) setAlgebra17FunComposition/ur fn 2 2.pg
This problem tests calculating new functions from old ones.
From the table below calculate the quantities asked for:
9.(1 pt) setAlgebra17FunComposition/sw4 7 1.pg
Given that f (x) = x2 − 15x and g(x) = x + 9, find
(a) f + g=
and its domain is (
,
)
(b) f − g=
and its domain is (
,
)
(c) f g=
and its domain is (
,
)
(d) f /g=
and its domain is x 6=
Note: If needed enter ∞ as infinity and −∞ as -infinity .
x
−14
2
4
−1 −2
f (x) −2940
4
48
−2 −12
g(x) 5698 −14 −116 4
22
( f + g)(−1) =
f ( f (−1)) =
( f g)(2) =
2.(1 pt) setAlgebra17FunComposition/s0 1
Let f (x) = 2x + 3 and g(x) = 4x2 + 2x.
( f + g)(3) =
83.pg
3.(1 pt) setAlgebra17FunComposition/s0 1
Let f (x) = 5x + 4 and g(x) = 3x2 + 5x.
After simplifying,
( f + g)(x) =
83a.pg
4.(1 pt) setAlgebra17FunComposition/s0 1
Let f (x) = 5x + 5 and g(x) = 4x2 + 4x.
( f g)(3) =
84.pg
5.(1 pt) setAlgebra17FunComposition/s0 1
Let f (x) = 4x + 2 and g(x) = 5x2 + 4x.
After simplifying,
( f g)(x) =
84a.pg
10.(1 pt) setAlgebra17FunComposition/sw4
√
√ 7 3.pg
Given that f (x) = 2 + x and g(x) = 2 − x,
(a) the domain of f + g is [
,
]
(b) the domain of f − g is [
,
]
(c) the domain of f g is [
,
]
(d) one of the intervals (1) [−2, 2], (2) [−2, 2), (3) (−2, 2], or (4)
(−2, 2) is the domain of f /g.
(Input 1, 2, 3, or 4)
Which one is the answer?
11.(1 pt) setAlgebra17FunComposition/beth1.pg
Given that f (x) = x2 − 7x and g(x) = x + 5, find
(a) f + g=
(b) f − g=
(c) f g=
(d) f /g=
12.(1 pt) setAlgebra17FunComposition/beth2algfun.pg
For the function f (x) and g(x) given in the graph
6.(1 pt) setAlgebra17FunComposition/c0s1p9.pg
This problem gives you some practice identifying how more
complicated functions can be built from simpler functions.
Let f (x) = x3 − 27and let g(x) = x − 3. Match the functions
defined below with the letters labeling their equivalent expressions.
1. g(x) f (x)
2. (g(x))2
3. f (x)/g(x)
4. f (x2 )
A. 9 − 6x + x2
B. −27 + x6
C. 81 − 27x − 3x3 + x4
D. 9 + 3x + x2
7.(1 pt) setAlgebra17FunComposition/ns1 2 29.pg
Let f (x) = x3 + 9x2 and g(x) = 5x2 − 5.
f /g is undefined at two points A and B where A < B.
A is
,
and B is
8.(1 pt) setAlgebra17FunComposition/ns1 2 29-sol.pg
Let f (x) = x3 + 6x2 and g(x) = 9x2 − 11.
f /g is undefined at two points A and B where A < B.
A is
,
and B is
find the corresponding function values. If there is no function
value, type DNE in the answer blank.
( f + g)(−1) =
( f − g)(1)) =
13.(1 pt) setAlgebra17FunComposition/p1.pg
11
12
and g(x) =
, find
Given that f (x) =
x − 10
x+6
(a) f +g=
and its domain is
(b) f −g=
and its domain is
(c) f g=
and its domain is
1
(d) f /g=
and its domain is
Note: If the answer includes more than one interval write the
intervals separated by the ”union” symbol, U. If needed enter ∞
as infinity and −∞ as -infinity .
14.(1 pt) setAlgebra17FunComposition/srw2 8 1.pg
Given that f (x) = x2 − 8x and g(x) = x + 14, find
(a) f + g=
and its domain is (
,
)
(b) f − g=
and its domain is (
,
)
(c) f g=
and its domain is (
,
)
(d) f /g=
and its domain is x 6=
Note: If needed enter ∞ as infinity and −∞ as -infinity .
15.(1 pt) setAlgebra17FunComposition/srw2
√
√ 8 3.pg
Given that f (x) = 15 + x and g(x) = 15 − x,
(a) the domain of f + g is [
,
]
,
]
(b) the domain of f − g is [
(c) the domain of f g is [
,
]
(d) one of the intervals (1) [−15, 15], (2) [−15, 15), (3)
(−15, 15], or (4) (−15, 15) is the domain of f /g.
Which one is the answer?
(Input 1, 2, 3, or 4)
find the corresponding function values.
f (g(2)) =
f (g(−2)) =
18.(1 pt) setAlgebra17FunComposition/s0
Let f (x) = 3x + 3 and g(x) = 2x2 + 5x.
( f ◦ g)(3) =
16.(1 pt) setAlgebra17FunComposition/ur fn 2 1.pg
Let f be the linear function (in blue) and let g be the parabolic
function (in red) below.
19.(1 pt) setAlgebra17FunComposition/s0
1 85.pg
1 85a.pg
Let f (x) = 3x + 4 and g(x) = 3x2 + 3x.
After simplifying,
( f ◦ g)(x) =
20.(1 pt) setAlgebra17FunComposition/s0 1 87.pg
Let f (x) = 5x + 3 and g(x) = 3x2 + 3x. Match the statements
defined below with the letters labeling their equivalent expressions.
You must get all of the answers correct to receive credit.
1. g ◦ g
2. f ◦ g
3. g ◦ f
4. f ◦ f
A. 75x2 + 105x + 36
B. 15x2 + 15x + 3
C. 25x + 18
D. 27x4 + 54x3 + 36x2 + 9x
21.(1 pt) setAlgebra17FunComposition/ns1 2 37.pg
1
Let f (x) = , g(x) = 8x2 + 10, and h(x) = 3x3 .
5x
Then f ◦ g ◦ h(6) =
Note: If the answer does not exist, enter ’DNE’:
1. ( f ◦ g)(2) =
2. (g ◦ f )(2) =
3. ( f ◦ f )(2) =
4. (g ◦ g)(2) =
5. ( f + g)(4) =
6. ( f /g)(2) =
22.(1 pt) setAlgebra17FunComposition/sw4 7 17.pg
Given that f (x) = 6x + 5 and g(x) = 7 − x2, calculate
(a) f (g(0))=
(b) g( f (0))=
17.(1 pt) setAlgebra17FunComposition/srw2 8 23.pg
For the function f (x) and g(x) given in the graph
2
23.(1 pt) setAlgebra17FunComposition/sw4 7 19.pg
Given that f (x) = 8x − 2 and g(x) = 8 − x2, calculate
(a) f ◦ g(−2)=
(b) g ◦ f (−2)=
(c) f ◦ f (x)=
(d) g ◦ g(x)=
24.(1 pt) setAlgebra17FunComposition/sw4 7 21.pg
Given that f (x) = 2x − 2 and g(x) = 6 − x2 , calculate
(a) f ◦ g(x)=
(b) g ◦ f (x)=
32.(1 pt) setAlgebra17FunComposition/faris1.pg
The number of bacteria in a refrigerated food product is given by
N(T ) = 22T 2 −15T +1, 1 < T < 31 where T is the temperature
of the food.
When the food is removed from the refrigerator, the temperature
is given by T (t) = 5t + 1.1 , where t is the time in hours.
Find the composite function N(T (t)):
N(T (t)) =
Find the time when the bacteria count reaches 14548
Time Needed =
25.(1 pt) setAlgebra17FunComposition/sw4 7 21.pg
Given that f (x) = 4x − 9 and g(x) = 5 − x2 , calculate
(a) f ◦ g(x)=
(b) g ◦ f (x)=
26.(1 pt) setAlgebra17FunComposition/sw4 7 29.pg
Given that f (x) = 7x + 7 and g(x) = 2x − 8, calculate
(a) f ◦ g(x)=
, its domain is (
,
)
(b) g ◦ f (x)=
, its domain is (
,
)
(c) f ◦ f (x)=
, its domain is (
,
)
(d) g ◦ g(x)=
, its domain is (
,
)
Note: If needed enter ∞ as infinity and −∞ as -infinity .
33.(1 pt) setAlgebra17FunComposition/pcomp.pg
Given that f (x) = x2 + 3x and g(x) = x − 9, calculate
(a) f ◦ g(x)=
,
(b) g ◦ f (x)=
,
(c) f ◦ f (x)=
,
(d) g ◦ g(x)=
,
27.(1 pt) setAlgebra17FunComposition/sw4 7 31.pg
Given that f (x) = x2 + 5 and g(x) = x + 8, calculate
(a) f ◦ g(x)=
, its domain is (
,
)
(b) g ◦ f (x)=
, its domain is (
,
)
(c) f ◦ f (x)=
, its domain is (
,
)
(d) g ◦ g(x)=
, its domain is (
,
)
Note: If needed enter ∞ as infinity and −∞ as -infinity .
34.(1 pt) setAlgebra17FunComposition/pcomp2.pg
Given that f (x) = x2 − 8x and g(x) = x + 8, calculate
(a) f ◦ g(x)=
,
(b) g ◦ f (x)=
,
(c) f ◦ f (x)=
,
(d) g ◦ g(x)=
,
28.(1 pt) setAlgebra17FunComposition/sw4 7 33.pg
Given that f (x) = 1x and g(x) = 4x + 3, calculate
(a) f ◦ g(x)=
, its domain is all real numbers except
(b) g ◦ f (x)=
, its domain is all real numbers except
(c) f ◦ f (x)=
, its domain is all real numbers except
35.(1 pt) setAlgebra17FunComposition/srw2 8 17.pg
Given that f (x) = 7x − 1 and g(x) = 8 − x2, calculate
(a) f (g(0))=
(b) g( f (0))=
36.(1 pt) setAlgebra17FunComposition/srw2 8 19.pg
Given that f (x) = 7x + 3 and g(x) = 1 − x2, calculate
(a) f ◦ g(−2)=
(b) g ◦ f (−2)=
(d) g ◦ g(x)=
, its domain is (
,
)
Note: If needed enter ∞ as infinity and −∞ as -infinity .
37.(1 pt) setAlgebra17FunComposition/srw2 8 21.pg
Given that f (x) = 5x − 3 and g(x) = 2 − x2, calculate
(a) f ◦ g(x)=
(b) g ◦ f (x)=
29.(1 pt) setAlgebra17FunComposition/sw4 7 35.pg
Given that f (x) = |x| and g(x) = 3x − 3, calculate
(a) f ◦ g(x)=
, its domain is (
,
)
(b) g ◦ f (x)=
, its domain is (
,
)
(c) f ◦ f (x)=
, its domain is (
,
)
(d) g ◦ g(x)=
, its domain is (
,
)
Note: If needed enter ∞ as infinity and −∞ as -infinity .
38.(1 pt) setAlgebra17FunComposition/srw2 8 29.pg
Given that f (x) = 5x + 6 and g(x) = 5x + 6, calculate
(a) f ◦ g(x)=
, its domain is (
,
)
(b) g ◦ f (x)=
, its domain is (
,
)
(c) f ◦ f (x)=
, its domain is (
,
)
(d) g ◦ g(x)=
, its domain is (
,
)
Note: If needed enter ∞ as infinity and −∞ as -infinity .
30.(1 pt) setAlgebra17FunComposition/beth3algfun.pg
Given that f (x) = 3x + 1 and g(x) = 4x − 8, calculate
(a) f ◦ g(x)=
(b) g ◦ f (x)=
(c) f ◦ f (x)=
(d) g ◦ g(x)=
31.(1 pt) setAlgebra17FunComposition/beth4algfun.pg
Use abs(x) for |x|.
Given that f (x) = |x| and g(x) = 2x − 7, calculate
(a) f ◦ g(x)=
(b) g ◦ f (x)=
39.(1 pt) setAlgebra17FunComposition/srw2 8 31.pg
Given that f (x) = x2 − 4 and g(x) = x + 4, calculate
(a) f ◦ g(x)=
, its domain is (
,
)
(b) g ◦ f (x)=
, its domain is (
,
)
(c) f ◦ f (x)=
, its domain is (
,
)
(d) g ◦ g(x)=
, its domain is (
,
)
Note: If needed enter ∞ as infinity and −∞ as -infinity .
3
40.(1 pt) setAlgebra17FunComposition/srw2 8 33.pg
Given that f (x) = 1x and g(x) = 6x + 5, calculate
(a) f ◦ g(x)=
, its domain is all real numbers except
(b) g ◦ f (x)=
, its domain is all real numbers except
(c) f ◦ f (x)=
, its domain is all real numbers except
balloon is increasing at the rate of 4 cm per second. Express the
surface area of the balloon as a function of time t (in seconds).
If needed you can enter π as pi.
Your answer is
.
48.(1 pt) setAlgebra17FunComposition/ur
fn 2 4.pg
√
Let f (x) = 90 − x and g(x) = x2 − x.
Then the domain of f ◦ g is equal to [a, b] for
a=
and
b=
49.(1 pt) setAlgebra17FunComposition/ur fn 2 5.pg
1
1
Let f (x) =
and g(x) =
.
x−2
x−7
Then the domain of f ◦ g is equal to all reals except for two
values, a and b with a < b and
a=
b=
50.(1 pt) setAlgebra17FunComposition/ur fn 2 6.pg
Let f (x) = 4x + 3 and g(x) = 2x2 + 4x.
Then ( f ◦ g)(−3) =
,
( f ◦ g)(x) =
.
(d) g ◦ g(x)=
, its domain is (
,
)
Note: If needed enter ∞ as infinity and −∞ as -infinity .
41.(1 pt) setAlgebra17FunComposition/srw2 8 35.pg
Given that f (x) = |x| and g(x) = 2x − 3, calculate
(a) f ◦ g(x)=
, its domain is (
,
)
(b) g ◦ f (x)=
, its domain is (
,
)
(c) f ◦ f (x)=
, its domain is (
,
)
(d) g ◦ g(x)=
, its domain is (
,
)
Note: If needed enter ∞ as infinity and −∞ as -infinity .
42.(1 pt) setAlgebra17FunComposition/srw2
√ 8 43.pg
If f (x) = x4 + 9, g(x) = x + 7, h(x) = x, then
.
f ◦ g ◦ h(x)=
43.(1 pt) setAlgebra17FunComposition/sw4 7 45.pg
Express the function f (x) = (x − 3)2 in the form f ◦ g. If
f (x) = x2 , find the function g(x).
Your answer is g(x)=
,
51.(1 pt) setAlgebra17FunComposition/ur
1
Let f (x) =
and g(x) = 2x + 10.
x−4
,
Then ( f ◦ g)(5) =
( f ◦ g)(x) =
.
44.(1 pt) setAlgebra17FunComposition/sw4 7 48.pg
1 in the form f ◦ g. If g(x) =
Express the function f (x) = x+2
x + 2, find the function f (x).
Your answer is f (x)=
,
fn 2 7.pg
52.(1 pt) setAlgebra17FunComposition/ur
fn 2
√
Let f (x) = 6 − x2 + 1 and g(x) = x − 2.
Then ( f ◦ g)(1) =
,
( f ◦ g)(x) =
.
45.(1 pt) setAlgebra17FunComposition/srw2 8 45.pg
Express the function h(x) = (x + 3)8 in the form f ◦ g. If
f (x) = x8 , find the function g(x).
Your answer is g(x)=
,
53.(1 pt) setAlgebra17FunComposition/ur fn 2
Let f (x) = 3x − 5 and g(x) = x2 − 2x + 4.
Then ( f ◦ g)(x) =
,
(g ◦ f )(x) =
.
46.(1 pt) setAlgebra17FunComposition/srw2 8 48.pg
1
Express the function h(x) = x−9
in the form f ◦ g. If g(x) =
x − 9, find the function f (x).
,
Your answer is f (x)=
54.(1 pt) setAlgebra17FunComposition/ur
1
6
Let f (x) =
and g(x) = + 6.
x−6
x
,
Then ( f ◦ g)(x) =
(g ◦ f )(x) =
.
47.(1 pt) setAlgebra17FunComposition/srw2 8 57.pg
A spherical weather balloon is being inflated. The radius of the
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
4
8.pg
9.pg
fn 2 10.pg
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra18FunInverse due 1/18/10 at 2:00 AM
6.(1 pt) setAlgebra18FunInverse/mec7.pg
If f (x) = 6x − 2, then
f −1 (y) =
f −1 (1) =
1.(1 pt) setAlgebra18FunInverse/ur inv 1.pg
Enter T or F depending on whether the function is one-to-one or
not. (You must enter T or F – True and False will not work.)
3
1. b(x) = 1x
√ − 3x
2. e(x) = 3 x + 3
x−3
3. c(x) =
3+x
4. d(x) = (3x − 1)2 + 3
5. a(x) = 3x4 − 3x
7.(1 pt) setAlgebra18FunInverse/srw2
If f (x) = x2 , x ≥ 0,
then f −1 (5) =
8.(1 pt) setAlgebra18FunInverse/srw2 10 21.pg
Let
f (x) = 5x + 6
f −1 (x) =
2.(1 pt) setAlgebra18FunInverse/srw2 10 7-12a.pg
Enter a Y (for Yes) or an N (for No) in each answer space below
to indicate whether the corresponding function is one-to-one or
not.
You must get all of the answers correct to receive credit.
1.
2.
3.
4.
5.
6.
9.(1 pt) setAlgebra18FunInverse/mec1.pg
Let
f (x) = 10 − x
−1
f (x) =
f (x) = sin x, 0 ≤ x ≤ π
f (t) = 2t
k(x) = cosx, 0 ≤ x ≤ π
h(t) = 6t 2 + 4, t ≤ 0
h(x) = |x| + 3
k(x) = (x − 4)2 , 3 ≤ x ≤ 5
10.(1 pt) setAlgebra18FunInverse/ur fn 4 1.pg
Let f (x) = −3x + 6. Find f −1 (x).
f −1 (x) =
.
Now for fun, verify that ( f ◦ f −1 )(x) = ( f −1 ◦ f )(x) = x
11.(1 pt) setAlgebra18FunInverse/mec2.pg
Let
1
f (x) =
x + 15
f −1 (x) =
3.(1 pt) setAlgebra18FunInverse/osu fn 4 1.pg
Enter a T or an F in each answer space below to indicate whether
or not the given function has an inverse. Unless otherwise indicated, assume the domain of the function is as large as possible.
You must get all of the answers correct to receive credit.
1.
2.
3.
4.
5.
6.
12.(1 pt) setAlgebra18FunInverse/mec3.pg
Let
f (x) = 14 − x2, x ≥ 0
−1
f (x) =
16 ln(x)
2x3 − 27x2 + 108x + 11 on the interval [0, 6]
80x + 7 sin(10x)
6 sin(x) − 8 cos(5x)
2x3 − 27x2 + 108x + 11 on the interval [6, ∞)
ln(x16 )
13.(1 pt) setAlgebra18FunInverse/ur fn 4 2.pg
x
. Find f −1 (x).
Let f (x) =
x−6
f −1 (x) =
.
Now for fun, verify that ( f ◦ f −1 )(x) = ( f −1 ◦ f )(x) = x
4.(1 pt) setAlgebra18FunInverse/srw2 10 17.pg
If f is one-to-one and f (−10) = 9, then
f −1 (9) =
and ( f (−10))−1 =
.
If g is one-to-one and g(0) = 3, then
g−1 (3) =
and (g(0))−1 =
.
If h is one-to-one and h(−5) = 12, then
h−1 (12) =
and (h(−5))−1 =
.
5.(1 pt) setAlgebra18FunInverse/srw2 10 17a.pg
(a) If f is one-to-one and f (−15) = 2, then f −1 (2) =
( f (−15))−1 =
.
(b) If g is one-to-one and g(9) = 11, then g−1 (11) =
(g(9))−1 =
.
10 20.pg
14.(1 pt) setAlgebra18FunInverse/mec4.pg
Let
x+3
f (x) =
x+7
−1
f (−2) =
15.(1 pt) setAlgebra18FunInverse/ur
fn 4 3.pg
√
Let f (x) = 7 + x − 5. Find f −1 (x).
f −1 (x) =
.
Now for fun, verify that ( f ◦ f −1 )(x) = ( f −1 ◦ f )(x) = x
16.(1 pt) setAlgebra18FunInverse/mec5.pg
Let
1
f (x) = x + 7, −1 ≤ x ≤ 0
6
The domain of f −1 is the interval [A, B]
where A =
and B =
and
and
1
26.(1 pt) setAlgebra18FunInverse/sw4√
8 41.pg
Find the inverse function of f (x) = 8x + 3.
f −1 (x) =
17.(1 pt) setAlgebra18FunInverse/mec6.pg
Let
f (x) = 3 + 2x + 3ex
f −1 (6) =
27.(1 pt) setAlgebra18FunInverse/sw4 8 45.pg
√
Find the inverse function of f (x) = 2 + 3 x.
f −1 (x) =
18.(1 pt) setAlgebra18FunInverse/ur inv 2.pg
Find the inverse for each of the following functions.
f (x) = 13x + 7
f −1 (x) =
g(x) = 1x3 − 9
g−1 (x) =
13
h(x) =
x+9
=
h−1 (x) √
j(x) = 3 x + 1
j −1 (x) =
19.(1 pt) setAlgebra18FunInverse/osu
f (x) =
28.(1 pt) setAlgebra18FunInverse/sw4 8 51.pg
(a) Find the inverse function of f (x) = 3x − 3.
f −1 (x) =
(b) The graphs of f and f −1 are symmetric with respect to the
line defined by y =
fn 4 2.pg
29.(1 pt) setAlgebra18FunInverse/ur inv
Below is the graph of a function f :
2ex − 3
22ex + 15
f −1 (x) =
The domain of f −1 (x) is the open interval (a, b), where
a=
and b =
20.(1 pt) setAlgebra18FunInverse/sw4 8 17.pg
Assume that the function f is a one-to-one function.
(a) If f (8) = 6, find f −1 (6).
Your answer is
(b) If f −1 (−6) = −7, find f (−7).
Your answer is
21.(1 pt) setAlgebra18FunInverse/sw4
If f (x) = 6 − 8x, find f −1 (6).
Your answer is
8 19.pg
22.(1 pt) setAlgebra18FunInverse/sw4 8 21.pg
If f (x) = x + 2 and g(x) = x − 2,
(a) f (g(x)) =
(b) g( f (x)) =
(c) Thus g(x) is called an
function of f (x)
23.(1 pt) setAlgebra18FunInverse/sw4 8 23.pg
x+7
If f (x) = 7x − 7 and g(x) =
,
7
(a) f (g(x)) =
(b) g( f (x)) =
(c) Thus g(x) is called an
function of f (x)
24.(1 pt) setAlgebra18FunInverse/sw4 8 31.pg
Find the inverse function of f (x) = 8x + 9.
f −1 (x) =
25.(1 pt) setAlgebra18FunInverse/sw4
Find the inverse function of f (x) =
f −1 (x) =
8 37.pg
1
.
x+5
Graph A
2
3.pg
Graph B
Graph A
Graph C
The inverse of the function f is (A, B or C):
Graph B
30.(1 pt) setAlgebra18FunInverse/ur inv 6.pg
Below is the graph of a function f :
3
2.
Graph C
3.
Graph D
The inverse of the function f is (A, B, C or D):
31.(1 pt) setAlgebra18FunInverse/ur
Match each function to its inverse.
inv 7.pg
4.
1.
4
5.
C.
D.
A.
E.
B.
5
32.(1 pt) setAlgebra18FunInverse/ur
33.(1 pt) setAlgebra18FunInverse/ur
fn 4 4.pg
A function f (x) is graphed in plane A. It is clearly a 1:1 function, so it must have an inverse.
Enter the color (”red”, ”green”, or ”blue”) of this inverse function which is graphed in plane B. Use what you know about
the graphs of inverse functions rather than algebraic calculations
based on what you might guess the function to be.
Color of f −1 graph =
Important!! You only have 2 attempts to get this problem right!
fn 4 5.pg
A function f (x) is graphed in plane A. It is clearly a 1:1 function, so it must have an inverse.
Enter the color (”red”, ”green”, ”blue”, or ”yellow”) of this inverse function which is graphed in plane B. Use what you know
about the graphs of inverse functions rather than algebraic calculations based on what you might guess the function to be.
Color of f −1 graph =
Important!! You only have 2 attempts to get this problem right!
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
6
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra19FunTransforms due 1/19/10 at 2:00 AM
1.(1 pt) setAlgebra19FunTransforms/c0s2p1.pg
Relative to the graph of
y = x2
the graphs of the following equations have been changed in what
way?
1.
2.
3.
4.
y = (x2 )/5
y = 5x2
y = x2 − 11
y = (x + 11)2
A.
B.
C.
D.
compressed vertically by the factor 5
shifted 11 units left
stretched vertically by the factor 5
shifted 11 units down
Enter the letter of the graph below which corresponds to the
transformation of the function.
1.
2.
3.
4.
2.(1 pt) setAlgebra19FunTransforms/c0s2p1b.pg
Relative to the graph of
y = x2
F(x − 3)
5F(x)
F(x + 3)
F(x2 )
the graphs of the following equations have been changed in what
way?
1.
2.
3.
4.
A.
B.
C.
D.
y = (x − 9)2
y = 9x2
y = (x + 9)2
y = x2 − 9
A
shifted 9 units left
shifted 9 units down
stretched vertically by the factor 9
shifted 9 units right
B
C
D
5.(1 pt) setAlgebra19FunTransforms/scaling.pg
Let g be the function below.
3.(1 pt) setAlgebra19FunTransforms/c0s2p3.pg
Relative to the graph of
y = x2
the graphs of the following equations have been changed in what
way?
1.
2.
3.
4.
y = x2 + 2
y = (x2 )/2
y = (x − 2)2
y = 2x2
A.
B.
C.
D.
stretched vertically by the factor 2
shifted 2 units right
compressed vertically by the factor 2
shifted 2 units up
The domain of g(x) is of the form [a, b], where a is
.
The range of g(x) is of the form [c, d], where c is
.
4.(1 pt) setAlgebra19FunTransforms/c0s2p2/c0s2p2.pg
This is a graph of the function F(x):
1
and b is
and d is
7.(1 pt) setAlgebra19FunTransforms/ur
Enter the letter of the graph which corresponds to each new
function defined below:
1. g(x − 2) + 2 is
.
2. g(2x) is
.
3. 2 + g(−x) is
.
4. g(x + 2) − 2 is
.
A
B
C
D
E
F
G
H
6.(1 pt) setAlgebra19FunTransforms/ur
fn 3 2.pg
For each of the following graph transformations, give the x and
y coordinates of the point on the new graph which corresponds
to the point P = (9, 2) on the original graph.
Shift Left by 10: (
,
)
Shrink Vertically by 5: (
,
)
Flip Horizontally: (
,
)
8.(1 pt) setAlgebra19FunTransforms/ur
fn 3 1.pg
fn 3 3.pg
For each of the following graph transformations, give the x and
y coordinates of the point on the new graph which corresponds
to the point P = (4, −3) on the original graph.
,
)
Shift Down by 5 and then Flip Vertically: (
Flip Vertically and then Shift Down by 5: (
,
)
Shrink Horizontally by 6 and then Shift Right by 4: (
,
)
Shift Right by 4 and then Shrink Horizontally by 6: (
,
)
For each of the following graph transformations, give the x and
y coordinates of the point on the new graph which corresponds
to the point P = (−3, −7) on the original graph.
Shift Up by 3: (
,
)
Strech Horizontally by 2: (
,
)
Flip Vertically: (
,
)
2
9.(1 pt) setAlgebra19FunTransforms/ur
fn 3 4.pg
Important!! You only have 3 attempts to get this problem right!
For each of the following graph transformations, give the x and
y coordinates of the point on the new graph which corresponds
to the point P = (−1, 1) on the original graph.
Shift Right by 5 and then Flip Horizontally: (
,
)
Flip Horizontally and then Shift Right by 5: (
,
)
Stretch Vertically by 6 and then Shift Down by 9: (
,
)
Shift Down by 9 and then Stretch Vertically by 6: (
,
)
10.(1 pt) setAlgebra19FunTransforms/ur fn 3 5.pg
Each of the four graphs in plane B below comes from the original graph in plane A via exactly one transformation. Match each
transformation of the original graph in plane A with the color of
the graph in plane B which is the result.
1.
2.
3.
4.
Shift Down
Stretch Horizontally
Shift Right
Shrink Vertically
A.
B.
C.
D.
red
yellow
green
blue
11.(1 pt) setAlgebra19FunTransforms/ur fn 3 6.pg
Each of the four graphs in plane B comes from the original graph
in plane A via exactly one transformation. Match each transformation of the original graph in plane A with the color of the
graph in plane B which is the result.
3
Important!! You only have 3 attempts to get this problem right!
Important!! You only have 3 attempts to get this problem right!
1.
2.
3.
4.
Shift Up
Shrink Horizontally
Stretch Vertically
Flip Vertically
1.
2.
3.
4.
f (x) = −(x + 3)x2
2
f (x) = ( 2x + 3) x4
f (x) = x(x − 3)2
f (x) = (−x + 3)x2
A.
B.
C.
D.
yellow
green
blue
red
A.
B.
C.
D.
red
yellow
blue
green
12.(1 pt) setAlgebra19FunTransforms/ur fn 3 7.pg
The graph in plane A is of the function f (x) = x2 (x + 3).
Match the color of each graph in plane B with the equation that
fits it. Use the fact that each graph in B can be obtained from the
original by applying just one of the basic transformations which
we have learned.
13.(1 pt) setAlgebra19FunTransforms/ur fn 3 8.pg
The graph in plane A is of the equation x = 2y .
Match the color of each graph in plane B with the equation that
fits it. Use the fact that each graph in B can be obtained from the
original by applying just one of the basic transformations which
we have learned.
4
The graph of f (x) = x2 is sketched in red and the graph of g(x)
is sketched in blue. Use the translation rule and f (x) = x2 to
identify the function g(x);
g(x) =
Important!! You only have 3 attempts to get this problem right!
1.
2.
3.
4.
x = −2y
x = 25y
x = 2y − 2
x = 2−y
A.
B.
C.
D.
red
blue
yellow
green
14.(1 pt) setAlgebra19FunTransforms/lh2-4
16.(1 pt) setAlgebra19FunTransforms/lh2-4
9a.pg
The graph of f (x) = x2 − 4 is sketched in red and the graph of
g(x) is sketched in blue. Use the translation rule and f (x) =
x2 − 4 to identify the function g(x);
g(x) =
The graph of f (x) = x2 is sketched in red and the graph of g(x)
is sketched in blue. Use the translation rule and f (x) = x2 to
identify the function g(x);
g(x) =
15.(1 pt) setAlgebra19FunTransforms/lh2-4
9b.pg
17.(1 pt) setAlgebra19FunTransforms/lh2-4
9d.pg
5
10a.pg
The graph of f (x) = x3 is sketched in red and the graph of g(x)
is sketched in blue. Use the translation rule and f (x) = x3 to
identify the function g(x);
g(x) =
18.(1 pt) setAlgebra19FunTransforms/lh2-4
The graph of f (x) = x3 is sketched in red and the graph of g(x)
is sketched in blue. Use the translation rule and f (x) = x3 to
identify the function g(x);
g(x) =
20.(1 pt) setAlgebra19FunTransforms/lh2-4
10b.pg
The graph of f (x) = |x| is sketched in red and the graph of g(x)
is sketched in blue. Use the translation rule and f (x) = |x| to
identify the function g(x);
g(x) =
You may use abs(.) for |.|, e.g. write
abs(5) for |5|.
The graph of f (x) = x3 − 2 is sketched in red and the graph of
g(x) is sketched in blue. Use the translation rule and f (x) =
x3 − 2 to identify the function g(x);
g(x) =
19.(1 pt) setAlgebra19FunTransforms/lh2-4
11a.pg
21.(1 pt) setAlgebra19FunTransforms/lh2-4
10c.pg
6
11b.pg
√
The graph of f (x) = x is sketched in red and the graph of
g(x) is sketched in green (click on the graph to see an enlarged
√
image). Use the translation rule and f (x) = x to identify the
function g(x);
√
g(x) =
You may use sqrt(.) for ., e.g. write
√
sqrt(5) for 5.
The graph of f (x) = |x| − 7 is sketched in red and the graph of
g(x) is sketched in blue. Use the translation rule and f (x) =
|x| − 7 to identify the function g(x);
g(x) =
You may use abs(.) for |.|, e.g. write
abs(5) for |5|.
22.(1 pt) setAlgebra19FunTransforms/lh2-4
11c.pg
24.(1 pt) setAlgebra19FunTransforms/lh2-4
The graph of f (x) = |x| is sketched in red and the graph of g(x)
is sketched in blue. Use the translation rule and f (x) = |x| to
identify the function g(x);
g(x) =
You may use abs(.) for |.|, e.g. write
abs(5) for |5|.
23.(1 pt) setAlgebra19FunTransforms/lh2-4
12b.pg
√
The graph of f (x) = x + 4 is sketched in red and the graph
of g(x)√is sketched in green. Use the translation rule and
f (x) = x + 4 to identify the function g(x);
√
g(x) =
You may use sqrt(.) for ., e.g. write
√
sqrt(5) for 5.
25.(1 pt) setAlgebra19FunTransforms/lh2-4
12a.pg
7
12c.pg
√
The graph of f (x) = x is sketched in red and the graph of g(x)
√
is sketched in green. Use the translation rule and f (x) = x to
identify the function g(x);
√
g(x) =
You may use sqrt(.) for ., e.g. write
√
sqrt(5) for 5.
26.(1 pt) setAlgebra19FunTransforms/lh2-4
The graph of f (x) = x2 is sketched in black and it had undergone a series of translations to graphs of functions f 1 sketched
in green, f 2 sketched in blue, and f 3 sketched in red. f → f 1 →
f 2 → f 3 . Use the translation rule and f (x) = x2 to identify the
function f 1 (x);
f 1 (x) =
Use the translation rule and f 1 (x) to identify the function f 2 (x);
f 2 (x) =
Use the translation rule and f 2 (x) to identify the function f 3 (x);
f 3 (x) =
23.pg
28.(1 pt) setAlgebra19FunTransforms/ns1
The graph of f (x) = x2 is sketched in black and it had undergone a series of translations to graphs of functions f 1 sketched
in green, f 2 sketched in blue, and f 3 sketched in red. f → f 1 →
f 2 → f 3 . Use the translation rule and f (x) = x2 to identify the
function f 1 (x);
f 1 (x) =
Use the translation rule and f 1 (x) to identify the function f 2 (x);
f 2 (x) =
Use the translation rule and f 2 (x) to identify the function f 3 (x);
f 3 (x) =
27.(1 pt) setAlgebra19FunTransforms/lh2-4
2 3.pg
Match the functions shown in the graph above with their formulas:
1. −x3
2. x3
3. −x + 2
29.(1 pt) setAlgebra19FunTransforms/beth1algfun.pg
The graph of y = f (x) is given below:
36.pg
8
A
C
D
31.(1 pt) setAlgebra19FunTransforms/lance1.pg
The graph of y = x3 − 9x2 is given below:
On a piece of paper sketch the graph of y = f (x + 4) and determine the new coordinates of points A, B and C.
A=
B=
C=
On a piece of paper sketch the graph of y = − f (x) − 6 and determine the new coordinates of points A, B and C.
A=
B=
C=
30.(1 pt) setAlgebra19FunTransforms/SRW2
Click on image for a larger view
For the function f (x) given in the graph
B
5 11/srw2 5 11.pg
Find a formula for the transformation whose graph is given below.
Match the following functons with their graphs. Enter the letter
of the graph below which corresponds to the function.
1.
2.
3.
4.
5.
6.
y = f (2x)
y = − f (x) + 3
y = f (x − 2)
y = 2 f (x)
y = f (−x)
y = f (x) − 2
y=
32.(1 pt) setAlgebra19FunTransforms/p1.pg
The graph of y = x2 is given below:
9
E
Find a formula for each of the transformations whose graphs are
given below.
a)
y=
33.(1 pt) setAlgebra19FunTransforms/p2.pg
The graph of y = x2 is given below:
y=
b)
Find a formula for the transformation whose graph is given below.
10
y=
y=
35.(1 pt) setAlgebra19FunTransforms/p4.pg
√
The graph of y = x is given below:
34.(1 pt) setAlgebra19FunTransforms/p3.pg
The graph of y = x2 is given below:
Find a formula for each of the transformations whose graphs are
given below.
Recall that square root is entered as sqrt.
a)
Find a formula for the transformation whose graph is given below.
11
y=
b)
Find a formula for each of the transformations whose graphs are
given below.
Recall that absolute value is entered as abs.
a)
y=
y=
b)
36.(1 pt) setAlgebra19FunTransforms/p5.pg
The graph of y = |x| is given below:
12
On a piece of paper sketch the graph of y = f
mine the new coordinates of points A, B and C.
A(
,
)
B(
,
)
C(
,
)
1
x and deter4
39.(1 pt) setAlgebra19FunTransforms/p8.pg
The graph of y = f (x) is given below:
y=
37.(1 pt) setAlgebra19FunTransforms/p6.pg
The graph of y = f (x) is given below:
On a piece of paper sketch the graph of y = f (−6x) and determine the new coordinates of points A, B and C.
A(
,
)
B(
,
)
C(
,
)
40.(1 pt) setAlgebra19FunTransforms/p9.pg
The graph of y = f (x) is given below:
On a piece of paper sketch the graph of y = f (6x) and determine
the new coordinates of points A, B and C.
A(
,
)
B(
,
)
C(
,
)
38.(1 pt) setAlgebra19FunTransforms/p7.pg
The graph of y = f (x) is given below:
On a piece of paper sketch the graph of y = −3 f (3x) and determine the new coordinates of points A, B and C.
A(
,
)
B(
,
)
C(
,
)
41.(1 pt) setAlgebra19FunTransforms/ptransf1.pg
Describe a function g(x) in terms of f (x) if the graph of g is
obtained by reflecting the graph of f about the x-axis and if it
is horizontally stretched by a factor of 2 when compared to the
graph of f .
g(x) = A f (Bx) +C where
A=
13
B=
C=
42.(1 pt) setAlgebra19FunTransforms/ptransf2.pg
Describe a function g(x) in terms of f (x) if the graph of g is
obtained by shifting the graph of f to the right 4 units and upward 4 units and if it is vertically stretched by a factor of 8 when
compared to f .
g(x) = A f (x + B) +C where
A=
B=
C=
43.(1 pt) setAlgebra19FunTransforms/srw2 5 1.pg
The graph of the function y = f (x) + 63 can be obtained from
the graph of y = f (x) by one of the following actions:
(a) shifting the graph of f (x) to the right 63 units;
(b) shifting the graph of f (x) to the left 63 units;
(c) shifting the graph of f (x) upward 63 units;
(d) shifting the graph of f (x) downward 63 units;
Your answer is (input a, b, c, or d)
Your answer is (input e, f, g, or h)
47.(1 pt) setAlgebra19FunTransforms/srw2 5 9.pg
The graph of the function y = f (88x) can be obtained from the
graph of y = f (x) by one of the following actions:
(a) horizontally stretching the graph of f (x) by a factor 88;
(b) horizontally shrinking the graph of f (x) by a factor 1/88;
(c) vertically stretching the graph of f (x) by a factor 88;
(d) vertically shrinking the graph of f (x) by a factor 1/88;
Your answer is (input a, b, c, or d)
48.(1 pt) setAlgebra19FunTransforms/srw2 5 15.pg
(a) The graph of f (x) = (x + 92)2 can be obtained from shifting
the graph of f (x) = x2 to the
92 units.
(b) The graph of f (x) = x2 + 92 can be obtained from shifting
the graph of f (x) = x2
√ 92 units.
(c) The graph of f (x) = 92 x can be obtained from
√
the graph of f (x) = x vertically
by a factor 92.
√
(d) The graph of f (x) = 92x can be obtained from
√
1
the graph of f (x) = x horizontally by a factor 92
.
44.(1 pt) setAlgebra19FunTransforms/srw2 5 3.pg
The graph of the function y = 65 f (x) can be obtained from the
graph of y = f (x) by one of the following actions:
(a) horizontally stretching the graph of f (x) by a factor 65;
(b) horizontally shrinking the graph of f (x) by a factor 65;
(c) vertically stretching the graph of f (x) by a factor 65;
(d) vertically shrinking the graph of f (x) by a factor 65;
Your answer is (input a, b, c, or d)
49.(1 pt) setAlgebra19FunTransforms/srw2 5 19.pg
Given f (x) = x2 , after performing the following transformations: shift upward 40 units and shift 15 units to the right, the
new function g(x) =
50.(1 pt) setAlgebra19FunTransforms/srw2 5 23.pg
Given f (x) = |x|, after performing the following transformations: shift to the left 32 units, shrink vertically by a factor
1
, and shift downward 53 units, the new function g(x) =
of
57
45.(1 pt) setAlgebra19FunTransforms/srw2 5 5.pg
The graph of the function y = −26 f (x) can be obtained from
the graph of y = f (x) by one of the following actions:
(a) horizontally stretching the graph of f (x) by a factor 26;
(b) horizontally shrinking the graph of f (x) by a factor 26;
(c) vertically stretching the graph of f (x) by a factor 26;
(d) vertically shrinking the graph of f (x) by a factor 26;
Your answer is (input a, b, c, or d)
Then followed by one of the following actions:
(e) reflecting the resulting graph in x-axis;
(f) reflecting the resulting graph in y-axis;
Your answer is (input e or f)
Use abs(x) for |x|.
51.(1 pt) setAlgebra19FunTransforms/srw2
5 31.pg
√
The graph of √
the function y = 70 + x can be obtained from the
graph of y = x by one of the following actions:
(a) shifting the graph of f (x) downward 70 units;
(b) shifting the graph of f (x) upward 70 units;
(c) horizontally stretching the graph of f (x) by a factor 70;
(d) horizontally shrinking the graph of f (x) by a factor 1/70;
Your answer is (input a, b, c, or g)
52.(1 pt) setAlgebra19FunTransforms/srw2 5 35.pg
The graph of the function y = 27 + (x + 36)2 can be obtained
from the graph of y = x2 by one of the following actions:
(a) shifting the graph of f (x) to the right 36 units;
(b) shifting the graph of f (x) to the left 36 units;
(c) vertically stretching the graph of f (x) by a factor 36;
(d) vertically shrinking the graph of f (x) by a factor 36;
Your answer is (input a, b, c, or d)
Then followed by one of the following actions:
(e) shifting the resulting graph upward 27 units;
(f) shifting the resulting graph downward 27 units;
(g) horizontally stretching the resulting graph by a factor 27;
(h) horizontally shrinking the resulting graph by a factor 27;
Your answer is (input e, f, g, or h)
46.(1 pt) setAlgebra19FunTransforms/srw2 5 7.pg
The graph of the function y = f (x − 48) + 19 can be obtained
from the graph of y = f (x) by one of the following actions:
(a) shifting the graph of f (x) to the right 48 units;
(b) shifting the graph of f (x) to the left 48 units;
(c) vertically stretching the graph of f (x) by a factor 48;
(d) vertically shrinking the graph of f (x) by a factor 48;
Your answer is (input a, b, c, or d)
Then followed by one of the following actions:
(e) shifting the resulting graph upward 19 units;
(f) shifting the resulting graph downward 19 units;
(g) horizontally stretching the resulting graph by a factor 19;
(h) horizontally shrinking the resulting graph by a factor
1/19;
14
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
15
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra20QuadraticFun due 1/20/10 at 2:00 AM
3.(1 pt) setAlgebra20QuadraticFun/lh3-1 6-8.pg
Attention: you are allowed to submit your answer two times
only for this problem!
1.(1 pt) setAlgebra20QuadraticFun/lh3-1 1-3.pg
Attention: you are allowed to submit your answer two times
only for this problem!
Identify the graphs A (blue), B (red) and C (green):
is the graph of the function f (x) = −(x − 5)2
is the graph of the function g(x) = −(x − 2)2 − 5
is the graph of the function h(x) = (x + 5)2 − 2
Identify the graphs A (blue), B (red) and C (green):
is the graph of the function f (x) = (x − 4)2
is the graph of the function g(x) = (x + 5)2
is the graph of the function h(x) = x2 − 5
4.(1 pt) setAlgebra20QuadraticFun/lh3-1 13-16.pg
Consider the Quadratic function f (x) = 4x2 − 1.
Its vertex is (
,
).
Its x-intercepts are x =
.
Note: If there is more than one answer enter them separated by
commas.
Its y-intercept is y =
.
2.(1 pt) setAlgebra20QuadraticFun/lh3-1 4-6.pg
Attention: you are allowed to submit your answer two times
only for this problem!
5.(1 pt) setAlgebra20QuadraticFun/lh3-1 19-20.pg
Consider the Quadratic function f (x) = x2 − 3x − 28.
Its vertex is (
,
).
Its x-intercepts are x =
.
Note: If there is more than one answer enter them separated by
commas.
Its y-intercept is y =
.
6.(1 pt) setAlgebra20QuadraticFun/lh3-1 23-24.pg
Consider the Quadratic function f (x) = −x2 + 11x − 28.
Its vertex is (
,
).
Its x-intercepts are x =
.
Note: If there is more than one answer enter them separated by
commas.
Its y-intercept is y =
.
Identify the graphs A (blue), B (red) and C (green):
is the graph of the function f (x) = 3 − x2
is the graph of the function g(x) = 3 − (x − 3)2
is the graph of the function h(x) = (x + 3)2 − 3
7.(1 pt) setAlgebra20QuadraticFun/lh3-1 25-26.pg
Consider the Quadratic function f (x) = 3x2 − 16x − 12.
Its vertex is (
,
);
its x-intercepts are x =
1
Note: If there is more than one answer enter them separated by
commas.
its y-intercept is y =
.
The graph of a quadratic function f (x) is shown above. It has a
vertex at (−1, −4) and passes the point (0, −3). Find the quadratic function.
f (x) =
8.(1 pt) setAlgebra20QuadraticFun/lh3-1 28.pg
Consider the Quadratic function f (x) = 2x2 − 13x − 7.
Its vertex is (
,
).
Its x-intercepts are x =
.
Note: If there is more than one answer enter them separated by
commas.
Its y-intercept is y =
.
9.(1 pt) setAlgebra20QuadraticFun/lh3-1
11.(1 pt) setAlgebra20QuadraticFun/lh3-1
41.pg
38.pg
The graph of a quadratic function f (x) is shown above. It has a
vertex at (−0, 1) and passes the point (0, 1). Find the quadratic
function.
f (x) =
12.(1 pt) setAlgebra20QuadraticFun/lh3-1
42.pg
The graph of a quadratic function f (x) is shown above. It has a
vertex at (2, 2) and passes the point (0, −2). Find the quadratic
function.
f (x) =
10.(1 pt) setAlgebra20QuadraticFun/lh3-1
40.pg
The graph of a quadratic function f (x) is shown above. It has
a vertex at (2, 0) and passes the point (0, 8). Find the quadratic
function.
f (x) =
13.(1 pt) setAlgebra20QuadraticFun/lh3-2 27.pg
Find all real zeros of the function f (x) = x2 − 64.
Zeros are x =
.
2
Note: If there is more than one answer enter them separated by
commas.
23.(1 pt) setAlgebra20QuadraticFun/sw3 3 69.pg
A box with a square base and no top is to be made from a square
piece of carboard by cutting 4 in. squares from each corner and
folding up the sides. The box is to hold 10816 in3 . How big a
piece of cardboard is needed?
Your answer is:
in. by
in.
14.(1 pt) setAlgebra20QuadraticFun/lh3-2 29.pg
Finf all real zeros of f (x) = x2 − 6x − 7.
Zeros are x =
.
Note: If there is more than one answer enter them separated by
commas.
24.(1 pt) setAlgebra20QuadraticFun/standardform.pg
Given the function f (x) = 5x2 − 70x + 208 find all of the following:
The vertex of the function is
.
Does the function have a minimum or a maximum? (Type minimum or maximum)
.
Find the extreme value of the function.
The smallest root is
.
The largest root is
.
15.(1 pt) setAlgebra20QuadraticFun/findroots.pg
Find the roots of g(k) = (98kx)2 + 53kx − 52
The smaller root is
.
The larger root is
.
16.(1 pt) setAlgebra20QuadraticFun/findroots2.pg
x
Find the roots of g(k) = 5(k − 58x)2 − 98
The smaller root is
.
The larger root is
.
25.(1 pt) setAlgebra20QuadraticFun/vertexform.pg
Given the function f (x) = 2(x − 18)2 − 4 find all of the following:
The vertex of the function is
.
Does the function have a minimum or a maximum? (Type minimum or maximum)
.
Find the extreme value of the function.
The smallest root is
.
The largest root is
.
17.(1 pt) setAlgebra20QuadraticFun/findroots3.pg
Find the roots of h(t) = (160kt)2 − 110t + 191
The smaller root is
.
The larger root is
.
What positive value of k will result in exactly one real root?
k=
18.(1 pt) setAlgebra20QuadraticFun/findstandard.pg
A quadratic function has its vertex at the point (0, −9). The
function passes through the point (8, 7). Find the quadratic and
linear coefficients and the constant term of the function.
The quadratic coefficient is
.
The linear coefficient is
.
The constant term is
.
26.(1 pt) setAlgebra20QuadraticFun/lh3-1 77.pg
A rancher has 248 feet of fencing to enclose two adjacent rectangular corrals. What dimensions will produce the largest total
area?
Your answer is:
(Enter length and width separated by commas.)
What is the maximum total area?
Your answer is:
27.(1 pt) setAlgebra20QuadraticFun/lh3-1 79.pg
The revenue function in terms of the number of units sold ,x, is
given as
R = 260x − 0.3x2
where R is the total revenue in dollars. Find the number of units
sold x that produces a maximum revenue?
Your answer is x =
What is the maximum revenue?
28.(1 pt) setAlgebra20QuadraticFun/lh3-1 85.pg
The height y (in feet) of a ball thrown by a child is
1
y = − x2 + 2x + 3
14
where x is the horizontal distance in feet from the point at which
the ball is thrown.
(a) How high is the ball when it leaves the child’s hand? (Hint:
Find y when x = 0)
Your answer is y =
(b) What is the maximum height of the ball?
(c) How far from the child does the ball strike the ground?
19.(1 pt) setAlgebra20QuadraticFun/findvertex.pg
A quadratic function has its vertex at the point (−10, −2). The
function passes through the point (−5, −1). When written in
vertex form, the function is f (x) = a(x − h)2 + k, where:
a=
.
h=
.
.
k=
20.(1 pt) setAlgebra20QuadraticFun/givencoeff.pg
Write a quadratic function with a linear coefficient of 15, a constant term of −4, and a quadratic term of 28b2.
f (b) =
21.(1 pt) setAlgebra20QuadraticFun/givenroots.pg
Write a quadratic function that has roots of 38 and P.
f (m) =
22.(1 pt) setAlgebra20QuadraticFun/sw3 1 11.pg
For the function y = (x − 8)(x + 7),
its y-intercept is
its x-intercepts are x =
Note: If there is more than one x-intercept write the x-values
separated by commmas.
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
3
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
4
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra21PolynomialFun due 1/21/10 at 2:00 AM
its x-intercepts are x1 =
, x2 =
and x3 =
x1 ≤ x 2 ≤ x 3
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
1.(1 pt) setAlgebra21PolynomialFun/srw3 1 13.pg
For the function y = x(x − 8)(x + 8),
its y-intercept is y =
its x-intercepts are x =
,
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
8.(1 pt) setAlgebra21PolynomialFun/sw5 1 15.pg
Given the function P(x) = (x − 2)2(x − 4), find
its y-intercept is
its x-intercepts are x1 =
and x2 =
with x1 < x2
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
2.(1 pt) setAlgebra21PolynomialFun/srw3 1 15.pg
For the function y = (x − 8)(x + 5)(4x − 3),
its y-intercept is y =
its x-intercepts are x =
.
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
9.(1 pt) setAlgebra21PolynomialFun/sw5 1 25.pg
Given the function P(x) = x4 − 1x3 − 56x2 , find
its y-intercept is
its x-intercepts are x1 =
, x2 =
and x3 =
x1 < x 2 < x 3
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
3.(1 pt) setAlgebra21PolynomialFun/srw3 1 17.pg
For the function y = (x − 3)2(x − 4),
its y-intercept is y =
its x-intercepts are x =
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
10.(1 pt) setAlgebra21PolynomialFun/sw5 1 27.pg
Given the function P(x) = x3 + 1x2 − 42x, find
its y-intercept is
its x-intercepts are x1 =
, x2 =
and x3 =
x1 < x 2 < x 3
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
with
with
11.(1 pt) setAlgebra21PolynomialFun/srw3 1 29.pg
For the function y = x3 + 3x2 − 28x,
its y-intercept is y =
its x-intercepts are x =
.
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
4.(1 pt) setAlgebra21PolynomialFun/srw3 1 27.pg
For the function y = x4 − 7x3 − 8x2 ,
its y-intercept is y =
its x-intercepts are x =
.
Note: If there is more than one answer enter them separated by
commas. If there are none, enter none .
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
12.(1 pt) setAlgebra21PolynomialFun/srw3 1 43.pg
Given P(x) = 20x3 − 6x2 + 8x + 8,
P(x) →
if x → −∞,
P(x) →
if x → ∞,
If your answer is −∞, input -infinity; if your answer is ∞, input
infinity.
5.(1 pt) setAlgebra21PolynomialFun/sw5 1 9.pg
Given the function P(x) = (x − 4)(x + 5), find
its y-intercept is
its x-intercepts are x1 =
and x2 =
with x1 ≤ x2
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
6.(1 pt) setAlgebra21PolynomialFun/sw5 1 11.pg
Given the function P(x) = x(x − 3)(x + 5), find
its y-intercept is
its x-intercepts are x1 =
, x2 =
and x3 =
x1 ≤ x 2 ≤ x 3
When x → ∞, y →
∞ (Input + or - for the answer)
When x → −∞, y →
∞ (Input + or - for the answer)
with
13.(1 pt) setAlgebra21PolynomialFun/srw3 1 47.pg
Given P(x) = 24x10 − 2x7 + 4x + 10,
P(x) →
if x → −∞,
P(x) →
if x → ∞,
If your answer is −∞, input -infinity; if your answer is ∞, input
infinity.
with
7.(1 pt) setAlgebra21PolynomialFun/sw5 1 13.pg
Given the function P(x) = (x − 8)(x + 2)(5x − 2), find
its y-intercept is
1
14.(1 pt) setAlgebra21PolynomialFun/SRW3 1 37 42/c3s1p37 42.pg
Match the functons with their graphs. Enter the letter of the
graph below which corresponds to the function.
1. −x2 (x2 − 4)
2. −x5 + 5x3 − 4x
3. x(x2 − 4)
4. −x3 + 2x2
15.(1 pt) setAlgebra21PolynomialFun/p1.pg
A.
To get a better look at the graph, you can click on it.
The curve above is the graph of a degree 3 polynomial. It goes
through the point (5, −3). Find the polynomial.
f (x) =
16.(1 pt) setAlgebra21PolynomialFun/p2.pg
B.
C.
The curve above is the graph of a degree 4 polynomial. It goes
through the point (5, −87.5). Find the polynomial.
f (x) =
17.(1 pt) setAlgebra21PolynomialFun/srw2 6 47.pg
For the function f (x) = x3 − 12x,
its local maximum is the point: (
,
its local minimum is the point: (
,
D.
18.(1 pt) setAlgebra21PolynomialFun/srw3 1 59.pg
The polynomial P(x) = 8x3 + 4x2 − 3x has
and minima.
2
);
).
local maxima
19.(1 pt) setAlgebra21PolynomialFun/srw3 1 63.pg
The polynomial P(x) = (x − 8)5 − 14 has
and minima.
20.(1 pt) setAlgebra21PolynomialFun/srw3 1 64.pg
The polynomial P(x) = (x − 7)6 + 18 has
and minima.
21.(1 pt) setAlgebra21PolynomialFun/sw1 6 53.pg
How many real solutions does the equation x3 = 27 have?
Input your answer here:
How many real solutions does the equation x3 = −27 have?
Input your answer here:
local maxima
local maxima
22.(1 pt) setAlgebra21PolynomialFun/regress.pg
Let x be the number of units (in thousands) that a company produces and let p(x) be the profit (in tens of thousands of dollars). The
following table gives the profit for different levels of production.
x
3
5
7
9
11
13
15
17
19
21
p(x) 2.4 3.4 8.3 20.4 31.9 62.4 85.8 119 176.7 261
Use the cubic regression program to find a mathematical model for p(x).
p(x) =
26.(1 pt) setAlgebra21PolynomialFun/modelling.pg
Given the table below, find a quartic formula for g(x).
23.(1 pt) setAlgebra21PolynomialFun/classify poly.pg
Classify the following polynomial according to its degree and
number of terms:
x
-4
-2
1
2
4
g(x) 1456.62 84.98 19.82 148.02 1834.22
f (x) = 9x2
g(x) =
f (x) is a ? ? .
NOTE: You have only one attempt at this problem.
.
27.(1 pt) setAlgebra21PolynomialFun/polyn.pg
A box without a lid is constructed from a 26 inch by 26 inch
piece of cardboard by cutting x in. squares from each corner
and folding up the sides.
a) Determine the volume of the box as a function of the variable
x.
V (x) =
b) Use a graphing calculator to approximate the values of x that
produce a volume of 1235.8125.
Note: There are 3 values of x that produce the given value but
only two of them are acceptable in the context of the problem.
List the two answers, to at least one decimal place, separated by
commas.
x=
24.(1 pt) setAlgebra21PolynomialFun/end behavior.pg
Given the following function, describe its end behavior.
f (x) = 8x5 + 7x4 − 7x3 − 5x2 − 2x + 8
To the left, f (x) ? .
To the right, f (x) ? .
NOTE: You have only one attempt at this problem.
25.(1 pt) setAlgebra21PolynomialFun/evaluate.pg
Evaluate f (x) = 9x5 − 8x4 − 8x3 − 2x2 − 5x + 9 when x = −3.
f (−3) =
.
NOTE: Your answer must be a plain number.
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
3
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra22PolynomialDivision due 1/22/10 at 2:00 AM
8.(1 pt) setAlgebra22PolynomialDivision/srw3 2 25.pg
Use synthetic division and the Remainder Theorem to evaluate
P(c), where
1.(1 pt) setAlgebra22PolynomialDivision/srw3 2 1.pg
Find the quotient and remainder using long division for
x2 + 9x + 20
.
x+3
P(x) = x2 + 7x + 11,
The quotient is
The remainder is
P(c) =
The quotient is
The remainder is
2.(1 pt) setAlgebra22PolynomialDivision/srw3 2 3.pg
Find the quotient and remainder using long division for
The quotient is
The remainder is
9.(1 pt) setAlgebra22PolynomialDivision/srw3 2 27.pg
Use long division and the Remainder Theorem to evaluate P(c),
where
P(x) = x3 − 8x2 + 20x − 24, c = 2.
x3 − 10x2 + 29x − 27
.
x−4
The quotient is
The remainder is
P(c) =
3.(1 pt) setAlgebra22PolynomialDivision/srw3 2 5.pg
Find the quotient and remainder using long division for
The quotient is
The remainder is
10.(1 pt) setAlgebra22PolynomialDivision/srw3 2 31.pg
Use synthetic division and the Remainder Theorem to evaluate
P(c), where
x3 − 9x2 + 18x − 9
.
x2 − 2x + 2
P(x) = x4 + 7x3 + 5x2 + 43x + 60,
The quotient is
The remainder is
P(c) =
4.(1 pt) setAlgebra22PolynomialDivision/srw3 2 7.pg
Find the quotient and remainder using long division for
c = −7.
11.(1 pt) setAlgebra22PolynomialDivision/srw3 2 39.pg
Use the Factor Theorem to show that x − 1 is a factor of
2x3 − 12x2 + 7x − 22
.
2x2 + 5
P(x) = x3 − 6x2 + 6x − 1.
The quotient is
The remainder is
The function value P(1) =
.
of P(x).
Thus, x − 1 is a
5.(1 pt) setAlgebra22PolynomialDivision/srw3 2 15.pg
Find the quotient and remainder using synthetic division for
12.(1 pt) setAlgebra22PolynomialDivision/srw3 2 41.pg
Use the Factor Theorem to show that x − 1/2 is a factor of
x3 + 4x2 + 9x + 16
.
x+2
P(x) = 2x3 − 4x2 + 5.5x − 2.
The quotient is
The remainder is
The function value P(1/2) =
.
Thus, x − 1/2 is a
of P(x).
6.(1 pt) setAlgebra22PolynomialDivision/srw3 2 19.pg
Find the quotient and remainder using synthetic division for
13.(1 pt) setAlgebra22PolynomialDivision/sw5 2 1.pg
Find the quotient and remainder using long division for
x5 − x4 + 5x3 − 5x2 + 9x − 16
.
x−1
x2 + 4x + 7
x+3
The quotient is
The remainder is
The quotient is
The remainder is
7.(1 pt) setAlgebra22PolynomialDivision/srw3 2 23.pg
Find the quotient and remainder using synthetic division for
The quotient is
The remainder is
c = −1.
14.(1 pt) setAlgebra22PolynomialDivision/sw5 2 3.pg
Find the quotient and remainder using long division for
x5 − x4 + 4x3 − 4x2 + 3x − 9
.
x−1
The quotient is
The remainder is
1
x3 − 9x2 + 22x − 10
.
x−4
15.(1 pt) setAlgebra22PolynomialDivision/sw5 2 5.pg
Find the quotient and remainder using long division for
The quotient is
The remainder is
x3 − 10x2 + 20x − 9
.
x2 − 2x + 2
21.(1 pt) setAlgebra22PolynomialDivision/jj1.pg
Use synthetic division to show that
The quotient is
The remainder is
16.(1 pt) setAlgebra22PolynomialDivision/sw5 2 15.pg
Find the quotient and remainder using synthetic division for
is a root of the equation
x3 + 12x2 + 11x − 168 = 0.
x3 + 6x2 + 15x + 21
.
x+2
Then, use the result to factor the polynomial completely into the
form
(x + A)(x + B)(x +C)
The quotient is
The remainder is
17.(1 pt) setAlgebra22PolynomialDivision/sw5 2 19.pg
Find the quotient and remainder using synthetic division for
Then give the list of values A, B,C you obtain:
22.(1 pt) setAlgebra22PolynomialDivision/long div easy.pg
Use long division to find the quotient and remainder when
f (x) = 25x4 + 30x3 − 15x2 + 20x + 20 is divided by g(x) =
−5x − 1.
The quotient is
.
The remainder is
.
x5 − x4 + 7x3 − 7x2 + 6x − 13
.
x−1
The quotient is
The remainder is
18.(1 pt) setAlgebra22PolynomialDivision/sw5 2 27.pg
Use synthetic division and the Remainder Theorem to evaluate
P(c), where
P(x) = x3 − 6x2 + 12x − 15,
23.(1 pt) setAlgebra22PolynomialDivision/long division 0.pg
Use long division to find the quotient and remainder when
f (x) = −2x4 + 5x3 − 3x2 − 3x + 6 is divided by g(x) = 1x − 6.
The quotient is
.
The remainder is
.
c = 2.
The quotient is
The remainder is
P(c) =
24.(1 pt) setAlgebra22PolynomialDivision/long division 1.pg
Use long division to find the quotient and remainder when
19.(1 pt) setAlgebra22PolynomialDivision/sw5 2 31.pg
Use synthetic division and the Remainder Theorem to evaluate
P(c), where
P(x) = x4 + 7x3 + 4x2 + 33x + 37,
The quotient is
The remainder is
P(c) =
f (x) = −2x5 − 2x4 + 7x3 + 8x2 − 4x − 4
is divided by
c = −7.
The quotient is
The remainder is
g(x) = −4x2 − 6x − 2.
.
.
25.(1 pt) setAlgebra22PolynomialDivision/synth division.pg
Use synthetic division to find the quotient and remainder when
f (x) = 3x4 + 6x3 + 7x2 + 5x − 8 is divided by g(x) = x + 4.
The quotient is
.
The remainder is
.
20.(1 pt) setAlgebra22PolynomialDivision/sw5 2 39.pg
Use the Factor Theorem to show that x − c is a factor of P(x) for
the given values of c, where
P(x) = x3 − 7x2 + 9x − 3,
x = −7
c = 1.
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
2
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra23PolynomialZeros due 1/23/10 at 2:00 AM
Between the first two roots, is the graph of f (x) above or below
the x-axis? Answer above or below:
.
Between the last two roots, is the graph of f (x) above or below
the x-axis? Answer above or below:
.
After the last root, is the graph of f (x) above or below the xaxis? Answer above or below:
5.(1 pt) setAlgebra23PolynomialZeros/roots1.pg
1.(1 pt) setAlgebra23PolynomialZeros/rational roots.pg
List all possible rational roots for the function
f (x) = 5x4 + 8x3 − 8x2 − 1x + 65.
Give your list in increasing order. Beside each possible rational
root, type ”yes” if it is a root and ”no” if it is not a root. Leave
any unnecessary answer blanks empty.
Possible rational root:
Is it a root?
.
Possible rational root:
Is it a root?
.
Is it a root?
.
Possible rational root:
Possible rational root:
Is it a root?
.
Possible rational root:
Is it a root?
.
Possible rational root:
Is it a root?
.
Possible rational root:
Is it a root?
.
Is it a root?
.
Possible rational root:
Possible rational root:
Is it a root?
.
Possible rational root:
Is it a root?
.
Possible rational root:
Is it a root?
.
Possible rational root:
Is it a root?
.
Match the polynomial function to its correct roots
Place the letter of the list of correct roots next to each function
listed below:
1. f (x) = x4 + 4x3 − 16x − 16
2. f (x) = x4 + 8x3 + 24x2 + 32x + 16
3. f (x) = x4 − 8x3 + 24x2 − 32x + 16
4. f (x) = x4 + 4x3 + 8x2 + 16x + 16
A. x = −2, −2, −2, −2
B. x = −2, −2, −2, 2
C. x = −2i, −2i, −2, −2
D. x = 2, 2, 2, 2
6.(1 pt) setAlgebra23PolynomialZeros/jay1.pg
2.(1 pt) setAlgebra23PolynomialZeros/describe graph.pg
Given f (x) = (x + 8)(x − 6)(x − 3), find the roots in increasing
order.
The roots are
,
, and
.
To the left of the first root, is the graph of f (x) above or below
the x-axis? Answer above or below:
.
Between the first two roots, is the graph of f (x) above or below
the x-axis? Answer above or below:
.
Between the last two roots, is the graph of f (x) above or below
the x-axis? Answer above or below:
.
After the last root, is the graph of f (x) above or below the xaxis? Answer above or below:
Match the polynomial function to its correct roots
Place the letter of the list of correct roots next to each function
listed below:
1. f (x) = 16x4 − 16x3 + 4x − 1
2. f (x) = 4x4 + 4x3 + 5x2 + 4x + 1
3. f (x) = 16x4 + 32x3 + 24x2 + 8x + 1
4. f (x) = 16x4 − 8x2 + 1
A. x = 0.5, with multiplicity 3, x = −0.5
B. x = −0.5, with multiplicity 2, x = i, x = −i
C. x = 0.5, with multiplicity 2, x = −0.5, with multiplicity
2
D. x = −0.5, with multiplicity 4
3.(1 pt) setAlgebra23PolynomialZeros/describe graph a.pg
Given f (x) = (x + 4)(x + 3)(x − 3), find the roots in increasing
order.
The roots are
,
, and
.
To the left of the first root, is the graph of f (x) above or below
the x-axis? Answer above or below:
.
Between the first two roots, is the graph of f (x) above or below
the x-axis? Answer above or below:
.
Between the last two roots, is the graph of f (x) above or below
the x-axis? Answer above or below:
.
After the last root, is the graph of f (x) above or below the xaxis? Answer above or below:
7.(1 pt) setAlgebra23PolynomialZeros/jay2.pg
Match the polynomial function to its correct roots
Place the letter of the list of correct roots next to each function
listed below:
1. f (x) = x4 − 12x3 + 54x2 − 108x + 81
2. f (x) = x4 − 18x2 + 81
3. f (x) = x4 + 12x3 + 54x2 + 108x + 81
4. f (x) = x4 + 6x3 + 18x2 + 54x + 81
A. x = 3, with multiplicity 2, x = −3, with multiplicity 2
B. x = −3, with multiplicity 4
C. x = −3, with multiplicity 2, x = 3i, x = −3i
D. x = 3, with multiplicity 4
4.(1 pt) setAlgebra23PolynomialZeros/describe graph b.pg
Given f (x) = 4(x+6)9 (x+4)4 (x−2)7 , find the roots in increasing order.
The roots are
,
, and
.
To the left of the first root, is the graph of f (x) above or below
the x-axis? Answer above or below:
.
8.(1 pt) setAlgebra23PolynomialZeros/jay3.pg
1
Match the polynomial function to its correct roots
Place the letter of the list of correct roots next to each function
listed below:
1. f (x) = x4 + 4x3 + 6x2 + 4x + 1
2. f (x) = x4 − 2x3 + 2x − 1
3. f (x) = x4 + 2x3 − 2x − 1
4. f (x) = x4 − 4x3 + 6x2 − 4x + 1
A. x = −1, with multiplicity 4
B. x = −1, x = 1, with multiplicity 3
C. x = 1, with multiplicity 4
D. x = −1, with multiplicity 3, x = 1
9.(1 pt) setAlgebra23PolynomialZeros/srw3
Find all zeros of the polynomial
14.(1 pt) setAlgebra23PolynomialZeros/sw5
Find all rational zeros of the polynomial
P(x) = x3 − 3x2 − 13x + 15.
, x2 =
and x3 =
with
Its rational zeros are x1 =
x1 ≤ x 2 ≤ x 3 .
Note: If the polynomial has only two rational zeros, input them
at x1 and x2 ; if the polynomial has only one rational zero, input
it at x1 .
15.(1 pt) setAlgebra23PolynomialZeros/sw5
Find all rational zeros of the polynomial
3 9.pg
, x2 =
, x3 =
and
Its rational zeros are x1 =
x4 =
with x1 ≤ x2 ≤ x3 ≤ x4 .
Note: If the polynomial has only three rational zeros, input
them at x1 , x2 and x3 ; If the polynomial has only two rational
zeros, input them at x1 and x2 ; if the polynomial has only one
rational zero, input it at x1 .
Its zeros are x1 =
, x2 =
and x3 =
with
x1 ≤ x 2 ≤ x 3 .
Note: If the polynomial has only two rational zeros, input them
at x1 and x2 ; if the polynomial has only one rational zero, input
it at x1 .
4
16.(1 pt) setAlgebra23PolynomialZeros/sw5
Find all rational zeros of the polynomial
3 15.pg
P(x) = x − 8x − 9.
Its rational zeros are x1 =
, x2 =
, x3 =
and
x4 =
with x1 ≤ x2 ≤ x3 ≤ x4 .
Note: If the polynomial has only three rational zeros, input
them at x1 , x2 and x3 ; If the polynomial has only two rational
zeros, input them at x1 and x2 ; if the polynomial has only one
rational zero, input it at x1 .
, x2 =
, x3 =
and
Its rational zeros are x1 =
x4 =
with x1 ≤ x2 ≤ x3 ≤ x4 .
Note: If the polynomial has only three rational zeros, input
them at x1 , x2 and x3 ; If the polynomial has only two rational
zeros, input them at x1 and x2 ; if the polynomial has only one
rational zero, input it at x1 .
4
3
17.(1 pt) setAlgebra23PolynomialZeros/sw5
Find all rational zeros of the polynomial
3 21.pg
P(x) = 2x + 5x − x + 5x − 3.
, x2 =
, x3 =
and
Its rational zeros are x1 =
x4 =
with x1 ≤ x2 ≤ x3 ≤ x4 .
Note: If the polynomial has only three rational zeros, input
them at x1 , x2 and x3 ; If the polynomial has only two rational
zeros, input them at x1 and x2 ; if the polynomial has only one
rational zero, input it at x1 .
Give a comma separated list of the rational zeros. If there are no
rational zeros, enter the word none .
3 28.pg
18.(1 pt) setAlgebra23PolynomialZeros/srw3
Find all rational zeros of the polynomial
P(x) = 2x4 − 6x3 − 6x2 − 6x − 8.
Its rational zeros are x1 =
, x2 =
, x3 =
and
x4 =
with x1 ≤ x2 ≤ x3 ≤ x4 .
Note: If the polynomial has only three rational zeros, input
them at x1 , x2 and x3 ; If the polynomial has only two rational
zeros, input them at x1 and x2 ; if the polynomial has only one
rational zero, input it at x1 .
13.(1 pt) setAlgebra23PolynomialZeros/srw3
Find all rational zeros of the polynomial
3 29.pg
P(x) = x4 − x3 − 18x2 − 2x − 40.
2
12.(1 pt) setAlgebra23PolynomialZeros/srw3
Find all rational zeros of the polynomial
3 28.pg
P(x) = 2x4 − 6x3 − 18x2 − 6x − 20.
2
11.(1 pt) setAlgebra23PolynomialZeros/srw3
Find all rational zeros of the polynomial
3 15.pg
P(x) = x4 − 3x3 − 9x2 − 3x − 10.
P(x) = x3 − 3x2 − 9x − 5.
10.(1 pt) setAlgebra23PolynomialZeros/srw3
Find all rational zeros of the polynomial
3 9.pg
3 65.pg
P(x) = 3x4 + 6x3 − 10x2 − 2x + 3,
and then find the irrational zeros, if any.
Its real zeros are x1 =
, x2 =
, x3 =
and x4 =
with x1 ≤ x2 ≤ x3 ≤ x4
Note: If the polynomial has only three rational zeros, input
them at x1 , x2 and x3 ; If the polynomial has only two rational
zeros, input them at x1 and x2 ; if the polynomial has only one
rational zero, input it at x1 . Give EXACT answers. No decimals.
3 29.pg
P(x) = x4 − 2x3 − 13x2 − 4x − 30.
Enter the rational zeros in a comma separated list. If there are
none, enter the word none .
2
19.(1 pt) setAlgebra23PolynomialZeros/jay4.pg
For the function
y = x5 − 10x3 + 25x,
Note: If the polynomial has only two real zeros, input them at
x1 and x2 ; if the polynomial has only one real zero, input it at
x1 . Give EXACT answers. No decimals.
When x → ∞, P(x) →
When x → −∞, P(x) →
If your answer is ∞, enter infinity; if your answer is −∞, enter
-infinity.
find all distinct real zeros and enter them as a comma separated
list. If there are no real zeros, enter the word none .
The distinct real zeros are x =
.
20.(1 pt) setAlgebra23PolynomialZeros/sw5 3 33.pg
Find all the real zeros of the polynomial
P(x) = x3 − 4x2 − 13x + 6.
Its real zeros are x1 =
, x2 =
and x3 =
with
x1 ≤ x 2 ≤ x 3 .
Note: If the polynomial has only two real zeros, input them at
x1 and x2 ; if the polynomial has only one real zero, input it at
x1 .
21.(1 pt) setAlgebra23PolynomialZeros/sw5
Find all the real zeros of the polynomial
27.(1 pt) setAlgebra23PolynomialZeros/srw3
Find all the real zeros of the polynomial
P(x) = x4 − 13x2 + 12x.
, x2 =
, x3 =
and x4 =
Its real zeros are x1 =
with x1 ≤ x2 ≤ x3 ≤ x4 .
Note: If the polynomial has only three real zeros, input them
at x1 , x2 and x3 ; if the polynomial has only two real zeros, input
them at x1 and x2 ; if the polynomial has only one real zero, input
it at x1 . Give EXACT answers. No decimals.
When x → ∞, P(x) →
When x → −∞, P(x) →
If your answer is ∞, enter infinity; if your answer is −∞, enter
-infinity.
3 35.pg
P(x) = x4 + x3 − 13x2 − x + 12.
Its real zeros are x1 =
, x2 =
, x3 =
and x4 =
with x1 ≤ x2 ≤ x3 ≤ x4 .
Note: If the polynomial has only three real zeros, input them
at x1 , x2 and x3 ; if the polynomial has only two real zeros, input
them at x1 and x2 ; if the polynomial has only one real zero, input
it at x1 .
22.(1 pt) setAlgebra23PolynomialZeros/srw3
Find all the real zeros of the polynomial
28.(1 pt) setAlgebra23PolynomialZeros/jay5.pg
For the function y = x3 + 5x2 − 14x, find all real zeros.
If there is more than one real zero, separate the answers by commas. Also, if you want to enter the square root of a number, like
two, enter sqrt(2)
The real zeros are x =
.
3 33.pg
P(x) = x3 − x2 − 7x + 3.
Give them as a comma separated list, and give exact answers no decimals.
The real zeros of P(x) have x =
.
29.(1 pt) setAlgebra23PolynomialZeros/jay6.pg
Find all real zeros of the function y = x3 − 8x2 − 9x + 72.
Give your answer as a comma separated list. If there are no real
zeros, type none .
The real zeros are x =
.
23.(1 pt) setAlgebra23PolynomialZeros/srw3 2 43.pg
c = 2 is a zero of P(x) = x3 − 15x2 + 66x − 80. Find all other
zeros of P(x).
x1 =
and x2 =
with x1 < x2 .
24.(1 pt) setAlgebra23PolynomialZeros/srw3
Find all the real zeros of the polynomial
30.(1 pt) setAlgebra23PolynomialZeros/jay7.pg
For the function y = x5 − 1x3 − 20x, find all real zeros.
Note: If there is more than one real zero, separate the answers
by commas. Also, if you want to enter the square root of a number, like two, enter sqrt(2).
The real zeros are x =
.
3 35.pg
P(x) = x4 − 26x2 + 25.
Give them as a comma separated list, and give exact answers no decimals.
Its real zeros are x =
.
25.(1 pt) setAlgebra23PolynomialZeros/srw3
Find all the real zeros of the polynomial
31.(1 pt) setAlgebra23PolynomialZeros/srw3 3 51.pg
By Descarte’s rule of signs, P(x) = x3 − 2x2 − 3x − 3
has
positive real zero(s);
and has
or
negative real zero(s) (please enter the
smaller number first).
3 37.pg
P(x) = x4 + 2x3 − 5x2 − 6x.
32.(1 pt) setAlgebra23PolynomialZeros/Descartes.pg
f (x) = x8 − 4x7 − 125x6 + 728x5 + 3444x4 − 32096x3 +
26380x2 + 120832x1 + 195840
What is the maximum number of positive real roots for f (x)?
If there is more than one zero write them separated by commas.
Give EXACT answers. No decimals.
Its real zeros are x =
.
26.(1 pt) setAlgebra23PolynomialZeros/srw3
Find all the real zeros of the polynomial
3 43.pg
P(x) = x3 − 4x2 − 13x + 6.
Its real zeros are x1 =
x1 ≤ x 2 ≤ x 3
, x2 =
and x3 =
3 47.pg
What is the maximum number of negative real roots for f (x)?
33.(1 pt) setAlgebra23PolynomialZeros/descartes2.pg
Use Descartes’ Rule of Signs to analyze the number of positive
with
3
and negative real roots and the number of non-real roots of the
function:
41.(1 pt) setAlgebra23PolynomialZeros/srw3 5 27.pg
The zeros of P(x) = x5 + 2x3 + 1x are
with multiplicity
;
x1 =
x2 =
+
i with negative imaginary part,
its multiplicity is
; and
x3 =
+
i with positive imaginary part,
its multiplicity is
.
h(x) = 9x6 − 6x5 + 19x4 − 6x3 − 17x2 + 2x + 19
There are at least
and at most
positive real roots.
There are at least
and at most
negative real roots.
There are at least
and at most
non-real roots.
34.(1 pt) setAlgebra23PolynomialZeros/srw3 5 3.pg
Give all of the zeros of the polynomial
42.(1 pt) setAlgebra23PolynomialZeros/srw3 5 39.pg
The zeros of P(x) = x3 − 5x2 + 9x − 45 are
x1 =
;
x2 =
+
i with negative imaginary part,
x3 =
+
i with positive imaginary part.
P(x) = x3 − x2 − 6x − 24.
as a comma separated list.
35.(1 pt) setAlgebra23PolynomialZeros/srw3 5 7.pg
Find all zeros of the polynomial P(x) = x4 − 256.
Its zeros are
x1 =
, x2 =
with x1 < x2 ,
x3 =
+
i with negative imaginary part and
+
i with positive imaginary part.
x4 =
43.(1 pt) setAlgebra23PolynomialZeros/srw3 5 45.pg
The zeros of P(x) = x3 + 6x2 + 12x + 9 are
x1 =
;
x2 =
+
i with negative imaginary part,
x3 =
+
i with positive imaginary part.
36.(1 pt) setAlgebra23PolynomialZeros/srw3 5 9.pg
Find all zeros of the polynomial P(x) = x6 − 729.
Its zeros are
x1 =
, x2 =
with x1 < x2 ,
x3 =
+
i with both negative real and imaginary parts,
x4 =
+
i with negative real part and positive imaginary part,
x5 =
+
i with positive real part and negative imaginary part,
x6 =
+
i with both positive real and imaginary parts.
44.(1 pt) setAlgebra23PolynomialZeros/FindAllRoots.pg
f (x) = x6 + 37x5 + 624x4 + 5908x3 + 31040x2 + 72000x1 + 0
Given that -6+8i and -8+4i are roots of f (x), find all the other
roots.
Give real roots first, in increasing order. Then give the complex
roots so that the imaginary parts are increasing.
DO NOT USE THE GIVEN ROOTS IN YOUR ANSWER.
The roots are:
37.(1 pt) setAlgebra23PolynomialZeros/srw3 5 11.pg
Give all of the zeros of P(x) = x2 + 25 as a comma separated
list.
45.(1 pt) setAlgebra23PolynomialZeros/FindAllRoots0.pg
38.(1 pt) setAlgebra23PolynomialZeros/srw3 5 13.pg
The zeros of P(x) = x2 + 3x + 4 are
x1 =
+
i with negative imaginary part,
its multiplicity is
; and
x2 =
+
i with positive imaginary part,
.
its multiplicity is
f (x) = x4 + 27x3 + 283x2 + 1349x1 + 2340
Given that -7+4i is roots of f (x), find all of the roots, giving
real roots in increasing order, followed by complex roots with
increasing imaginary parts.
,
,
,
.
The roots are:
46.(1 pt) setAlgebra23PolynomialZeros/srw3 5 55.pg
Factor P(x) = x3 + 3x2 + 7x + 5 into linear and irreducible quadratic factors with real coefficients.
Let P(x) = (x + a)(x2 + bx + c). Then
a=
b=
c=
39.(1 pt) setAlgebra23PolynomialZeros/srw3 5 15.pg
The zeros of P(x) = x3 + 1x are
x1 =
with multiplicity
;
x2 =
+
i with negative imaginary part,
its multiplicity is
; and
x3 =
+
i with positive imaginary part,
its multiplicity is
.
48.(1 pt) setAlgebra23PolynomialZeros/jj1.pg
The polynomial
40.(1 pt) setAlgebra23PolynomialZeros/srw3 5 21.pg
The zeros of P(x) = x3 + 1x2 + 1x + 1 are
x1 =
with multiplicity
;
x2 =
+
i with negative imaginary part,
its multiplicity is
; and
x3 =
+
i with positive imaginary part,
its multiplicity is
.
f (x) = 8x3 − 3x2 + 128x − 48
has 4i as a root. Give all of the roots of f in a comma-separated
list, including the given one.
Roots:
4
49.(1 pt) setAlgebra23PolynomialZeros/find
Let
f (x) =
all roots.pg
56.(1 pt) setAlgebra23PolynomialZeros/bounds on zeros.pg
Determine if 9 is an upper bound, lower bound or no bound for
the roots of f (x) = −5x4 − 4x3 + 1x2 + 3x + 6.
9 is (a/an)
bound for the roots of f (x).
g(x) = −20x5 + 232x4 − 1048x3 − 3792x2 − 1812x + 680.
Given that 7+6i is a root of g(x) and that g(x) has at least one
rational root, find all the real roots of g(x) in increasing order.
The roots are:
,
,
,
,
.
Leave any unneeded answer spaces blank.
If a root has multiplicity greater than 1, enter it into your list
multiple times.
50.(1 pt) setAlgebra23PolynomialZeros/find
Let
57.(1 pt) setAlgebra23PolynomialZeros/count zeros.pg
Given the following table of values, what is the minimum number of roots that f (x) can have?
x
-4 -2 0 2 4
f(x) 7 -1 -6 2 5
all roots 2.pg
f (x) has at least
g(x) = −36x6 − 120x5 + 93x4 + 1074x3 − 663x2 − 876x + 468.
59.(1 pt) setAlgebra23PolynomialZeros/find
The polynomial p(x) has exactly one positive real root. Between
what two consecutive integers does it lie?
The positive root is between
and
.
60.(1 pt) setAlgebra23PolynomialZeros/factor checking.pg
Is (x − 5) a factor of f (x) = 6x4 − 9x3 + 2x2 − 6x + 6?
Answer yes or no:
52.(1 pt) setAlgebra23PolynomialZeros/FindPositiveRoot.pg
61.(1 pt) setAlgebra23PolynomialZeros/find degree.pg
Suppose p(x) is a polynomial with real coefficients that bounces
off the x-axis at 76, bounces off the x-axis at −61, and breaks
through the x-axis at −71. If p(−4 − 7i) = p(−5 + 7i) = 0.
What is the smallest possible degree that p(x) could have?
If p(x) has no other roots than those described above, could the
degree of p(x) be 39? (yes or no)
f (x) = x5 + 3x4 − 114x3 − 292x20x1 − 5700
The function f (x) has only one positive real root. Between what
two consecutive integers does the root lie?
The root is between
and
.
53.(1 pt) setAlgebra23PolynomialZeros/GivenRoots.pg
Find an equation with real coefficients for the polynomial that
passes through the point (0, 51) that has the following roots with
the given multiplicities.
62.(1 pt) setAlgebra23PolynomialZeros/find eqn.pg
Write the equation
of a polynomial with real coefficients that has
√
roots at ± 7 and 3 − 7i that passes through the point (0, 111).
y=
Root
-1 -3 4+2i -4+3i
Multiplicity 3 2
2
2
.
63.(1 pt) setAlgebra23PolynomialZeros/find funct comp.pg
Given that f (x) is a cubic function with zeros at −4 and −4i + 2
, find an equation for f (x) given that f (5) = −2.
f (x) =
54.(1 pt) setAlgebra23PolynomialZeros/GivenRoots1.pg
Write the equation, in standard form, of a polynomial with real
coefficients that has roots at 1, 2 − 3i, and −4i, and passes
through the point (0, 38).
Your answer should include only (decimal) numbers, the letter
”x”, and the characters ”+”, ”-”, and ”ˆ”
f (x) =
.
55.(1 pt) setAlgebra23PolynomialZeros/GivenRootsAndBehavior.pg
Find an equation for f (x), the polynomial of smallest degree
with real coefficients such that f (x) bounces off of the x-axis at
−1, bounces off of the x-axis at 3, has complex roots of −2 − 3i
and −2i and passes through the point (0, −46).
bounds.pg
f (x) = −14x5 + x4 − 13x3 − 10x2 + 7x − 15
What is the smallest positive integer that is an upper bound to
all the roots of f (x)?
What is the largest negative integer that is a lower bound to all
the roots of f (x)?
positive.pg
p(x) = x6 + 18x5 + 160x4 + 684x3 + 1276x2 − 1440x − 3200.
f (x) =
root(s).
58.(1 pt) setAlgebra23PolynomialZeros/find real roots.pg
Write the equation
√ of a polynomial with real coefficients that
has roots at ± 29 and −3 + 5i that passes through the point
(0, 109).
y=
Given that -3-2i is a root of g(x) and that g(x) has at least one
rational root, find all the real roots of g(x) in increasing order.
The roots are:
,
,
,
,
,
.
Leave any unneeded answer spaces blank.
If a root has multiplicity greater than 1, enter it in to your list
multiple times.
51.(1 pt) setAlgebra23PolynomialZeros/find
Let
.
64.(1 pt) setAlgebra23PolynomialZeros/find function.pg
Given that f (x) is a cubic function with zeros at −8, 3, and 7,
find an equation for f (x) given that f (6) = −5.
f (x) =
65.(1 pt) setAlgebra23PolynomialZeros/quadfromroots1.pg
Enter a quadratic polynomial which has zeros at 9 and 4.
Enter a quadratic polynomial which has zeros at -9 and 10.
5
74.(1 pt) setAlgebra23PolynomialZeros/srw3 2 49.pg
Find a degree 3 polynomial that has zeros −1, 4 and 7 and in
which the coefficient of x2 is −20.
The polynomial is
Enter a quadratic polynomial which has a double root at -3.
66.(1 pt) setAlgebra23PolynomialZeros/quadfromroots2.pg
75.(1 pt) setAlgebra23PolynomialZeros/srw3 5 31.pg
Find a degree 3 polynomial with coefficient of x3 equal to 1 and
zeros 4, −2i, and 2i. Simplify your answer so that it has only
real numbers as coefficients.
Your answer is
.
76.(1 pt) setAlgebra23PolynomialZeros/srw3 5 35.pg
A degree 4 polynomial with integer coefficients has zeros −4 −
4i and 1, with 1 a zero of multiplicity 2. If the coefficient of x4
is 1,
then the polynomial is
.
Enter a quadratic polynomial which has roots at -4/15 and -1.
67.(1 pt) setAlgebra23PolynomialZeros/sw5 2 45.pg
Find a degree 3 polynomial having zeros -6, 1 and 5 and the
coefficient of x3 equal 1.
The polynomial is
68.(1 pt) setAlgebra23PolynomialZeros/sw5 2 47.pg
Find a degree 4 polynomial having zeros -8, -3, 1 and 7 and the
coefficient of x4 equal 1.
The polynomial is
77.(1 pt) setAlgebra23PolynomialZeros/p1.pg
69.(1 pt) setAlgebra23PolynomialZeros/srw3 2 45.pg
Find a degree 3 polynomial having zeros -4, 2 and 7 and the
coefficient of x3 equal 1.
The polynomial is
A degree 4 polynomial P(x) with integer coefficients has zeros
3i and 2, with 2 being a zero of multiplicity 2. Moreover, the
coefficient of x4 is 1. Find the polynomial.
P(x) =
70.(1 pt) setAlgebra23PolynomialZeros/p3.pg
The polynomial of degree 5, P(x) has leading coefficient 1, has
roots of multiplicity 2 at x = 3 and x = 0, and a root of multiplicity 1 at x = −3
Find a possible formula for P(x).
P(x) =
,
78.(1 pt) setAlgebra23PolynomialZeros/p10.pg
Find a polynomial with integer coefficients, with leading coefficient 1, degree 5, zeros i and 8 − i, and passing through the
origin.
P(x) =
71.(1 pt) setAlgebra23PolynomialZeros/p4.pg
The polynomial of degree 4, P(x) has a root of multiplicity 2 at
x = 3 and roots of multiplicity 1 at x = 0 and x = −4. It goes
through the point (5, 36).
Find a formula for P(x).
P(x) =
,
79.(1 pt) setAlgebra23PolynomialZeros/srw3 3 83.pg
A grain silo consists of a cylindrical main section and a hemispherical roof. If the total volume of the silo (including the part
inside the roof section) is 15000 ft3 and the cylindrical part is
30 ft tall, what is the radius of the silo?
your answer is
72.(1 pt) setAlgebra23PolynomialZeros/p5.pg
The polynomial of degree 3, P(x), has a root of multiplicity 2 at
x = 3 and a root of multiplicity 1 at x = −3. The y-intercept is
y = −13.5.
Find a formula for P(x).
,
P(x) =
80.(1 pt) setAlgebra23PolynomialZeros/beth1polydiv.pg
A grain silo consists of a cylindrical main section and a hemispherical roof. If the total volume of the silo (including the part
inside the roof section) is 15000 ft3 and the cylindrical part is
30 ft tall, what is the radius of the silo?
Note: The following formulas may be useful:
Volume of a Cylinder = πr 2 h
4
Volume of a Sphere = πr 3
3
Radius =
73.(1 pt) setAlgebra23PolynomialZeros/srw3 2 47.pg
Find a degree 4 polynomial having zeros −7, −3, 4 and 5 and
the coefficient of x4 equal 1.
The polynomial is
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
6
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra24Variation due 1/24/10 at 2:00 AM
1.(1 pt) setAlgebra24Variation/ur alg 11 1.pg
Suppose r varies directly with t and that r = 16 when t = 4.
What is the value of r when t = 11?
r=
2.(1 pt) setAlgebra24Variation/ur alg 11 2.pg
Suppose p varies directly with q and that p = 48 when q = 12.
What is the value of p when q = 4?
p=
12.(1 pt) setAlgebra24Variation/variation eqn2.pg
If q varies jointly as r and the square root of t and inversely as
p, then find an equation for q if q = 2 when t = 1, r = 3, and
p = 9.
q=
13.(1 pt) setAlgebra24Variation/lh3-5 48.pg
Suppose p varies inversely as the square root of q. If p = −8
when q = 1, what is p if q is 11?
p=
3.(1 pt) setAlgebra24Variation/ur alg 11 3.pg
Suppose z varies inversely with t and that z = 12 when t = 8.
What is the value of z when t = 3?
z=
4.(1 pt) setAlgebra24Variation/ur alg 11 4.pg
Suppose f varies inversely with g and that f = 24 when g = 4.
What is the value of f when g = 8?
f=
14.(1 pt) setAlgebra24Variation/lh3-5 50.pg
Suppose p varies jointly as the cube of q and the cube of r. If
p = 10 when q = 6 and r = 15, what is p if q = 5 and r = 13?
p=
5.(1 pt) setAlgebra24Variation/ur alg 11 5.pg
Suppose z varies directly with x and inversely with the square of
y. If z = 20 when x = 5 and y = 3, what is z when x = 12 and
y = 8?
z=
6.(1 pt) setAlgebra24Variation/ur alg 11 6.pg
Suppose z varies directly with y and directly with the cube of x.
If z = 810 when x = 3 and y = 10, what is z when x = 7 and
y = 4?
z=
7.(1 pt) setAlgebra24Variation/ur ab 6 2.pg
At 3:00 PM a man 137 cm tall casts a shadow 133 cm long. At
the same time, a tall building nearby casts a shadow 178 m long.
How tall is the building?
Give your answer in meters. (You may need the fact that 100
cm = 1 m.)
15.(1 pt) setAlgebra24Variation/lh3-5 54.pg
If t varies jointly as p and q and inversely as r, then find an
equation for t if t = 9 when p = 4, q = −1, and r = −7.
t=
16.(1 pt) setAlgebra24Variation/ur alg 11 7.pg
Match each equation with the way in which r varies with respect
to t in that equation.
IMPORTANT!! You only have 3 attempts to get this problem
right!
1.
2.
3.
8.(1 pt) setAlgebra24Variation/find const.pg
Suppose p varies directly as the square root of q. If p = −1
when q = 1, what is p if q is 3?
p=
4.
9.(1 pt) setAlgebra24Variation/joint.pg
Suppose p varies jointly as the cube root of q and the cube of r.
If p = 8 when q = 10 and r = 12, what is p if q = 15 and r = 1?
p=
5.
10.(1 pt) setAlgebra24Variation/simple variation.pg
Suppose p varies directly as the square of q. If p = 11 when
q = 2, what is p if q is 14?
p=
A.
B.
C.
D.
E.
F.
6.
11.(1 pt) setAlgebra24Variation/variation eqn.pg
If t varies jointly as p and q and inversely as r, then find an
equation for t if t = 8 when p = −2, q = 6, and r = 3.
t=
1
4
v
= 2 3
27t
s r
25rs4 vt 2 = 100
1
1
= 2
5r 2t
5s v
r
8v2
=
s3
t3
st 2
= 2vs
8r
3r
= v2 s
t3
directly with the square of t
inversely with the square of t
directly with the cube of t
directly with the cube root of t
inversely with the cube of t
inversely with the square root of t
17.(1 pt) setAlgebra24Variation/ur
For each power function, choose (by letter) the graph which
most closely resembles the graph of that function. You may
always assume that the constant of variation k is positive.
Warning: You have only 4 attempts at this problem so make
them count!
5
y = kx
y = kx 2
k
y = x2.5
y = kx1.05
alg 11 8.pg
y = kx3
19.(1 pt) setAlgebra24Variation/lh3-5 26.pg
State sales tax y is directly proportional to retail price x. An item
that sells for 142 dollars has a sales tax of 14.42 dollars. Find a
mathematical model that gives the amount of sales tax y in terms
of the retail price x.
Your answer is y =
What is the sales tax on a 400 dollars purchase.
Your answer is:
20.(1 pt) setAlgebra24Variation/lh3-5 60.pg
The stopping distance d of an automobile is directly proportional to the square of its speed v. A car required 35 feet to
stop when its speed was 50 miles per hour. Find a mathematical
model that gives the stopping distance d in terms of its speed v.
Your answer is d =
Estimate the stopping distance if the brakes are applied when
the car is traveling at 74 miles per hour.
Your answer is:
21.(1 pt) setAlgebra24Variation/lh3-5 62.pg
A company has found that the demand for its product varies inversely as the price of the product. When the price x is 3.25
dollars, the demand y is 400 units. Find a mathematical model
that gives the demand y in terms of the price x in dollars.
Your answer is y =
Approximate the demand when the price is 9.5 dollars.
Your answer is:
For each power function, choose (by letter) the graph which
most closely resembles the graph of that function. You may
always assume that the constant of variation k is positive.
Warning: You have only 4 attempts at this problem so make
them count!
y = kx5
y = xk3
1
y = kx 2
y = kx
y=
k
3
x8
y = kx4.95
18.(1 pt) setAlgebra24Variation/ur
1
y = kx 4
alg 11 9.pg
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
2
ARNOLD PIZER
rochester problib from CVS June 25, 2004
Rochester WeBWorK Problem Library
WeBWorK assignment Algebra25RationalFun due 1/25/10 at 2:00 AM
The root(s) of f (x), in increasing order, is/are:
.
f (x) has hole(s) when x is:
,
,
.
f (x) has vertical asymptotes when x is:
,
f (x) has a horizontal asymptote at y =
1.(1 pt) setAlgebra25RationalFun/FindInfo.pg
Leave any unneeded answer spaces blank.
1x3 + 9x2 + 20x + 12
1x3 + 6x2
The domain of the function f (x), in interval notation from left
to right, is:
∪
∪
.
(Type -inf for −∞ and inf for ∞) Do not use any spaces in your
answer. Don’t forget to use parentheses.
The root(s) of f (x), in increasing order, is/are:
,
,
.
f (x) has hole(s) at the point(s): (
,
),(
,
).
f (x) has vertical asymptotes when x is:
,
,
.
f (x) has a horizontal asymptote at y =
f (x) =
,
−9x3 + 75x2 − 108x − 108
−3x3 + 42x2 − 153x + 162
The domain of the function f (x), in interval notation from left
∪
∪
∪
.
to right, is:
(Type -inf for −∞ and inf for ∞) Do not use any spaces in your
answer. Don’t forget to use parentheses.
,
,
The root(s) of f (x), in increasing order, is/are:
.
f (x) has one hole at the point: (
,
).
f (x) has vertical asymptotes when x is:
,
.
f (x) has a horizontal asymptote at y =
,
6.(1 pt) setAlgebra25RationalFun/cross
.
f (x) =
OA.pg
8x3 − 3x2 − 3x + 7
−9x2 − 2x + 1
.
Find the equation of the non-vertical asymptote.
y=
Does f (x) intersect its non-vertical asymptote? (yes or no)
What is the smallest value of x at which f (x) intersects its nonvertical asymptote? (Leave this question blank if you answered
.
no above.)
3.(1 pt) setAlgebra25RationalFun/FindInfo2 r.pg
Leave any unneeded answer spaces blank.
1x3 + 1x2 − 14x − 24
1x3 − 5x2 − 4x + 20
The domain of the function f (x), in interval notation from left
to right, is:
∪
∪
∪
.
(Type -inf for −∞ and inf for ∞) Do not use any spaces in your
answer. Don’t forget to use parentheses.
The root(s) of f (x), in increasing order, is/are:
,
,
.
f (x) has one hole at the point: (
,
).
f (x) has vertical asymptotes when x is:
,
.
f (x) has a horizontal asymptote at y =
f (x) =
7.(1 pt) setAlgebra25RationalFun/cross
asymptote.pg
7x4 − 14x3 + 7x2 − 2x + 8
1x4 − 2x3 − 9x2 − 2x − 7
What is the equation of the horizontal asympotote? y =
.
Does the graph of f (x) intersect its horizontal asymptote?
(yes or no)
.
At what x-values does f (x) intersect its horizontal asymptote? Give your answers in increasing order. If f (x) does not intersect its horizontal asymptote, leave this question blank.
,
f (x) =
4.(1 pt) setAlgebra25RationalFun/FindInfo3.pg
Leave any unneeded answer spaces blank.
f (x) =
.
f (x) =
f (x) =
,
,
,
5.(1 pt) setAlgebra25RationalFun/FindInfo3 r.pg
Leave any unneeded answer spaces blank.
2.(1 pt) setAlgebra25RationalFun/FindInfo2.pg
Leave any unneeded answer spaces blank.
1x3 + 12x2 − 1x − 252
1x3 − 8x2 − 29x + 180
The root(s) of f (x), in increasing order, is/are:
.
f (x) has hole(s) when x is:
,
,
.
f (x) has vertical asymptotes when x is:
,
f (x) has a horizontal asymptote at y =
,
−6x3 + 36x2 − 48x
−3x3 + 6x2 + 27x − 54
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
1
ARNOLD PIZER
rochester problib from CVS June 25, 2004
Rochester WeBWorK Problem Library
1.(1 pt) setAlgebra26PartialFraction/srw8
The partial fraction decomposition of
WeBWorK assignment Algebra26PartialFraction due 1/26/10 at 2:00 AM
f (x) =
g(x) =
9 1.pg
9
can be
(x − 1)(x + 2)
g(x)
f (x)
+
.
x−1 x+2
The possible anwsers for f (x) and g(x) are (a) A, a constant, or
(b) Ax + B, a linear function.
f (x) is in the form of (input a or b)
and g(x) is in the form of (input a or b)
.
written in the form of
2.(1 pt) setAlgebra26PartialFraction/srw8
The partial fraction decomposition of
5.(1 pt) setAlgebra26PartialFraction/srw8
The partial fraction decomposition of
the form of
f (x) =
g(x) =
9 5.pg
x2 + 8
can be
(x − 3)(x2 + 4)
f (x)
g(x)
+
, where
x−2 x+2
,
.
6.(1 pt) setAlgebra26PartialFraction/srw8
f (x)
g(x)
+
.
x − 3 x2 + 4
The possible anwsers for f (x) and g(x) are (a) A, a constant, or
(b) Ax + B, a linear function.
f (x) is in the form of (input a or b)
and g(x) is in the form of (input a or b)
.
written in the form of
,
.
The partial fraction decomposition of
ten in the form of
f (x) =
g(x) =
3.(1 pt) setAlgebra26PartialFraction/srw8 9 9.pg
How many fraction terms are there in the partial fraction decomx3 + x2 + 13
position of
?
x(2x − 5)3(x2 + 2x + 5)2
Your answer is
.
4.(1 pt) setAlgebra26PartialFraction/srw8 9 11.pg
28
The partial fraction decomposition of
can be
(x − 1)(x + 1)
f (x)
g(x)
written in the form of
+
, where
x−1 x+1
,
.
9 17.pg
36
can be written in
x2 − 4
9 21.pg
50x
8x2 − 10x + 3
can be writ-
f (x)
g(x)
+
, where
2x − 1 4x − 3
7.(1 pt) setAlgebra26PartialFraction/srw8 9 25.pg
x2 + 31
The partial fraction decomposition of 3
can be written in
x + x2
f (x) g(x) h(x)
the form of
+ 2 +
, where
x
x
x+1
,
f (x) =
g(x) =
,
h(x) =
.
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
1
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra27Conics due 1/27/10 at 2:00 AM
1.(1 pt) setAlgebra27Conics/hyperinfo.pg
A hyperbola has a vertical transverse axis of length 18 and
asymptotes of y = 23 x − 9 and y = − 23 x + 9. Find the center
of the hyperbola, its focal length, and its eccentricity.
The center of the hyperbola is (
,
).
The focal length is
.
The eccentricity is
.
3.
2.(1 pt) setAlgebra27Conics/matching.pg
Match each graph to its equation.
(For all graphs on this page, if you are having a hard time seeing
the picture clearly, click on it. It will expand to a larger picture
on its own page so that you can inspect it more closely.)
4.
1.
5.
2.
1
6.
9.
7.
10.
A.
B.
C.
D.
E.
F.
G.
8.
H.
I.
J.
x2 y2
−
=1
4 16
(y − 1)2
=1
x2 +
4
y2 x2
−
=1
4 16
2
2
x
y
+
=1
4 16
2
(x − 1) = 2(y + 1)
(x − 1)2 (y + 1)2
+
=1
4
16
2
y
=1
x2 +
16
2
y
x2 + = 1
4
(y − 1)2 = −2(x − 1)
(x − 1)2 = −2(y − 1)
3.(1 pt) setAlgebra27Conics/ur geo 3 1.pg
Match each graph to its equation.
(For all graphs on this page, if you are having a hard time seeing
the picture clearly, click on it. It will expand to a larger picture
on its own page so that you can inspect it more closely.)
2
1.
4.
2.
5.
3.
6.
A.
B.
C.
D.
E.
F.
y2 = −2x
y2 = 2x
(x − 1)2 = −2(y − 1)
x2 = 2y
(y − 1)2 = 2(x + 1)
(y − 1)2 = −2(x − 1)
4.(1 pt) setAlgebra27Conics/ur geo 3 2.pg
Find an equation of the parabola that has a focus at (3, 13) and
a vertex at (3, 4):
3
y=
Find an equation of its directrix:
y=
5.(1 pt) setAlgebra27Conics/ur geo 3 3.pg
Find the vertex, focus, and directrix for the following functions.
(a)
(y − 7)2 = 12(x − 4)
vertex : (
,
)
focus : (
,
)
directrix x =
(b)
y2 − 8y = 20x − 42
vertex : (
,
)
focus : (
,
)
directrix x =
(c)
(x − 3)2 = 20(y − 3)
vertex : (
,
)
focus : (
,
)
directrix y =
(d)
x2 + 24x = 4y − 28
vertex : (
,
)
focus : (
,
)
directrix y =
(y − K)2 = A(y − H)
where K =
where H =
where A =
6.(1 pt) setAlgebra27Conics/ur geo 3 4.pg
Write equations for each parabola (If you have a hard time seeing the picture clearly, click on the picture so that you can inspect it more closely.)
(a)
7.(1 pt) setAlgebra27Conics/ur geo 3 5.pg
Match each graph to its equation.
(For all graphs on this page, if you are having a hard time seeing
the picture clearly, click on it. It will expand to a larger picture
on its own page so that you can inspect it more closely.)
where K =
where H =
where A =
(b)
(y − K)2 = A(x − H)
1.
4
2.
5.
3.
6.
A.
B.
C.
D.
E.
F.
4.
x2 y2
+ =1
16 4
2
x
+ y2 = 1
4
(x − 1)2 (y + 1)2
+
=1
4
16
2
y
=1
x2 +
16
(x + 1)2
+ (y − 1)2 = 1
4
x2 y2
+
=1
4 16
8.(1 pt) setAlgebra27Conics/ur geo 3 6.pg
Find the center, vertices, and foci of each ellipse.
x2 y2
+
=1
(a)
36 25
,
)
Center: (
Right vertex: (
,
)
Left vertex: (
,
)
Top vertex: (
,
)
Bottom vertex: (
,
)
Right focus: (
,
)
Left focus: (
,
)
5
(x + 17)2 (y − 9)2
+
=1
9
36
Center: (
,
)
Right vertex: (
,
)
Left vertex: (
,
)
Top vertex: (
,
)
Bottom vertex: (
,
)
Top focus: (
,
)
Bottom focus: (
,
)
(c) 4x2 + 9y2 − 56x − 90y + 385 = 0
Center: (
,
)
Right vertex: (
,
)
Left vertex: (
,
)
Top vertex: (
,
)
Bottom vertex: (
,
)
Right focus: (
,
)
Left focus: (
,
)
where C =
where D =
(b)
(b)
9.(1 pt) setAlgebra27Conics/ur geo 3 7.pg
The equation of the ellipse that has a center at (5, 3), a focus at
(8, 3), and a vertex at (10, 3), is
(x −C)2 (y − D)2
+
=1
A2
B2
(y − A)2 (x −C)2
+
=1
B2
D2
where
A=
B=
C=
D=
where A =
where B =
where C =
where D =
10.(1 pt) setAlgebra27Conics/ur geo 3 8.pg
Write equations for each ellipse (If you have a hard time seeing
the picture clearly, click on the picture so that you can inspect it
more closely.)
(a)
where A =
where B =
11.(1 pt) setAlgebra27Conics/ur geo 3 9.pg
Match each graph to its equation.
(For all graphs on this page, if you are having a hard time seeing
the picture clearly, click on it. It will expand to a larger picture
on its own page so that you can inspect it more closely.)
(y − A)2 (x −C)2
+
=1
B2
D2
1.
6
2.
5.
3.
6.
A.
B.
C.
D.
E.
F.
x2
− y2 = 1
4
y2
x2 − = 1
4
y2 − 4x2 = 1
4x2 − y2 = 1
y2 x2
−
=1
4 16
2
2
x − 4y = 1
12.(1 pt) setAlgebra27Conics/ur geo 3 10.pg
The equation of the hyperbola that has a center at (8, 9), a focus
at (13, 9), and a vertex at (4, 9), is
(x −C)2 (y − D)2
−
=1
A2
B2
4.
where
A=
B=
C=
D=
13.(1 pt) setAlgebra27Conics/ur geo 3 11.pg
Write equations for each hyperbola (If you have a hard time
seeing the picture clearly, click on the picture so that you can
inspect it more closely.)
7
(a)



x=
y=
−12
+ 12
x+1
y − 23=(x + 3)2
15.(1 pt) setAlgebra27Conics/ur geo 3 13.pg
The parabola given by the equation x = y2 + 6y + 1 has its vertex
at (h, k) for:
h=
and
k=
16.(1 pt) setAlgebra27Conics/ur geo 3 14.pg
The parabola given by the equation y = −x2 + 8x − 1 has its
vertex at (h, k) for:
h=
and
k=
17.(1 pt) setAlgebra27Conics/ur geo 3 15.pg
The parabola given by the equation x = −2y2 + 24y − 75 has its
vertex at (h, k) for:
h=
and
k=
18.(1 pt) setAlgebra27Conics/ur geo 3 16.pg
The parabola given by the equation y = 3x2 − 18x + 10 has its
vertex at (h, k) for:
h=
and
k=
19.(1 pt) setAlgebra27Conics/ur geo 3 17.pg
The parabola given by the equation 2y − 24 = x2 + 8x has its
vertex at (h, k) for:
h=
and
k=
20.(1 pt) setAlgebra27Conics/ur geo 3 18.pg
The parabola given by the equation 5x − 2y = y2 + 36 has its
vertex at (h, k) for:
h=
and
k=
21.(1 pt) setAlgebra27Conics/ur geo 3 19.pg
Match each equation for a parabola with the direction that the
parabola opens.
IMPORTANT!! You only have 4 attempts to get this problem
right!
1. y = −10(x + 1)2 − 2
2. y = 10(x + 1)2 − 2
3. x = 10(y + 1)2 − 2
4. x = −10(y + 1)2 − 2
A. up
(x − A)2 (y −C)2
−
=1
B2
D2
where A =
where B =
where C =
where D =
(b)
(y − A)2 (x −C)2
−
=1
B2
D2
where A =
where B =
where C =
where D =
14.(1 pt) setAlgebra27Conics/ur


y=
geo 3 12.pg
Solve the system by graphing each equation and finding the
point of intersection.
8
B. right
C. left
D. down
3.
4.
A.
B.
C.
D.
22.(1 pt) setAlgebra27Conics/ur geo 3 20.pg
Match each equation for a parabola with the direction that the
parabola opens.
IMPORTANT!! You only have 4 attempts to get this problem
right!
1. y = − 15 (x − 7)2 + 2
2. x = − 15 (y − 7)2 + 2
3. y = 15 (x − 7)2 + 2
4. x = 15 (y − 7)2 + 2
A. down
B. up
C. right
D. left
x = − 19 (y + 6)2 + 10
y = 19 (x + 6)2 + 10
up
down
right
left
25.(1 pt) setAlgebra27Conics/ur geo 3 23.pg
x2 +y2 +12x−16y+0 = 0 is the equation of a circle with center
(h, k) and radius r for:
h=
and
k=
and
r=
26.(1 pt) setAlgebra27Conics/ur geo 3 24.pg
x2 + y2 − 12x + 16y + 19 = 0 is the equation of a circle with
center (h, k) and radius r for:
h=
and
k=
and
r=
27.(1 pt) setAlgebra27Conics/ur geo 3 25.pg
3x2 + 3y2 − 12x − 24y + 57 = 0 is the equation of a circle with
center (h, k) and radius r for:
h=
and
k=
and
r=
28.(1 pt) setAlgebra27Conics/ur geo 3 26.pg
2x2 + 2y2 + 12x + 8y + 18 = 0 is the equation of a circle with
center (h, k) and radius r for:
h=
and
k=
and
r=
23.(1 pt) setAlgebra27Conics/ur geo 3 21.pg
Match each equation for a parabola with the direction that the
parabola opens.
IMPORTANT!! You only have 4 attempts to get this problem
right!
1. y = 5(x − 6)2 − 3
2. x = 5(y − 6)2 − 3
3. y = −5(x − 6)2 − 3
4. x = −5(y − 6)2 − 3
A. right
B. left
C. up
D. down
24.(1 pt) setAlgebra27Conics/ur geo 3 22.pg
Match each equation for a parabola with the direction that the
parabola opens.
IMPORTANT!! You only have 4 attempts to get this problem
right!
1. x = 19 (y + 6)2 + 10
2. y = − 19 (x + 6)2 + 10
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
9
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra28ExpFunctions due 1/28/10 at 2:00 AM
1.(1 pt) setAlgebra28ExpFunctions/sw6 1 3.pg
For the function f (x) = 2x , calculate the following function values:
f (−3) =
f (−1) =
f (0) =
f (1) =
f (3) =
2.(1 pt) setAlgebra28ExpFunctions/sw6
1 5.pg
x
1
For the function f (x) =
, calculate the following function
9
values:
f (−3) =
f (−1) =
f (0) =
f (1) =
f (3) =
C.
3.(1 pt) setAlgebra28ExpFunctions/c6s1p15 20/c6s1p15 20.pg
Match the functions with their graphs. Enter the letter of the
graph below which corresponds to the function.
1.
2.
3.
4.
5.
D.
f (x) = 5x + 3
f (x) = 5x+1 − 4
f (x) = 5x−3
f (x) = −5x
f (x) = 5x
E.
A.
4.(1 pt) setAlgebra28ExpFunctions/c4s1p13 18/c4s1p13 18.pg
Match the functions with their graphs. Enter the letter of the
graph below which corresponds to the function.
1.
2.
3.
4.
5.
B.
1
f (x) = 5−x
f (x) = −5x
f (x) = 5x−3
f (x) = 5x+1 − 4
f (x) = 5x
A.
E.
5.(1 pt) setAlgebra28ExpFunctions/ur log 1 3.pg
Starting with the graph of f (x) = 8x , write the equation of the
graph that results from
(a) shifting f (x) 8 units upward. y =
(b) shifting f (x) 1 units to the left. y =
(c) reflecting f (x) about the x-axis and the y-axis. y =
(d) reflecting f (x) about the line x = −1. y =
6.(1 pt) setAlgebra28ExpFunctions/srw4 1 1.pg
For the function f (x) = 10x , calculate the following function
values:
f (−3) =
f (−1) =
f (0) =
f (1) =
f (3) =
B.
7.(1 pt) setAlgebra28ExpFunctions/srw4
1 3.pg
x
For the function f (x) = 12 , calculate the following function
values:
f (−3) =
f (−1) =
f (0) =
f (1) =
f (3) =
C.
8.(1 pt) setAlgebra28ExpFunctions/srw4 1 5.pg
For the function f (x) = 9ex , calculate the following function
values:
f (−3) =
f (−1) =
f (0) =
f (1) =
f (3) =
D.
9.(1 pt) setAlgebra28ExpFunctions/srw4 1 9.pg
Find the exponential function f (x) = ax whose graph goes
through the point (2, 4).
a=
.
10.(1 pt) setAlgebra28ExpFunctions/srw4 1 11.pg
Find the exponential function f (x) = ax whose graph goes
through the point 2, 1/25.
a=
.
2
11.(1 pt) setAlgebra28ExpFunctions/srw4 1 21.pg
The graph of the function f (x) = 9x − 5 can be obtained from
the graph of g(x) = 9x by one of the following actions:
(a) shifting the graph of g(x) to the right 5 units;
(b) shifting the graph of g(x) to the left 5 units;
(c) shifting the graph of g(x) upward 5 units;
(d) shifting the graph of g(x) downward 5 units;
(e) reflecting the graph of g(x) in the x-axis;
(f) reflecting the graph of g(x) in the y-axis;
Your answer is (input a, b, c, d, e, or f)
Is the domain of the function f (x) still (−∞, ∞)?
Your answer is (input Yes or No)
The range of the function f (x) is (A, ∞),
the value of A is
(b) shifting the graph of g(x) to the left 5 units;
(c) shifting the graph of g(x) upward 5 units;
(d) shifting the graph of g(x) downward 5 units;
(e) reflecting the graph of g(x) in the x-axis;
(f) reflecting the graph of g(x) in the y-axis;
Your answer is (input a, b, c, d, e, or f)
Is the domain of the function f (x) still (−∞, ∞)?
Your answer is (input Yes or No)
The range of the function f (x) is (−∞, A),
the value of A is
16.(1 pt) setAlgebra28ExpFunctions/sw6 1 23.pg
The graph of the function f (x) = 9x − 9 can be obtained from
the graph of g(x) = 9x by one of the following actions:
(a) shifting the graph of g(x) to the right 9 units;
(b) shifting the graph of g(x) to the left 9 units;
(c) shifting the graph of g(x) upward 9 units;
(d) shifting the graph of g(x) downward 9 units;
(e) reflecting the graph of g(x) in the x-axis;
(f) reflecting the graph of g(x) in the y-axis;
Your answer is (input a, b, c, d, e, or f)
Is the domain of the function f (x) still (−∞, ∞)?
Your answer is (input Yes or No)
The range of the function f (x) is (A, ∞),
the value of A is
12.(1 pt) setAlgebra28ExpFunctions/srw4 1 29.pg
The graph of the function f (x) = e−x − 5 can be obtained from
the graph of g(x) = ex by one of the following actions:
(a) reflecting the graph of g(x) in the x-axis;
(b) reflecting the graph of g(x) in the y-axis;
your answer is (input a or b)
then, by one of the following actions:
(a) shifting the resulting graph to the right 5 units;
(b) shifting the resulting graph to the left 5 units;
(c) shifting the resulting graph upward 5 units;
(d) shifting the resulting graph downward 5 units;
Your answer is (input a, b, c, or d)
Is the domain of the function f (x) still (−∞, ∞)?
Your answer is (input Yes or No)
The range of the function f (x) is (A, ∞),
the value of A is
17.(1 pt) setAlgebra28ExpFunctions/sw6 1 24.pg
The graph of the function f (x) = 8x−6 can be obtained from the
graph of g(x) = 8x by one of the following actions:
(a) shifting the graph of g(x) to the right 6 units;
(b) shifting the graph of g(x) to the left 6 units;
(c) shifting the graph of g(x) upward 6 units;
(d) shifting the graph of g(x) downward 6 units;
(e) reflecting the graph of g(x) in the x-axis;
(f) reflecting the graph of g(x) in the y-axis;
Your answer is (input a, b, c, d, e, or f)
Is the domain of the function f (x) still (−∞, ∞)?
Your answer is (input Yes or No)
The range of the function f (x) is (A, ∞),
the value of A is
13.(1 pt) setAlgebra28ExpFunctions/srw4 1 31.pg
The graph of the function f (x) = 6x−7 can be obtained from the
graph of g(x) = 6x by one of the following actions:
(a) shifting the graph of g(x) to the right 7 units;
(b) shifting the graph of g(x) to the left 7 units;
(c) shifting the graph of g(x) upward 7 units;
(d) shifting the graph of g(x) downward 7 units;
(e) reflecting the graph of g(x) in the x-axis;
(f) reflecting the graph of g(x) in the y-axis;
Your answer is (input a, b, c, d, e, or f)
Is the domain of the function f (x) still (−∞, ∞)?
Your answer is (input Yes or No)
The range of the function f (x) is (A, ∞),
the value of A is
18.(1 pt) setAlgebra28ExpFunctions/sw6 2 3.pg
The graph of the function f (x) = −ex can be obtained from the
graph of g(x) = ex by one of the following actions:
(a) reflecting the graph of g(x) in the y-axis;
(b) reflecting the graph of g(x) in the x-axis;
Your answer is (input a or b)
The range of the function f (x) is f (x) < A, find A.
The value of A is
Is the domain of the function f (x) still (−∞, ∞)?
Your answer is (input Yes or No)
14.(1 pt) setAlgebra28ExpFunctions/srw4 1 33.pg
Find the exponential function f (x) = Cax whose graph goes
through the points (0, 5) and (2, 20).
a=
,
C=
.
19.(1 pt) setAlgebra28ExpFunctions/sw6 2 5.pg
The graph of the function f (x) = e−x − 6 can be obtained from
the graph of g(x) = ex by two of the following actions:
(a) reflecting the graph of g(x) in the y-axis;
(b) reflecting the graph of g(x) in the x-axis;
15.(1 pt) setAlgebra28ExpFunctions/sw6 1 21.pg
The graph of the function f (x) = −5x can be obtained from the
graph of g(x) = 5x by one of the following actions:
(a) shifting the graph of g(x) to the right 5 units;
3
(c) shifting the graph of g(x) to the right 6 units;
(d) shifting the graph of g(x) to the left 6 units;
(e) shifting the graph of g(x) upward 6 units;
(f) shifting the graph of g(x) downward 6 units;
Your answer: Apply the action
(input a, b, c, d, e, or f)
then apply the action
(Please give your answer in the order the changes are applied,
e.g. a first, then b second.)
The range of the function f (x) is f (x) > A, find A.
The value of A is
Is the domain of the function f (x) still (−∞, ∞)?
Your answer is (input Yes or No)
(c) How many bacteria will the culture contain at time t=5?
Your answer is
24.(1 pt) setAlgebra28ExpFunctions/sw6 2 23.pg
The population of the world in 1987 was 5 billion and the relative growth rate was estimated at 2 percent per year. Assuming
that the world population follows an exponential growth model,
find the projected world population in 1995.
Your answer is
billion
25.(1 pt) setAlgebra28ExpFunctions/sw6 2 27.pg
Certain radioactive material decays in such a way that the mass
remaining after t years is given by the function
20.(1 pt) setAlgebra28ExpFunctions/sw6 2 9.pg
If 8400 dollars is invested at an interest rate of 10 percent per
year, compounded semiannually, find the value of the investment after the given number of years.
(a) 5 years:
Your answer is
(b) 10 years:
Your answer is
(c) 15 years:
Your answer is
where m(t) is measured in grams.
(a) Find the mass at time t = 0.
Your answer is
(b) How much of the mass remains after 25 years?
Your answer is
26.(1 pt) setAlgebra28ExpFunctions/beth1.pg
Complete the table below giving the amount P that must be invested at interest rate 12 % compounded quarterly to obtain a
balance of A = $170000 in t years.
m(t) = 115e−0.02t
21.(1 pt) setAlgebra28ExpFunctions/sw6 2 11.pg
If 10300 dollars is invested at an interest rate of 5 percent per
year, find the value of the investment at the end of 5 years for
the following compounding methods.
(a) Annual:
Your answer is
(b) Semiannual:
Your answer is
(c) Monthly:
Your answer is
(d) Daily:
Your answer is
(e) Continuously:
Your answer is
t
1
10
20
30
40
50
P
27.(1 pt) setAlgebra28ExpFunctions/beth2.pg
Complete the table below giving the amount P that must be invested at interest rate 8.5 % compounded continuously to obtain
a balance of A = $180000 in t years.
t
1
10
20
30
40
50
22.(1 pt) setAlgebra28ExpFunctions/sw6 2 13.pg
Which of the given interest rates and compounding periods
would provide the best investment?
(a) 6 12 percent per year, compounded semiannually;
(b) 6 14 percent per year, compounded quarterly;
(c) 6 percent per year, compounded continuously.
Your answer is (input a, b, or c)
P
28.(1 pt) setAlgebra28ExpFunctions/ur le 2 13.pg
Starting with the graph of f (x) = 5x , write the equation of the
graph that results from
(a) shifting f (x) 7 units downward. y =
(b) shifting f (x) 2 units to the left. y =
(c) reflecting f (x) about the x-axis and the y-axis. y =
23.(1 pt) setAlgebra28ExpFunctions/sw6 2 17.pg
The number of bacteria in a culture is given by the function
n(t) = 950e0.3t
where t is measured in hours.
(a) What is the relative rate of growth of this bacterium population?
Your answer is
percent
(b) What is the initial population of the culture (at t=0)?
Your answer is
29.(1 pt) setAlgebra28ExpFunctions/pexp.pg
Starting with the graph of f (x) = 9x , write the equation of the
graph that results from
(a) shifting f (x) 2 units upward. y =
4
(b) shifting f (x) 5 units to the left. y =
(c) reflecting f (x) about the y-axis. y =
, which is approximately
32.(1 pt) setAlgebra28ExpFunctions/ur le 1 5.pg
Find the exponential function f (x) = a · 2bx whose graph is
shown below
30.(1 pt) setAlgebra28ExpFunctions/Test1 20.pg
You invest $ 10000 in Acme Inc. on January 1, 2000. Your
investment returns 3.5 % compounded monthly. How much
money will you have on June 30, 2008?
You will have $
After what month and year will you have at least $ 15,000?
You will have at least $ 15,000 after
(month)
(year).
Please capitalize the month and do not use any abbreviation.
31.(1 pt) setAlgebra28ExpFunctions/simplifying.pg
Simplify the following expressions. Give exact answers with
the fewest number of e’s possible. Then give a decimal approximation.
(a)
e6 e−3 e5 =
, which is approximately
.
(b)
e18 − e−18
=
e9 − e−9
, which is approximately
.
(c)
(e3 − 3)(e6 − 1) =
a=
b=
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
5
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra29LogFunctions due 1/29/10 at 2:00 AM
Your answer is
(c) log8 8
Your answer is
6.(1 pt) setAlgebra29LogFunctions/sw6 3 17.pg
Evaluate the expression, reduce to simplest form.
1
(a) log3 27
Your answer
is
√
(b) log 3 10
Your answer is
(c) log0.001
Your answer is
7.(1 pt) setAlgebra29LogFunctions/sw6 3 19.pg
Evaluate the expression, reduce to simplest form.
(a) 2log2 8
Your answer is
(b) 10log9
Your answer is
(c) eln 3
Your answer is
8.(1 pt) setAlgebra29LogFunctions/sw6 3 45.pg
The graph of the function f (x) = log2 (x − 5) can be obtained
from the graph of g(x) = log2 x by one of the following actions:
(a) shifting the graph of g(x) to the right 5 units;
(b) shifting the graph of g(x) to the left 5 units;
(c) shifting the graph of g(x) upward 5 units;
(d) shifting the graph of g(x) downward 5 units;
Your answer is (input a, b, c, or d)
The domain of the function f (x) is x > A, find A
The value of A is
Is the range of the function f (x) still (−∞, ∞)?
Your answer is (input Yes or No)
1.(1 pt) setAlgebra29LogFunctions/sw6 3 1.pg
Express the equation in exponential form
(a) log2 4 = 2.
That is, write your answer in the form 2A = B. Then
A=
and
B=
(b) log5 125 = 3.
That is, write your answer in the form 5C = D. Then
C=
and
D=
2.(1 pt) setAlgebra29LogFunctions/sw6 3 3.pg
Express the equation in exponential form
(a) log16 2 = 14 .
That is, write your answer in the form 16A = B. Then
A=
and
B=
(b) log2 18 = −3.
That is, write your answer in the form 2C = D. Then
C=
and
D=
3.(1 pt) setAlgebra29LogFunctions/sw6 3 7.pg
Express the equation in logarithmic form
(a) 24 = 16.
That is, write your answer in the form log2 A = B. Then
A=
and
B=
(b) 10−2 = 0.010000.
That is, write your answer in the form log10 C = D. Then
C=
and
D=
4.(1 pt) setAlgebra29LogFunctions/sw6
Evaluate the expression
(a) log3 35
Your answer is
(b) log2 8
Your answer is
(c) log2 2
Your answer is
9.(1 pt) setAlgebra29LogFunctions/sw6 3 49.pg
The graph of the function f (x) = 2+log3 x can be obtained from
the graph of g(x) = log3 x by one of the following actions:
(a) shifting the graph of g(x) to the right 2 units;
(b) shifting the graph of g(x) to the left 2 units;
(c) shifting the graph of g(x) upward 2 units;
(d) shifting the graph of g(x) downward 2 units;
Your answer is (input a, b, c, or d)
The domain of the function f (x) is x > A, find A
The value of A is
Is the range of the function f (x) still (−∞, ∞)?
Your answer is (input Yes or No)
3 13.pg
10.(1 pt) setAlgebra29LogFunctions/sw6 4 1.pg
Use the Laws of logarithms to rewrite the expression
5.(1 pt) setAlgebra29LogFunctions/sw6 3 15.pg
Evaluate the expression, reduce to simplest form.
(a) log6 68
Your answer is
(b) log8 4096
log2 (6x(x − 16))
in a form with no logarithm of a product, quotient or power.
After rewriting we will have:
1
log2 (6x(x − 16)) = log2 A + log2 x + log2 f (x)
with the constant
A=
and the function
f (x) =
After rewriting we have
√
ln 4 xy = A logx + B logy
with the constant
A=
and the constant
B=
16.(1 pt) setAlgebra29LogFunctions/sw6 4 25.pg
Evaluate
√ the expression, reducing to simplest form
log5 3125 =
11.(1 pt) setAlgebra29LogFunctions/sw6 4 3.pg
Use the Laws of logarithms to rewrite the expression
log1914
in a form with no logarithm of a product, quotient or power.
After rewriting we have:
17.(1 pt) setAlgebra29LogFunctions/sw6 4 27.pg
Evaluate the expression, reduce to simplest terms
log24 + log54 =
log1914 = A log 19
with the constant
A=
12.(1 pt) setAlgebra29LogFunctions/sw6 4 5.pg
Use the Laws of logarithms to rewrite the expression
18.(1 pt) setAlgebra29LogFunctions/sw6
Rewrite the expression
4 37.pg
log2 x + 5 log2 y − 5 log2 z
log2 (x5 y14 )
as a single logarithm log2 A. Then the function
A=
19.(1 pt) setAlgebra29LogFunctions/sw6 4 39.pg
Rewrite the expression
in a form with no logarithm of a product, quotient or power.
After rewriting we have
log2 (x5 y14 ) = A log2 x + B log2 y
with the constant
A=
and the constant
B=
13.(1 pt) setAlgebra29LogFunctions/sw6 4 7.pg
Use the Laws of logarithms to rewrite the expression
p
log3 (x6 3 y14 )
5 logx − 3 log(x2 + 1) + 4 log(x − 1)
as a single logarithm logA. Then the function
A=
20.(1 pt) setAlgebra29LogFunctions/sw6 4 40.pg
Rewrite the expression
ln(a + b) + 3ln(a − b) − 5 lnc
as a single logarithm lnA. Then the function
A=
21.(1 pt) setAlgebra29LogFunctions/sw6 4 41.pg
Rewrite the expression
in a form with no logarithm of a product, quotient or power.
After rewriting we have
p
log3 (x6 3 y14 ) = A log3 x + B log3 y
with the constant
A=
and the constant
B=
14.(1 pt) setAlgebra29LogFunctions/sw6 4 9.pg
Use the Laws of logarithms to rewrite the expression
p
4
log5 x2 + 4
ln5 + 3 lnx + 2 ln(x2 + 10)
as a single logarithm lnA. Then the function
A=
22.(1 pt) setAlgebra29LogFunctions/srw4 2 5.pg
Express the equation in exponential form
(a) ln4 = x is equivalent to eA = B.
A=
and
B=
(b) lnx = 2 is equivalent to eC = D.
C=
and
D=
23.(1 pt) setAlgebra29LogFunctions/srw4 2 9.pg
Express the equation in logarithmic form:
(a) 43 = 64 is equivalent to log4 A = B.
A=
and
B=
(b) 10−2 = 0.01 is equivalent to log10 C = D.
in a form with no logarithm of a product, quotient or power.
After rewriting we have
p
4
log5 x2 + 4 = A log5 f (x)
with the constant
A=
and the function
f (x) =
15.(1 pt) setAlgebra29LogFunctions/sw6 4 11.pg
Use the Laws of logarithms to rewrite the expression
√
ln 4 xy
in a form with no logarithm of a product, quotient or power.
2
30.(1 pt) setAlgebra29LogFunctions/srw4 3 19.pg
Use the Laws of logarithms to rewrite the expression
 s 
y16 
ln x16
z10
C=
and
D=
24.(1 pt) setAlgebra29LogFunctions/srw4 2 11.pg
Express the equation in logarithmic form:
(a) ex = 3 is equivalent to lnA = B. Then
A=
and
B=
(b) e2 = x is equivalent to lnC = D. Then
C=
and
D=
in a form with no logarithm of a product, quotient or power.
After rewriting we have
 s

16
y
 = A ln(x) + B ln(y) +C ln(z)
ln x16
z10
with the constant
A=
the constant
B=
and the constant
C=
31.(1 pt) setAlgebra29LogFunctions/srw4 3 23.pg
Use the Laws of logarithms to rewrite the expression
s
x2 + 19
log
2
(x + 7)(x3 − 2)20
25.(1 pt) setAlgebra29LogFunctions/srw4 2 21.pg
Evaluate the expression, reduce to simplest form.
(a) ln e−2
Your answer is
(b) ln e7
Your answer is
(c) ln(1/e)
Your answer is
26.(1 pt) setAlgebra29LogFunctions/srw4 2 45.pg
The graphs of the functions y = ax and y = loga x are symmetric
with respect to the line
y=
in a form with no logarithm of a product, quotient or power.
After rewriting we have
s
x2 + 19
log
= A log(x2 +19)+B log(x2 +7)+C log(x3 −2)
(x2 + 7)(x3 − 2)20
27.(1 pt) setAlgebra29LogFunctions/srw4 2 59.pg
The domain of the function g(x) = loga (x2 − 9) is
(−∞,
) and (
, ∞).
with the constant A =
the constant B =
and the constant C =
32.(1 pt) setAlgebra29LogFunctions/srw4
Evaluate the following expressions.
1
(a) log3 243
=
(b) log9 √
1=
(c) log4 64 =
(d) 9log9 2 =
28.(1 pt) setAlgebra29LogFunctions/srw4 3 15.pg
Use the Laws of logarithms to rewrite the expression
8 2
x y
log
z15
in a form with no logarithm of a product, quotient or power.
After rewriting we have
8 2
x y
log
= A log(x) + B log(y) +C log(z)
z15
33.(1 pt) setAlgebra29LogFunctions/srw4 3 25.pg
Use the Laws of logarithms to rewrite the expression
√
x6 x − 1
ln(
)
3x − 4
in a form with no logarithm of a product, quotient or power.
After rewriting we have
√
x6 x − 1
) = A ln x + B ln(x − 1) +C ln(3x − 4)
ln(
3x − 4
with the constant A =
the constant B =
and the constant C =
34.(1 pt) setAlgebra29LogFunctions/srw4 3 27-30.pg
Evaluate the following expressions.
log4 0.0625
log8 0.001953125
with
A=
B=
and
C=
29.(1 pt) setAlgebra29LogFunctions/srw4
Evaluate the following expressions.
(a) log6 614 =
(b) log2 8 =
(c) log4 64 =
(d) log4 45 =
3 23-26.pg
3 17-20.pg
3
√
log64 4
log4 8
35.(1 pt) setAlgebra29LogFunctions/srw4
Evaluate the following expressions.
(a) ln e1 =
5=
(b) eln √
ln
(c) e 2 =
(d) ln(1/e5 ) =
B. y lnx
C. x lny
D. lnx − lny
3 33-36.pg
44.(1 pt) setAlgebra29LogFunctions/srw4 4 45-50.pg
Enter a T or an F in each answer space below to indicate whether
the corresponding statement is true or false.
You must get all of the answers correct to receive credit.
1. lnab = b lna
2. loga b = logb a
3. ln(x − y) = ln x − lny
4. lnx
lny = lnx − lny
36.(1 pt) setAlgebra29LogFunctions/srw4 3 37.pg
Evaluate the expression, reducing to simplest form
log(log10000100000 ) =
+ log
Note. Your answers must
be integers.
37.(1 pt) setAlgebra29LogFunctions/srw4
Rewrite the expression
3 41.pg
log2 x + 3 log2 y − 3 log2 z
as a single logarithm log2 A. Then the function
A=
38.(1 pt) setAlgebra29LogFunctions/srw4 3 43.pg
Rewrite the expression
4 logx − 4 log(x2 + 1) + 3log(x − 1)
45.(1 pt) setAlgebra29LogFunctions/srw4
Rewrite the expression in terms of ln
log4 10 =
4 78.pg
46.(1 pt) setAlgebra29LogFunctions/srw4
Rewrite the expression in terms of ln
log27 2 =
4 80.pg
47.(1 pt) setAlgebra29LogFunctions/c4s2p39 44/c4s2p39 44.pg
Match the functions with their graphs. Enter the letter of the
graph below which corresponds to the function.
as a single logarithm logA. Then the function
A=
39.(1 pt) setAlgebra29LogFunctions/srw4 3 44.pg
Rewrite the expression
1.
2.
3.
4.
5.
ln(a + b) + 3ln(a − b) − 4lnc
as a single logarithm ln A. Then the function
A=
40.(1 pt) setAlgebra29LogFunctions/srw4 3 45.pg
Rewrite the expression
ln 7 + 6 lnx + 4 ln(x2 + 9)
as a single logarithm ln A. Then the function
A=
41.(1 pt) setAlgebra29LogFunctions/srw4 3 49.pg
Evaluate the expression, correct to six decimal places, by the
Change of Base Formula and a calculator.
log2 8 =
42.(1 pt) setAlgebra29LogFunctions/srw4 3 55.pg
Evaluate the expression, correct to six decimal places, by the
Change of Base Formula and a calculator.
log4 51 =
A.
43.(1 pt) setAlgebra29LogFunctions/srw4 4 1-5.pg
Match the statements defined below with the letters labeling
their equivalent expressions.
1. ln(xy)
2. ln xy
3. ln(yx )
4. ln(xy )
A. ln x + lny
B.
4
f (x) = − lnx
f (x) = ln(x − 2)
f (x) = ln(2 − x)
f (x) = 2 + lnx
f (x) = − ln(−x)
ln a3 b−1
(c)
=
ln(bc)−4
a 2
(d) (ln c2 ) ln −3 =
b
50.(1 pt) setAlgebra29LogFunctions/problem1.pg
Evaluate the following expressions. Your answers must be exact
and in simplest form.
(a) log17 1710 =
(b) log3 6561 =
(c) log6 7776 =
(d) log6 614 =
C.
51.(1 pt) setAlgebra29LogFunctions/problem2.pg
Evaluate the following expressions. Your answers must be exact
and in simplest
form.
1
(a) log5 125
=
(b) log3 √
1=
(c) log7 823543 =
(d) 7log7 14 =
52.(1 pt) setAlgebra29LogFunctions/problem3.pg
Evaluate the following expressions.
log2 0.125
log8 0.000244140625
√
log3125 5
log4 8
D.
53.(1 pt) setAlgebra29LogFunctions/problem4.pg
Evaluate the following expressions. Your answers must be exact
and in simplest form.
(a) lne−11 =
(b) eln5√ =
(c) eln 4 =
(d) ln(1/e4) =
E.
48.(1 pt) setAlgebra29LogFunctions/beth1logfun.pg
The graph of the function f (x) = log3 (x − 5) can be obtained
from the graph of g(x) = log3 x by one of the following actions:
(a) shifting the graph of g(x) to the right 5 units;
(b) shifting the graph of g(x) to the left 5 units;
(c) shifting the graph of g(x) upward 5 units;
(d) shifting the graph of g(x) downward 5 units;
Your answer is (input a, b, c, or d)
The domain of the function f (x) is
.
Note: Enter your answer using interval notation.
The range of the function f (x) is
.
Note: Enter your answer using interval notation.
The x-intercept of the function f (x) is
.
The vertical asymptote of the function f (x) has equation:
.
54.(1 pt) setAlgebra29LogFunctions/problem10.pg
If logb 2 = x and logb 3 = y, evaluate the following in terms of x
and y:
(a) logb 12 =
(b) logb 162 =
(c) logb 16
81 =
log 81
(d) logb 4 =
b
55.(1 pt) setAlgebra29LogFunctions/simplifying expressions.pg
Simplify the following expressions. Your answers must be exact
and in simplest form.
(a) log2 28x+3 =
(b) 10log10 6−5q =
(c) logk =
(d) 78 log7 3−3 log 7 8 =
49.(1 pt) setAlgebra29LogFunctions/evaluating expressions.pg
If ln a= 2, ln
b = 3, and ln c = 5, evaluate the following:
a−4
(a) ln 3 4 =
√b c
(b) ln b−2 c4 a4 =
56.(1 pt) setAlgebra29LogFunctions/ur
Simplify:
64log8 3 =
5log25 16 =
5
le 1 4.pg
57.(1 pt) setAlgebra29LogFunctions/srw4
is equal to
4 9.pg
√
7
ln(r 3 s9 r 7 s10 )
where A =
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
6
A ln r + B lns
and where B =
ARNOLD PIZER
Rochester WeBWorK Problem Library
1.(1 pt) setAlgebra30LogExpEqns/sw6
Find x.
(a) log4 x = 3
Your answer is
(b) log2 4 = x
Your answer is
3 23.pg
2.(1 pt) setAlgebra30LogExpEqns/sw6
Find x.
(a) logx = 4
x=
(b) log5 x = 2
x=
3 25.pg
3.(1 pt) setAlgebra30LogExpEqns/sw6
Find x.
(a) logx 81 = 2
x=
(b) logx 8 = 3
x=
3 27.pg
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra30LogExpEqns due 1/30/10 at 2:00 AM
9.(1 pt) setAlgebra30LogExpEqns/sw6 5 13.pg
Find the solution of the exponential equation
15−x/10 = 16
in terms of logarithms, or correct to four decimal places.
x=
10.(1 pt) setAlgebra30LogExpEqns/sw6 5 15.pg
Find the solution of the exponential equation
e2x+1 = 21
in terms of logarithms, or correct to four decimal places.
x=
11.(1 pt) setAlgebra30LogExpEqns/sw6 5 33.pg
Find the solution of the logarithmic equation
lnx = 3
in terms of logarithms, or correct to four decimal places.
Your answer is
x=
12.(1 pt) setAlgebra30LogExpEqns/sw6 5 37.pg
Find the solution of the logarithmic equation
4.(1 pt) setAlgebra30LogExpEqns/sw6 5 1.pg
Find the solution of the exponential equation
log(3x + 2) = 2
7x = 18
in terms of logarithms, or correct to four decimal places.
Your answer is
x=
13.(1 pt) setAlgebra30LogExpEqns/sw6 5 40.pg
Find the solution of the logarithmic equation
in terms of logaritms, or correct to at least four decimal places.
x=
5.(1 pt) setAlgebra30LogExpEqns/sw6 5 3.pg
Find the solution of the exponential equation
log2 (x2 + 3x − 50) = 2
131−x = 8
in terms of logarithms, or correct to four decimal places.
Your answers are x1 =
and x2 =
with x1 ≤ x2
in terms of logarithms, or correct to four decimal places.
x=
14.(1 pt) setAlgebra30LogExpEqns/sw6 5 43.pg
Find the solution of the logarithmic equation
6.(1 pt) setAlgebra30LogExpEqns/sw6 5 5.pg
Find the solution of the exponential equation
logx + log(x − 18) = log(17x)
13ex = 17
in terms of logarithms, or correct to four decimal places.
Your answer is
x=
15.(1 pt) setAlgebra30LogExpEqns/sw6 5 48.pg
Find the solution of the logarithmic equation
in terms of logarithms, or correct to four decimal places.
x=
7.(1 pt) setAlgebra30LogExpEqns/sw6 5 7.pg
Find the solution of the exponential equation
e1−4x = 12
ln(x + 5) + ln(x − 5) = 0
in terms of logarithms, or correct to four decimal places.
x=
in terms of logarithms, or correct to four decimal places.
Your answer is
x=
16.(1 pt) setAlgebra30LogExpEqns/srw4 1 27.pg
The equation x2 ∗ 6x − 8x ∗ 6x = 0 has two roots. The smaller
root is
and the bigger root is
8.(1 pt) setAlgebra30LogExpEqns/sw6 5 9.pg
Find the solution of the exponential equation
−4 + 65x = 24
in terms of logarithms, or correct to four decimal places.
x=
1
17.(1 pt) setAlgebra30LogExpEqns/srw4 1 29.pg
The equation 8x2 e−9x − 4x3 e−9x = 0 has two roots.
The smaller root is
and the bigger root is
28.(1 pt) setAlgebra30LogExpEqns/srw4 3
If ln(5x + 6) = 5, then x =
.
.
18.(1 pt) setAlgebra30LogExpEqns/srw4 2 3.pg
Express the equation in exponential form
(a) log16 2 = 14 .
That is, write your answer in the form 16A = B. Then
A=
and
B=
1
(b) log2 32
= −5.
That is, write your answer in the form 2C = D. Then
C=
and
D=
19.(1 pt) setAlgebra30LogExpEqns/srw4 2 23.pg
Find x.
(a) log6 x = 4
Your answer is
(b) log2 16 = x
Your answer is
20.(1 pt) setAlgebra30LogExpEqns/srw4 2 27.pg
Find x.
(a) logx = 4
x=
(b) log5 x = 2
x=
21.(1 pt) setAlgebra30LogExpEqns/srw4 2 30.pg
Find x.
(a) logx 125 = 3
x=
(b) logx 36 = 2
x=
22.(1 pt) setAlgebra30LogExpEqns/srw4 2 35.pg
The graph of the function y = loga x goes through (43, 1).
Then a =
23.(1 pt) setAlgebra30LogExpEqns/srw4 2 37.pg
The graph of the function y = loga x goes through (6, −1).
Then a = 1/
24.(1 pt) setAlgebra30LogExpEqns/srw4
(a) If log3 x = 2, then x =
.
(b) If log6 x = 4, then x =
.
3 37-38.pg
25.(1 pt) setAlgebra30LogExpEqns/srw4
(a) If logx 27 = 3, then x =
.
(b) If logx 256 = 4, then x =
.
3 41-42.pg
26.(1 pt) setAlgebra30LogExpEqns/srw4
(a) If 6x = 31, then x =
.
(b) If 14−x = 2, then x =
.
3 43-44.pg
27.(1 pt) setAlgebra30LogExpEqns/srw4
Solve the given equation for x.
43x−4 = 31
x=
3 46.pg
29.(1 pt) setAlgebra30LogExpEqns/srw4
If e3x = 22, then x =
.
48.pg
3 51.pg
30.(1 pt) setAlgebra30LogExpEqns/srw4 4 1.pg
Find the solution of the exponential equation
18x = 15
in terms of logaritms, or correct to at least four decimal places.
x=
31.(1 pt) setAlgebra30LogExpEqns/srw4 4 5.pg
Find the solution of the exponential equation
81−x = 18
in terms of logarithms, or correct to four decimal places.
x=
32.(1 pt) setAlgebra30LogExpEqns/srw4 4 7.pg
Find the solution of the exponential equation
6ex = 4
in terms of logarithms, or correct to four decimal places.
x=
33.(1 pt) setAlgebra30LogExpEqns/srw4 4 9.pg
Find the solution of the exponential equation
e1−4x = 2
in terms of logarithms, or correct to four decimal places.
x=
34.(1 pt) setAlgebra30LogExpEqns/srw4 4 11.pg
Find the solution of the exponential equation
−4 + 35x = 15
correct to at least four decimal places.
x=
35.(1 pt) setAlgebra30LogExpEqns/srw4 4 15.pg
Find the solution of the exponential equation
15−x/5 = 19
in terms of logarithms, or correct to four decimal places.
x=
36.(1 pt) setAlgebra30LogExpEqns/srw4 4 17.pg
Find the solution of the exponential equation
e2x+1 = 21
in terms of logarithms, or correct to four decimal places.
x=
37.(1 pt) setAlgebra30LogExpEqns/srw4 4 21.pg
Find the solution of the exponential equation
22x+20 = 3x−38
in terms of logarithms, or correct to four decimal places.
x=
2
38.(1 pt) setAlgebra30LogExpEqns/srw4 4 25.pg
Find the solution of the exponential equation
46.(1 pt) setAlgebra30LogExpEqns/srw4 4 50.pg
Find the solution(s) of the logarithmic equation
100(1.04)2t = 50000
ln(x + 4) + ln(x − 4) = 0
in terms of logarithms, or correct to four decimal places.
x=
39.(1 pt) setAlgebra30LogExpEqns/srw4 4 27.pg
Find the solutions of the exponential equation
x1 =
and x2 =
correct to four decimal places.
If there is more than one solution write them separated by commas.
x=
47.(1 pt) setAlgebra30LogExpEqns/srw4 4 51.pg
For what value of x is the following true?
x2 2x − 2x 2 = 0.
with x1 < x2 .
log(x + 3) = logx + log3.
40.(1 pt) setAlgebra30LogExpEqns/srw4 4 31.pg
Find the solutions of the exponential equation
Your answer is
x=
e2x − 5ex + 4 = 0.
48.(1 pt) setAlgebra30LogExpEqns/srw4
If ln x + ln(x − 2) = ln 6x, then x =
Enter your answer as a comma-separated list, and enter none if
there are no solutions.
41.(1 pt) setAlgebra30LogExpEqns/srw4 4 32-34.pg
Evaluate the following expressions.
(a) e2 ln 5 =
(b) 105 log10 5 =
(c) log3 274 =
[NOTE: Your answers cannot be algebraic expressions. ]
42.(1 pt) setAlgebra30LogExpEqns/srw4 4 33.pg
Find the solutions of the exponential equation
e2x + 1ex − 12 = 0.
x1 =
and x2 =
with x1 < x2 .
Note. If there is only one solution, input it at x1 .
43.(1 pt) setAlgebra30LogExpEqns/srw4 4 41.pg
Find the solution of the logarithmic equation
20 − ln(3 − x) = 0
4 58.pg
.
49.(1 pt) setAlgebra30LogExpEqns/srw4
Solve the given equation for x.
log10 x + log10 (x − 21) = 2
x=
4 62.pg
50.(1 pt) setAlgebra30LogExpEqns/srw4
Solve the given equation for x.
2x−8 = 2
x=
4 65.pg
51.(1 pt) setAlgebra30LogExpEqns/srw4
Solve the given equation for x.
2x/8 = 2
x=
4 68.pg
52.(1 pt) setAlgebra30LogExpEqns/srw4
Solve
the given equation for x.
3 x
=6
5
x=
4 72.pg
53.(1 pt) setAlgebra30LogExpEqns/mec1.pg
The equation e2x − 8ex + 15 = 0 has two solutions.
The smaller one is:
and the larger one is:
.
correct to four decimal places.
Your answer is
x=
44.(1 pt) setAlgebra30LogExpEqns/srw4 4 42.pg
Find the solution(s) of the logarithmic equation
54.(1 pt) setAlgebra30LogExpEqns/mec2.pg
If e2x + 1ex = +2, then x =
.
log2 (x2 + 1x − 68) = 2
55.(1 pt) setAlgebra30LogExpEqns/ur log 1 1.pg
For each of the following, find the base b if the graph of y = bx
contains the given point.
(−1, 0.333333333333333) b =
(−3, 0.008) b =
(−3, 1) b =
(4, 256) b =
(0.5, 1) b =
(1, 5) b =
(4, 16) b =
(−4, 0.0016) b =
(−1, 0.5) b =
(4, 81) b =
correct to four decimal places.
If there is more than one solution write them separated by commas.
x=
45.(1 pt) setAlgebra30LogExpEqns/srw4 4 45.pg
Find the solution(s) of the logarithmic equation
logx + log(x − 16) = log(12x)
correct to four decimal places.
If there is more than one solution write them separated by commas.
x=
3
56.(1 pt) setAlgebra30LogExpEqns/ur log 1 2.pg
Determine the smallest integer x that satisfies the given inequality.
√
3 2x > 28
x√>
x 4 > 138
x >√
6.4 3x > 26
xq
>
63.(1 pt) setAlgebra30LogExpEqns/problem4.pg
Solve for x:
9
log x = −1
8 4
x=
64.(1 pt) setAlgebra30LogExpEqns/problem5a.pg
Solve for x:
(log4 (log4 x)) = −4
3
x 4 > 26
x>
57.(1 pt) setAlgebra30LogExpEqns/beth2logfun.pg
Find the solution of the exponential equation
x=
65.(1 pt) setAlgebra30LogExpEqns/problem5.pg
(a) If log2 x = 7, then x =
.
(b) If log5 x = 4, then x =
.
10ex − 3 = 14
66.(1 pt) setAlgebra30LogExpEqns/problem6.pg
(a) If logx 81 = 2, then x =
.
(b) If logx 4096 = 4, then x =
.
in terms of logarithms, or correct to four decimal places.
x=
58.(1 pt) setAlgebra30LogExpEqns/beth3logfun.pg
Express the equation in exponential form
(a) log32 2 = 15 .
That is, write your answer in the form AB = C. Then
A=
,B=
, and C =
1 = −4.
(b) log2 16
That is, write your answer in the form DE = F. Then
D=
,E=
, and F =
67.(1 pt) setAlgebra30LogExpEqns/problem6a.pg
Solve for x:
log3 x7 = −9
x=
68.(1 pt) setAlgebra30LogExpEqns/problem7.pg
Solve for x:
59.(1 pt) setAlgebra30LogExpEqns/beth4logfun.pg
Express the equation in logarithmic form
(a) 24 = 16.
That is, write your answer in the form logA B = C. Then
A=
,B=
, and C =
(b) 10−4 = 0.000100.
That is, write your answer in the form logD = E. Then
D=
and E =
60.(1 pt) setAlgebra30LogExpEqns/problem1.pg
Solve for x:
8
7x−10
=
1
16
log2 x + log2 (x + 5) = log2 5
x=
69.(1 pt) setAlgebra30LogExpEqns/problem8.pg
Solve for x:
log4 x + log4 (x + 2) = 3
x=
70.(1 pt) setAlgebra30LogExpEqns/problem9.pg
Solve for x:
5x = 14
2x−2
x=
71.(1 pt) setAlgebra30LogExpEqns/problem10.pg
Solve for x:
x=
61.(1 pt) setAlgebra30LogExpEqns/problem2.pg
Solve for x:
x = log6 914
x=
72.(1 pt) setAlgebra30LogExpEqns/problem11.pg
Solve for x:
x = 10log10 6+log10 9
x=
Note: Your answer must be exact and in simplest form.
62.(1 pt) setAlgebra30LogExpEqns/problem3.pg
Solve for x:
9 · 25x−5 = 59
x=
73.(1 pt) setAlgebra30LogExpEqns/problem12.pg
Solve for x:
1
log4
=x
64
x=
Your answer must be exact an in simplest terms.
log (x6 ) = (log x)2
4
80.(1 pt) setAlgebra30LogExpEqns/solve difference of
Find the largest value of x that satisfies:
Note, there are 2 solutions, A and B, where A < B.
A=
B=
74.(1 pt) setAlgebra30LogExpEqns/problem13a.pg
Solve for x:
√
ln x
√ =6
ln x
Note, there are 2 possible solutions, A and B, where A < B.
A=
Is A a solution (yes or no)?
B=
Is B a solution (yes or no)?
log3 (x2 ) − log3 (x + 5) = 6
x=
81.(1 pt) setAlgebra30LogExpEqns/solve factoring.pg
The equation 3x2 e−7x − 5x3 e−7x = 0 has two roots.
The smaller root is
and the bigger root is
.
82.(1 pt) setAlgebra30LogExpEqns/solve in exponent.pg
Solve for x:
93x−2 = 67x−3
x=
.
83.(1 pt) setAlgebra30LogExpEqns/solve nested logs.pg
Solve for x:
(log4 (log4 x)) = 2
x=
84.(1 pt) setAlgebra30LogExpEqns/solve quad.pg
The equation e2x − 10ex + 16 = 0 has two solutions.
The smaller one is:
, and the larger one is:
.
75.(1 pt) setAlgebra30LogExpEqns/problem13.pg
Solve for x in terms of k.
x=
Find x if k = 3.
log10 x−4 − log10 x−5 = k.
76.(1 pt) setAlgebra30LogExpEqns/problem14.pg
Solve for x in terms of k.
log6 x + log6 (x + 3) = k.
x=
Find x if k = 8.
77.(1 pt) setAlgebra30LogExpEqns/problem15.pg
Solve for x in terms of k.
85.(1 pt) setAlgebra30LogExpEqns/solve sum of logs.pg
Solve for x:
log6 x + log6 (x + 3) = 2
There are two potential roots, A and B, where A ≤ B.
A=
B=
Is A actually a root? (yes or no)
Is B actually a root? (yes or no)
log2 x − log2 (x + 7) = log2 k.
x=
Find x if k = 1/8.
86.(1 pt) setAlgebra30LogExpEqns/simplify rules.pg
Solve for x:
√
3
2
x = (log6 108 − 3 log6 ( √ ) + 1 log6 9 + log6 128)
8
3
x=
.
Note: Your answer must be a decimal or fraction.
87.(1 pt) setAlgebra30LogExpEqns/ur le 1 6.pg
Solve the equation
ex+1 = ex + 4
x=
78.(1 pt) setAlgebra30LogExpEqns/problem16.pg
Solve for x in terms of a and b.
log x = −4(log a + log b) + 3log b8 + 9(log b − log a)
x=
79.(1 pt) setAlgebra30LogExpEqns/solve
Solve for x in each of the following.
(a) If logx 1024 = 34 , then x =
.
6x+9
= 2, then x =
.
(b) If 3
logs.pg
easy eqn.pg
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
5
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra31LogExpApplications due 1/31/10 at 2:00 AM
Your answer is P(t) =
(b) Use the function from part (a) to estimate the fox population
in the year 2008.
Your answer is (the answer must be an integer)
1.(1 pt) setAlgebra31LogExpApplications/srw4 2 1.pg
A bacteria culture initially contains 3000 bacteria and doubles
every half hour.
Find the size of the baterial population after 60 minutes.
9.(1 pt) setAlgebra31LogExpApplications/srw4 5 5.pg
The population of a certain city was 176000 in 1998, and the
observed relative growth rate is 4 percent per year.
(a) Find a function that models the population after t years.
Your answer is
(b) Find the projected population in the year 2004.
Your answer is
(c) In what year will the population reach 362730? Your answer
is
10.(1 pt) setAlgebra31LogExpApplications/srw4 5 8.pg
If a bateria culture starts with 5000 bateria and doubles every 25
minutes, how many minutes will it take the population to reach
41000?
11.(1 pt) setAlgebra31LogExpApplications/srw4 5 9.pg
A culture starts with 16500 bacteria. After one hour the count is
17600.
(a) Find the relative growth rate of the bacteria. Give your answer to at least 4 decimal places.
Your answer is
per hour.
(b) Find the number of bacteria after 2 hours.
Your answer is (your answer must be an integer)
(c) After how many hours will the number of bacteria double?
Your answer is
hours.
12.(1 pt) setAlgebra31LogExpApplications/srw4 5 10.pg
The count in a bateria culture was 200 after 15 minutes and 1100
after 40 minutes. What was the initial size of the culture?
Find the doubling period.
Find the population after 60
minutes.
When will the population reach 11000.
Find the size of the baterial population after 10 hours.
2.(1 pt) setAlgebra31LogExpApplications/srw4 2 2.pg
The doubling period of a baterial population is 15 minutes. At
time t = 80 minutes, the baterial population was 80000. What
was the initial population at time t = 0?
Find the size of the baterial population after 5 hours.
3.(1 pt) setAlgebra31LogExpApplications/srw4 2 7.pg
The half-life of Radium-226 is 1590 years. If a sample contains 200 mg, how many mg will remain after 1000 years?
4.(1 pt) setAlgebra31LogExpApplications/srw4 2 9.pg
The half-life of Palladium-100 is 4 days. After 12 days a sample of Palladium-100 has been reduced to a mass of 7 mg. What
was the initial mass (in mg) of the sample?
What is the mass 5 weeks after the start?
5.(1 pt) setAlgebra31LogExpApplications/srw4 2 13.pg
If 5000 dollars is invested in a bank account at an interest rate
of 7 per cent per year, find the amount in the bank after 15 years
if interest is compounded annually
Find the amount in the bank after 15 years if interest is compounded quaterly
Find the amount in the bank after 15 years if interest is compounded monthly
Finally, find the amount in the bank after 15 years if interest is
compounded continuously
13.(1 pt) setAlgebra31LogExpApplications/srw4 5 13.pg
An infectious strain of bacteria increases in number at a relative growth rate of 260 percent per hour. When a certain critical
number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level
is reached in 24 hours. How long will it take for the critical level
to be reached if the same person is infected with 10 bacteria?
Your answer is
hours.
14.(1 pt) setAlgebra31LogExpApplications/srw4 5 17.pg
The half-life of strontium-90 is 28 years. How long will it take
a 44 mg sample to decay to a mass of 11 mg?
Your answer is
years.
6.(1 pt) setAlgebra31LogExpApplications/srw4 4 57.pg
Find the time required for an investment of 5000 dollars to grow
to 6700 dollars at an interest rate of 7.5 percent per year, compounded quarterly.
Your answer is t =
years.
7.(1 pt) setAlgebra31LogExpApplications/srw4 5 2.pg
The pH scale for acidity is defined by pH = − log10 [H+ ] where
[H+ ]is the concentration of hydrogen ions measured in moles
per liter (M). A substance has a hydrogen ion concentration of
[H+ ] = 7.4 × 10−8M. Calculate the pH of the substance.
The pH is
8.(1 pt) setAlgebra31LogExpApplications/srw4 5 3.pg
The fox polulation in a certain region has a relative growth rate
of 6 percent per year. It is estimated that the population in the
year 2000 was 21800.
(a) Find a function that models the population t years after 2000
(t = 0 for 2000).
15.(1 pt) setAlgebra31LogExpApplications/srw4 5 21.pg
A wooden artifact from an ancient tomb contains 50 percent of
the carbon-14 that is present in living trees. How long ago was
the artifact made? (The half-life of carbon-14 is 5730 years.)
Your answer is
years.
1
16.(1 pt) setAlgebra31LogExpApplications/srw4 5 24.pg
The 1906 San Francisco earthquake had a magnitude of 8.3 on
the Richter scale. At the same time in South America there was
an eathquake with magnitude 4 that caused only minor damage. How many times more intense was the San Francisco earthquake than the South American one?
What is the rat population going to be in the year 2004 ?
17.(1 pt) setAlgebra31LogExpApplications/srw4 5 25.pg
A roasted turkey is taken from an oven when its temperature has
reached 185 Fahrenheit and is placed on a table in a room where
the temperature is 75 Fahrenheit.
(a) If the temperature of the turkey is 149 Fahrenheit after half
an hour, what is its temperature after 45 minutes?
Your answer is
Fahrenheit.
(b) When will the trukey cool to 100 Fahrenheit?
Your answer is
hours.
18.(1 pt) setAlgebra31LogExpApplications/srw4 5 29.pg
The pH reading of a sample of each substances is given. Calculate the hydrogen ion concentration of the substance.
(a) Vinegar: pH = 3.0.
Your answer is
.
(b) Milk: pH = 6.5.
Your answer is
.
19.(1 pt) setAlgebra31LogExpApplications/srw4 5 33.pg
If one earthquake is 30 times as intense as another, how much
larger is its magnitude on the Richter scale?
Your answer is
.
20.(1 pt) setAlgebra31LogExpApplications/srw4 5 37.pg
The 1985 Mexico City eqrthquake had a magnitude of 8.1 on
the Richter scale. The 1976 earthquake in Tangshan, China, was
1.26 times as intense. What was the magnitude of the Tangshan
eqrthquake?
.
Your answer is
21.(1 pt) setAlgebra31LogExpApplications/SRM srw4 2 1.pg
A bacteria culture initially contains 3000 bacteria and doubles
every half hour. The formula for the population is p(t) =
3000ekt for some constant k. (You will need to find k to answer
the following.)
Find the size of the baterial population after 20 minutes.
25.(1 pt) setAlgebra31LogExpApplications/jj1.pg
Students in a fifth-grade class were given an exam. During the
next 2 years, the same students were retested several times. The
average score was given by the model
24.(1 pt) setAlgebra31LogExpApplications/ur le 2 12.pg
A certain bacteria population is known to doubles every 150
minutes. Suppose that there are initially 190 bacteria.
What is the size of the population after t hours?
f (t) = 90 − 12log10 (t + 1),
0 ≤ t ≤ 24
where t is the time in months.
(a) What is the average score on the original exam?
(b) What was the average score after 6 months?
(c) What was the average score after 18 months?
26.(1 pt) setAlgebra31LogExpApplications/decay1.pg
At the beginning of an experiment, a scientist has 220 grams of
radioactive goo. After 255 minutes, her sample has decayed to
3.4375 grams.
What is the half-life of the goo in minutes?
Find a formula for G(t), the amount of goo remaining at time t.
G(t) =
How many grams of goo will remain after 74 minutes?
27.(1 pt) setAlgebra31LogExpApplications/decay2.pg
The half-life of Palladium-100 is 4 days. After 20 days a sample of Palladium-100 has been reduced to a mass of 1 mg. What
was the initial mass (in mg) of the sample?
What is the mass 6 weeks after the start?
28.(1 pt) setAlgebra31LogExpApplications/growth1.pg
The rule of 72 states that if an investment earns P % interest per
year, it will take approximately 72/P years for your money to
double.
You invest 5000 at 1.8 % interest annually.
According to the rule of 72, what is the doubling time, in years,
for this investment
Use the doubling time to find a formula for V (t), the value of
your investment at time t. V (t) =
According to the doubling time, how much will your investment
be worth after 49 years?
Use the compound interest formula to find how much the investment will be worth after 49 years.
.
You may notice that your two values for the investment’s worth
after 49 years are different. That is because the doubling time
you found with the rule of 72 is only an approximation. If the
approximation were better, the two values would be the same.
Find the size of the baterial population after 4 hours.
22.(1 pt) setAlgebra31LogExpApplications/SRM srw4 2 2.pg
The doubling period of a baterial population is 10 minutes. At
time t = 90 minutes, the baterial population was 50000. For
some constant A, the formula for the population is p(t) = Aekt
ln 2
where k =
. What was the initial population at time t = 0?
10
Find the size of the baterial population after 5 hours.
23.(1 pt) setAlgebra31LogExpApplications/ur le 2 11.pg
The rat population in a major metropolitan city is given by the
formula n(t) = 26e0.015t where t is measured in years since 1993
and n(t) is measured in millions.
What was the rat population in 1993 ?
29.(1 pt) setAlgebra31LogExpApplications/growth2.pg
The doubling period of a baterial population is 20 minutes. At
time t = 110 minutes, the baterial population was 90000. What
was the initial population at time t = 0?
2
Find the size of the baterial population after 3 hours.
To the nearest year, When will the two accounts have the same
balance?
The two accounts will have the same balance after
years.
30.(1 pt) setAlgebra31LogExpApplications/infection1.pg
The town of Sickville, with a population of 29500 is exposed to
the Blue Moon Virus, against which there is no immunity. The
number of people infected when the virus is detected is 30. Suppose the number of infections grows logistically, with k = 0.22.
Find A.
Find the formula for the number of people infected after t days.
N(t) =
Find the number of people infected after 15 days.
34.(1 pt) setAlgebra31LogExpApplications/problem9.pg
If log p = x and log q = y, evaluate the following in terms of x
and y:
7 −7
(a) log (p
pq )=
(b) log p−8 q−8 =
p2
(c) log 8 =
q
log p6
(d)
=
log q−4
(e) (log p1 )1 =
31.(1 pt) setAlgebra31LogExpApplications/terminalvelocity.pg
Let P(t) = 40(1 − e−kt ) + 59 represent the expected score for a
student who studies t hours for a test. Suppose k = 0.11 and test
scores must be integers.
What is the highest score the student can expect?
If the student does not study, what score can he expect?
35.(1 pt) setAlgebra31LogExpApplications/problem11.pg
The pH scale for acidity is defined by pH = − log10 [H + ] where
[H+ ]is the concentration of hydrogen ions measured in moles
per liter (M).
A solution has a pH of 1.
Calculate the concentration of hydrogen ions in moles per liter
(M).
moles per liter.
The concentration of hydrogen ions is
32.(1 pt) setAlgebra31LogExpApplications/cooling.pg
You are taking a road trip in a car without A/C. The temperture
in the car is 102 degrees F. You buy a cold pop at a gas station.
Its initial temperature is 45 degrees F. The pop’s temperature
reaches 60 degrees F after 24 minutes.
Given that
T −A
= e−kt
T0 − A
where T = the temperature of the pop at time t.
T0 = the initial temperature of the pop.
A = the temperature in the car.
k = a constant that corresponds to the warming rate.
and t = the length of time that the pop has been warming up.
How long will it take the pop to reach a temperature of 70.5
degrees F?
It will take
minutes.
36.(1 pt) setAlgebra31LogExpApplications/problem12.pg
If 2000 dollars is invested in a bank account at an interest
rate of 9 per cent per year, compounded continuously. How
many years will it take for your balance to reach 10000 dollars?
NOTE: Give your answer to the nearest tenth of a year.
37.(1 pt) setAlgebra31LogExpApplications/radioactive dye.pg
You go to the doctor and he gives you 10 milligrams of radioactive dye. After 12 minutes, 6 milligrams of dye remain in your
system. To leave the doctor’s office, you must pass through a
radiation detector without sounding the alarm. If the detector
will sound the alarm if more than 2 milligrams of the dye are in
your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived? Give your
answer to the nearest minute.
minutes at the doctor’s office.
You will spend
33.(1 pt) setAlgebra31LogExpApplications/investing equity.pg
7000 dollars is invested in a bank account at an interest rate
of 9 per cent per year, compounded continuously. Meanwhile,
18000 dollars is invested in a bank account at an interest rate of
4 percent compounded annually.
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
3
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra32EqnSystems due 2/1/10 at 2:00 AM
x=
y=
1.(1 pt) setAlgebra32EqnSystems/beth1sys2var.pg
Find all solutions of the system
7.(1 pt) setAlgebra32EqnSystems/srw8 1 3.pg
Use the substitution method to solve the system
y + x2 = 6x,
y + 6x = 36.
The solution of the system is:
If there is more than one point, type the points separated by a
comma (e.g.: (1,2),(3,4)).
If the system has no solutions, type none in the answer blank.
Your answer is
x1 =
, y1 =
x2 =
, y2 =
2.(1 pt) setAlgebra32EqnSystems/beth2sys2var.pg
Use the method of elimination to solve the system
y = x2 ,
y = 7x − 10.
and
with x1 < x2 .
8.(1 pt) setAlgebra32EqnSystems/srw8 1 7.pg
Use the substitution method to solve the system
−4x − 1y = 7,
4x + 3y = −13.
x + y2 = 0,
2x + 5y2 = 27.
Your answer is
If there is more than one point, type the points separated by a
comma (e.g.: (1,2),(3,4)).
If the system has no solutions, type none in the answer blank.
Your answer is
x1 =
, y1 =
x2 =
, y2 =
3.(1 pt) setAlgebra32EqnSystems/beth3sys2var.pg
Use a calculator solve the system
9.(1 pt) setAlgebra32EqnSystems/srw8 1 9.pg
Use the elimination method to find all solutions of the system
x2 + y2 = 21,
x + y = 1.
5x + 2y = 17,
7x + 3y = 24.
Your answer is
If there is more than one point, write the points separated by a
comma (e.g.: (1,2),(3,4)).
If the system has no solutions, type none in the answer blank.
Your answer is
x=
y=
10.(1 pt) setAlgebra32EqnSystems/srw8 1 11.pg
Use the elimination method to find all solutions of the system
4.(1 pt) setAlgebra32EqnSystems/beth4sys2var.pg
Solve the system
x2 − 2y = 6,
x2 + 5y = −1.
2x − 6y = 15,
−3x + 9y = −24.
The two solutions of the system are:
the one with x < 0 is
x=
y=
the one with x > 0 is
x=
y=
Your answer is
If there is more than one point, type the points separated by a
comma (e.g.: (1,2),(3,4)).
If the system has no solutions, type none in the answer blank.
5.(1 pt) setAlgebra32EqnSystems/beth5sys2var.pg
Use the method of substitution to solve the system
11.(1 pt) setAlgebra32EqnSystems/srw8 1 13.pg
Use the elimination method to find all solutions of the system
x2 + y2 = 11,
x + y = 1.
3x2 − y2 = 11,
x2 + 4y2 = 8.
Your answer is
If there is more than one point, write the points separated by a
comma (e.g.: (1,2),(3,4)).
If there is no solution, type none in the answer blank.
The four solutions of the system are: (−a, −b), (−a, b),(a, −b),
and (a, b) with positive numbers
a=
and b =
.
12.(1 pt) setAlgebra32EqnSystems/srw8 1 15.pg
Use the elimination method to find all solutions of the system
6.(1 pt) setAlgebra32EqnSystems/srw8 1 1.pg
Use the substitution method to solve the system
Your answer is
and
with y1 < y2 .
−x + y = −1,
4x − 3y = 2.
1
x2 − y2 + 3 = 0,
2x2 + y2 − 4 = 0.
The four solutions of the system are: (−a, −b), (−a,b), (a, −b),
and (a, b) with positive numbers
a=
and b =
.
The four solutions of the system are:
the one with x < 0, y < 0 is
x=
y=
the one with x < 0, y > 0 is
x=
y=
the one with x > 0, y < 0 is
x=
y=
the one with x > 0, y > 0 is
x=
y=
13.(1 pt) setAlgebra32EqnSystems/srw8 1 17.pg
Use the elimination method to find all solutions of the system
y + x2 = 4x,
y + 4x = 16.
The solution of the system is:
x=
y=
14.(1 pt) setAlgebra32EqnSystems/srw8
Find all solutions of the system
1 20.pg
18.(1 pt) setAlgebra32EqnSystems/srw8
Use a calculator solve the system
y = 100 − x2,
y = x2 − 100.
The two solutions of the system are:
the one with x < 0 is
x=
y=
the one with x > 0 is
x=
y=
1 31.pg
3x + 2y = 2,
x − 2y = 6.
Your answer is
x=
y=
19.(1 pt) setAlgebra32EqnSystems/srw8
Use a calculator solve the system
1 35.pg
x2 + y2 = 3,
x + y = 1.
15.(1 pt) setAlgebra32EqnSystems/srw8 1 21.pg
Use the substitution method to find all solutions of the system
Your answer is
x1 =
x2 =
y = x + 2,
xy = 3.
The solutions of the system are:
x1 =
, y1 =
and
x2 =
, y2 =
with x1 < x2 .
, y1 =
, y2 =
and
with x1 < x2 .
20.(1 pt) setAlgebra32EqnSystems/srw8 1 49.pg
Solve the system
x2 + xy = 1,
xy + y2 = 575.
Your answer is
, y1 =
and
x1 =
x2 =
, y2 =
with x1 < x2 .
16.(1 pt) setAlgebra32EqnSystems/srw8 1 25.pg
Use the elimination method to find all solutions of the system
x2 + y2 = 6,
x2 − y2 = 3.
The four solutions of the system are:
the one with x < 0, y < 0 is
x=
y=
the one with x < 0, y > 0 is
x=
y=
the one with x > 0, y < 0 is
x=
y=
the one with x > 0, y > 0 is
x=
y=
21.(1 pt) setAlgebra32EqnSystems/ur ab 9 1.pg
Solve the system
2
x − 12x + y2=−11
4x − 3y= 49
x=
y=
17.(1 pt) setAlgebra32EqnSystems/srw8 1 27.pg
Use the elimination method to find all solutions of the system
23.(1 pt) setAlgebra32EqnSystems/sw7 1 7.pg
Use the elimination method to find all solutions of the system
x2 + y2 = 9,
x2 − y2 = 2.
22.(1 pt) setAlgebra32EqnSystems/ur ab 9 2.pg
Solve the system
lnx y = 5
log8 y = 2 log8 x + 2
x=
y=
5x + 2y = 19,
7x + 3y = 27.
2
28.(1 pt) setAlgebra32EqnSystems/sw7 2 7.pg
Solve the system
−x + y = 3,
4x − 3y = −12.
If the system has infinitely many solutions, express your answer
in the form x = x and y as a function of x
Your answer is
x=
y=
Your answer is
x=
y=
24.(1 pt) setAlgebra32EqnSystems/sw7 1 9.pg
Use the elimination method to find all solutions of the system
x2 − 2y = 7,
x2 + 5y = −14.
The two solutions of the system are:
the one with x < 0 is
x=
y=
the one with x > 0 is
x=
y=
29.(1 pt) setAlgebra32EqnSystems/sw7 2 11.pg
Solve the system
x + 2y = −3,
5x − y = −15.
If the system has infinitely many solutions, express your answer
in the form x = x and y as a function of x
Your answer is
x=
y=
25.(1 pt) setAlgebra32EqnSystems/sw7 1 13.pg
Use the elimination method to find all solutions of the system
y + x2 = 4x,
y + 4x = 16.
30.(1 pt) setAlgebra32EqnSystems/sw7 2 13.pg
Solve the system
3x + 2y = −7,
x − 2y = −5.
If the system has infinitely many solutions, express your answer
in the form x = x and y as a function of x
Your answer is
x=
y=
The solution of the system is:
x=
y=
26.(1 pt) setAlgebra32EqnSystems/sw7
Find all solutions of the system
1 16.pg
y = 4 − x 2,
y = x2 − 4.
The two solutions of the system are:
the one with x < 0 is
x=
y=
the one with x > 0 is
x=
y=
31.(1 pt) setAlgebra32EqnSystems/sw7 2 15.pg
Solve the system
x + 4y = −5,
3x + 12y = −15.
If the system has infinitely many solutions, express your answer
in the form x = x and y as a function of x
Your answer is
x=
y=
27.(1 pt) setAlgebra32EqnSystems/sw7 1 21.pg
Use the elimination method to find all solutions of the system
32.(1 pt) setAlgebra32EqnSystems/sw7
Solve the system
x2 + y2 = 7,
x2 − y2 = 3.
2 17.pg
2x − 6y = 12,
−3x + 9y = −18.
The four solutions of the system are:
the one with x < 0, y < 0 is
x=
y=
the one with x < 0, y > 0 is
x=
y=
the one with x > 0, y < 0 is
x=
y=
the one with x > 0, y > 0 is
x=
y=
If the system has infinitely many solutions, express your answer
in the form x = x and y as a function of x
Your answer is
x=
y=
33.(1 pt) setAlgebra32EqnSystems/sw7 2 19.pg
Solve the system
6x + 4y = −6,
9x + 6y = −9.
If the system has infinitely many solutions, express your answer
in the form x = x and y as a function of x
3
Your answer is
x=
y=
requires 3 aspirin and 1 hours of sleep to recover. Each hour
with the chemistry tutor requires 1 aspirin and 3 hour of sleep to
recover. Charlie has only 30 aspirin left, and can only afford to
sleep for 15 hours this weekend. If each hour of math tutoring
will improve his grade by 2 points and each hour of chemistry
tutoring will improve his grade by 3 points, how many hours
should he spend with each tutor in order to improve his grades
the most?
Charlie should spend
hours with his math tutor and
hours with his chemistry tutor to improve his grades by a total
of
points.
34.(1 pt) setAlgebra32EqnSystems/linearsystem 2 2.pg
Solve the following system of equations. If there are no solutions, type ”No Solution” for both x and y. If there are infinitely
many solutions, type ”x” for x, and an expression in terms of x
for y.
−1x − 1y = 5
−2x + 3y = −7
.
x=
.
y=
39.(1 pt) setAlgebra32EqnSystems/beth1.pg
A company that makes thing-a-ma-bobs has a start up cost of $
29473. It costs the company $ 1.78 to make each thing-a-mabob. The company charges $ 3.76 for each thing-a-ma-bob. Let
x denote the number of thing-a-ma-bobs produced.
Write the cost function for this company.
C(x) =
35.(1 pt) setAlgebra32EqnSystems/linearsystem 2 2a.pg
Solve the following system of equations. If there are no solutions, type ”No Solution” for both x and y. If there are infinitely
many solutions, type ”x” for x, and an expression in terms of x
for y.
2x − 1y = −7
6x − 3y = 21
x=
.
y=
.
Write the revenue function for this company.
R(x) =
What is the minumum number of thing-a-ma-bobs that
the company must produce and sell to make a profit?
36.(1 pt) setAlgebra32EqnSystems/circleparab.pg
Solve the following system of equations.
If there are no solutions, type ”No Solution” for all x and y values.
If there are solutions, enter them in increasing order of the x
values. Type ”No Solution” for any entry that you do not need.
x2 + y2 = 64
y = 1.6x2 − 10
. y1 =
.
x1 =
x2 =
. y2 =
.
x3 =
. y3 =
.
x4 =
. y4 =
.
40.(1 pt) setAlgebra32EqnSystems/beth2.pg
Find the point of equilibrium for the following supply and demand equations where x is number of units and p is the price
per unit.
Demand: p = 281 − 0.000100x
Supply: p = 251 + 0.000200x
Number of units for equilibrium =
Price per unit at equilibrium =
41.(1 pt) setAlgebra32EqnSystems/srw8 1 41.pg
A rectangle has an area of 153 cm2 and a perimeter of 52 cm.
What are its dimensions?
Its length is
Its width is
42.(1 pt) setAlgebra32EqnSystems/jj1.pg
You are offered two different sales jobs. The first company offers a straight commission of 5% of the sales. The second company offers a salary of $ 330 per week plus 4% of the sales. How
much would you have to sell in a week in order for the straight
commission offer to be at least as good?
37.(1 pt) setAlgebra32EqnSystems/parabline.pg
Solve the following system of equations.
If there are no solutions, type ”No Solution” for all x and y values.
If there is only one solution, use x1 and y1 for your answers.
Type ”No Solution” for the other x and y values.
If there are two solutions, use x1 and y1 for the solution with the
smallest x value.
6x − 1y = −2
y = 2x2 − 4x + 7
x1 =
. y1 =
.
x2 =
. y2 =
.
43.(1 pt) setAlgebra32EqnSystems/sw7 1 37.pg
A rectangle has an area of 96 cm2 and a perimeter of 44 cm.
What are its dimensions?
Its length is
Its width is
44.(1 pt) setAlgebra32EqnSystems/sw7 1 39.pg
The perimeter of a rectangle is 70 and its diagonal is 25. Find
its length and width.
Its length is
Its width is
38.(1 pt) setAlgebra32EqnSystems/tutoring.pg
Charlie is trying to allocate his study time this weekend. He can
spend time working with either his math tutor or his chemistry
tutor to prepare for next week’s tests. His math tutor charges $
20 per hour. His chemistry tutor charges $ 40 per hour. He has
$ 240 to spend on tutoring, but each hour with the math tutor
4
Your answer is
number of children equals
number of adults equals
45.(1 pt) setAlgebra32EqnSystems/sw7 1 40.pg
A circular piece of sheet metal has a diameter of 20 in. The
edges are to be cut off to form a rectangle of area 30 in2 (see the
figure below). What are the dimensions of the rectangle?
Its length is
Its width is
49.(1 pt) setAlgebra32EqnSystems/sw7 2 37.pg
A man flies a small airplane from Fargo to Bismarck, North
Dakota — a distance of 180 miles. Because he is flying into a
head wind, the trip takes him 2 hours. On the way back, the
wind is still blowing at the same speed, so the return trip takes
only 1 hour 12 minutes. What is his speed in still air, and how
fast is the wind blowing?
Your answer is
his speed equals
the wind speed equals
50.(1 pt) setAlgebra32EqnSystems/sw7 2 45.pg
A man invests his savings in two accounts, one paying 6 percent
and the other paying 10 percent simple interest per year. He
puts twice as much in the lower-yielding account because it is
less risky. His annual interest is 9416 dollars. How much did he
invest at each rate?
Your answer is
total in the account paying 6 percent interest is
total in the account paying 10 percent interest is
46.(1 pt) setAlgebra32EqnSystems/sw7 2 33.pg
Find two numbers a and b whose sum a + b is 1 and whose
difference a − b is -9.
Your answer is
a=
b=
47.(1 pt) setAlgebra32EqnSystems/sw7 2 35.pg
A man has 21 coins in his pocket, all of which are dimes and
quarters. If the total value of his change is 315 cents, how many
dimes and how many quarters does he have?
Your answer is
number of dimes equals
number of quarters equals
51.(1 pt) setAlgebra32EqnSystems/investing.pg
Country Day’s scholarship fund receives a gift of $ 155000. The
money is invested in stocks, bonds, and CDs. CDs pay 2.5 %
interest, bonds pay 4.3 % interest, and stocks pay 8.7 % interest.
Country day invests $ 35000 more in bonds than in CDs. If the
annual income form the investments is $ 9295 , how much was
invested in each vehicle?
in stocks.
Country Day invested $
Country Day invested $
in bonds.
Country Day invested $
in CDs.
52.(1 pt) setAlgebra32EqnSystems/modelling.pg
Given the table below, find a cubic equation in standard form for
g(x).
48.(1 pt) setAlgebra32EqnSystems/sw7 2 36.pg
The admission fee at an amusement park is 1.5 dollars for children and 4 dollars for adults. On a certain day, 327 people entered the park, and the admission fees collected totaled 978 dollars. How many children and how many adults were admitted?
x
-2 2
8
-4
g(x) -59 21 1761 -351
g(x) =
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
5
.
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra33SystemsIneq due 2/2/10 at 2:00 AM
3.(1 pt) setAlgebra33SystemsIneq/feasible region 3.pg
Given the system of inequalities below, determine the shape of
the feasible region and find the vertices of the feasible region.
Give the shape as ”triangle”, ”quadrilateral”, ”pentagon”, or
”unbounded”. Report your vertices starting with the one which
has the smallest x-value. If more than one vertex has the same,
smallest x-value, start with the one that has the smallest y-value.
Proceed clockwise from the first vertex. Leave any unnecessary
answer spaces blank. Also give the value of the objective function P = −7x + 2y for each vertex.
x + y ≥ 13
3y − x ≤ 35
4x − y ≥ 2
x≥0
y≥0
.
The shape of the feasible region is (a)
The first vertex is ( , ). P =
The second vertex is ( , ).P =
The third vertex is ( , ).P =
The fourth vertex is ( , ).P =
The fifth vertex is ( , ). P =
1.(1 pt) setAlgebra33SystemsIneq/feasible region 1.pg
Given the system of inequalities below, determine the shape of
the feasible region and find the vertices of the feasible region.
Give the shape as ”triangle”, ”quadrilateral”, or ”unbounded”.
Report your vertices starting with the one which has the smallest x-value. If more than one vertex has the same, smallest xvalue, start with the one that has the smallest y-value. Proceed
clockwise from the first vertex. Leave any unnecessary answer
spaces blank.
x+y≥ 6
3x + y ≥ 11
x≥0
y≥0
.
The shape of the feasible region is
The first vertex is ( , ).
The second vertex is ( , ).
The third vertex is ( , ).
The fourth vertex is ( , ).
2.(1 pt) setAlgebra33SystemsIneq/feasible region 2.pg
Given the system of inequalities below, determine the shape of
the feasible region and find the vertices of the feasible region.
Give the shape as ”triangle”, ”quadrilateral”, or ”unbounded”.
Report your vertices starting with the one which has the smallest x-value. If more than one vertex has the same, smallest xvalue, start with the one that has the smallest y-value. Proceed
clockwise from the first vertex. Leave any unnecessary answer
spaces blank.
x+y≤ 7
2x + y ≥ 10
x + 3y ≥ 12
x≥0
y≥0
.
The shape of the feasible region is
The first vertex is ( , ).
The second vertex is ( , ).
The third vertex is ( , ).
The fourth vertex is ( , ).
4.(1 pt) setAlgebra33SystemsIneq/linear programming 1.pg
Given the system of inequalities below, determine the shape of
the feasible region and find the vertices of the feasible region.
Report your vertices starting with the one which has the smallest x-value. If more than one vertex has the same, smallest xvalue, start with the one that has the smallest y-value. Proceed
clockwise from the first vertex. Leave any unnecessary answer
spaces blank.
x+y≤ 8
8x + y ≥ 12
x≥0
y≥0
The feasible region is
.
The first vertex is ( , ).
The second vertex is ( , ).
The third vertex is ( , ).
The fourth vertex is ( , ).
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
1
ARNOLD PIZER
rochester problib from CVS June 25, 2004
Rochester WeBWorK Problem Library
1.(1 pt) setAlgebra34Matrices/id

8 -8
 8 6

 6 -8

 5 -5
A=
 5 8

 -2 1

 5 -3
4 5
then A53 is
.
WeBWorK assignment Algebra34Matrices due 2/3/10 at 2:00 AM


entry.pg
If
Then 2A − 4B =
-1 1 6
-2 1 9 

-3 8 9 

-3 9 6 

4 8 6 

1 2 -5 

9 3 -9 
1 -6 -7


and 5AT =
2.(1 pt) setAlgebra34Matrices/size.pg
-9 -4
A=
1 1
.
then the size of A is
7.(1
pt) setAlgebra34Matrices/matrixmult3.pg



-3 0 2
4 4 4
If A = 0 1 4  and B = -2 0 -3 
1 1 -3
4 1 3


3.(1 pt) setAlgebra34Matrices/defined ops.pg
1 6
A=
-1 -5
0 -8 -8 0
B=
2 5 9 -7
C = 5 -5
Then AB =

and BA =
and AT =


9.(1 pt) setAlgebra34Matrices/sw7 4 1.pg
Given
the matrices
2 3 5
4 -3 -2
B=
,
C=
,
2 1 2
5 5 -1
find B +C.
Write B +C as a11 a12 a13
B +C =
.
a21 a22 a23
Input your answer below:
a11 =
a12 =
a13 =
a21 =


5.(1
pt) setAlgebra34Matrices/scalarmult3a.pg



0 -1 1
2 -1 -2
If A = -3 4 -4  and B = -2 4 1 
1 2 3
-1 0 -3

If
4.(1
pt) setAlgebra34Matrices/scalarmult3.pg



-4 4 2
-2 4 -4
If A = 1 -4 0  and B = 2 -1 1 
4 1 4
-1 -2 -1




8.(1 pt) setAlgebra34Matrices/product size.pg


1 -4 1 2 -8
A =  4 -2 6 0 -7 
4 4 0 -1 0


9
 -8 



B=
 2 
 9 
-7
then the size of AB is
and the size of BA is
.
NOTE: If either of the products is not defined, type UNDEFINED for you answer.
then decide if each of the following operations is defined (answer yes or no)
A+B
A +C
B +C
AB
BA
AC
CA
BC
CB
Then 2A + B =

6.(1pt) setAlgebra34Matrices/matrixmult2.pg
3
0
1
-2-2i
If A =
and B =
3+4i -3-i
1+2i -2+4i
Then AB =
and BA =
If
If


1
14.(1 pt) setAlgebra34Matrices/sw7
4 17.pg
3 -2
Given the matrix A =
, find A3 .
0 -2
a11 a12
Write A3 as
.
a21 a22
Input your answer below:
a11 =
a12 =
a21 =
a22 =
a22 =
a23 =
10.(1 pt) setAlgebra34Matrices/sw7 4 3.pg
Given
the matrices -1 -2 4
0 -1 5
B=
,
C=
,
1 1 -3
2 -3 -2
find C −
B. Write C − B as a11 a12 a13
C−B =
.
a21 a22 a23
Input your answer below:
a11 =
a12 =
a13 =
a21 =
a22 =
a23 =
15.(1 pt) setAlgebra34Matrices/cubing
2x2.pg
-3 -4
Given the matrix A =
, find A3 .
0 -4
A3 =
.
11.(1 pt) setAlgebra34Matrices/sw7 4 5.pg
Given
the matrices 3 -1 -2
2 2 5
B=
,
C=
,
-2 -5 -2
4 -2 3
find 3B + 2C.
Write 3B + 2C as
a11 a12 a13
3B + 2C =
.
a21 a22 a23
Input your answer below:
a11 =
a12 =
a13 =
a21 =
a22 =
a23 =
12.(1 pt) setAlgebra34Matrices/sw7 4 10.pg
Given
the matrices -5 4 -2
3 4 0
B=
,
C=
,
2 -1 4
-2 -2 1
can the operation BC be performed?
Your answer is (input Yes or No)
Note: You have only one chance to input your answer.
13.(1 pt) setAlgebra34Matrices/sw7 4 11.pg
Given the matrices


2 -1 -3
-3 3 -2
B=
,
F = 0 -1 -2  ,
0 2 0
2 2 1
can the operation BF be performed?
Your answer is (input Yes or No)
If youranswer is Yes, calculate
BF. Write BF as
a11 a12 a13
.
BF =
a21 a22 a23
Input your answer below:
a11 =
a12 =
a13 =
a21 =
a22 =
a23 =
16.(1 pt) setAlgebra34Matrices/sw7
5 3.pg
2 1
Given the matrix
,
9 5
(a) does the inverse of the matrix exist?
Your answer is (input Yes or No) : a11
(b) if your answer is yes, write it as
a21
Input your answer below:
a11 =
a12 =
a21 =
a22 =
a12
a22
.
17.(1 pt) setAlgebra34Matrices/sw7
5 5.pg
2
9
Given the matrix
,
13 58
(a) does the inverse of the matrix exist?
Your answer is (input Yes or No) : a11
(b) if your answer is yes, write it as
a21
Input your answer below:
a11 =
a12 =
a21 =
a22 =
a12
a22
.
18.(1 pt) setAlgebra34Matrices/inverse2x2.pg
-1 2
Given the matrix
,
4 9
(a) does the inverse of the matrix exist?
Your answer is (input Yes or No) :
(b)
is yes, write the inverse here:
if your answer
.
19.(1 pt) setAlgebra34Matrices/inverse2x2a.pg
2 13
Given the matrix
,
1 6
(a) does the inverse of the matrix exist?
Your answer is (input Yes or No) :
2
(b)
is yes,
if your answer
.
write the inverse here:
Input your answer below:
a11 =
a12 =
a13 =
a21 =
a22 =
a23 =
a31 =
a32 =
a33 =
20.(1 pt) setAlgebra34Matrices/inverse3x3.pg


3 0 1
Given the matrix  -1 1 -1  ,
1 1 0
(a) does the inverse of the matrix exist?
Your answer is (input Yes or No):
(b)
 if your answer isYes, write the inverse as
a11 a12 a13
 a21 a22 a23  .
a31 a32 a33
Input your answer below:
a11 =
a12 =
a13 =
a21 =
a22 =
a23 =
a31 =
a32 =
a33 =
24.(1 pt) setAlgebra34Matrices/sw7
6 3.pg
0 4
Given the matrix
,
3 5
(a) find its determinant;
Your answer is :
(b) does the matrix have an inverse?
Your answer is (input Yes or No) :
25.(1 pt) setAlgebra34Matrices/determinant
2x2.pg
-1 -1
Given the matrix A =
, find its determinant.
2 4
The determinant of A is
If
21.(1 pt) setAlgebra34Matrices/inverse3x3a.pg


3 4 1
Given the matrix  -1 1 -1  ,
1 3 0
(a) does the inverse of the matrix exist?
Your answer is (input Yes or No):
(b)
 if your answer is Yes, write
 the inverse here:

A=
find |A| =
.
-4-i -1+2i
2-4i 1+i
28.(1 pt) setAlgebra34Matrices/sw7

6 17.pg
1 0 -5
Given the matrix  0 -5 -1  ,
2 0 1
(a) find its determinant;
Your answer is :
(b) does the matrix have an inverse?
Your answer is (input Yes or No) :
.
23.(1 pt) setAlgebra34Matrices/sw7
5 13.pg


1 2 3
Given the matrix  -3 3 -1  ,
-9 0 -11
(a) does the inverse of the matrix exist?
Your answer is (input Yes or No):
(b)
 if your answer isYes, write the inverse as
a11 a12 a13
 a21 a22 a23  .
a31 a32 a33
.
2x2a.pg
27.(1 pt) setAlgebra34Matrices/sw7

6 15.pg
0 0 -3
Given the matrix  2 0 3  ,
0 -2 3
(a) find its determinant;
Your answer is :
(b) does the matrix have an inverse?
Your answer is (input Yes or No) :
22.(1 pt) setAlgebra34Matrices/inverse3x3b.pg


1 2 3
Given the matrix  1 5 1  ,
-2 8 -14
(a) does the inverse of the matrix exist?
Your answer is (input Yes or No):
(b)
 the inverse here:
 if your answer is Yes, write

26.(1 pt) setAlgebra34Matrices/determinant
29.(1 pt) setAlgebra34Matrices/det

inv
0 -1 1
Given the matrix  5 0 4  ,
0 -1 -1
(a) find its determinant;
Your answer is :
(b) does the matrix have an inverse?
Your answer is (input Yes or No) :
3
3x3.pg
30.(1 pt) setAlgebra34Matrices/det
inv 3x3a.pg


3 0 4
Given the matrix  0 -5 3  ,
-1 0 -5
(a) find its determinant;
Your answer is :
(b) does the matrix have an inverse?
Your answer is (input Yes or No) :
32.(1 pt) setAlgebra34Matrices/determinant
3x3a.pg


a 6 4
Given the matrix A = a -8 1  ,
3 -2 a
find all values of a that make the |A| = 0.
Give your answers in increasing order.
a can be
,
, or
.
Note: Leave any unneeded answer spaces blank.
31.(1 pt) setAlgebra34Matrices/determinant
3x3.pg


3 -3 3
Given the matrix  0 0 1  , find its determinant.
0 4 -3
The determinant is :
33.(1 pt) setAlgebra34Matrices/determinant
3x3b.pg


0 4 4
Given the matrix  3 1 -2  , find its determinant. Do not
1 3 3
use a caculator.
The determinant is :
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
4
ARNOLD PIZER
rochester problib from CVS June 25, 2004
Rochester WeBWorK Problem Library
WeBWorK assignment Algebra35SystemMatrices due 2/4/10 at 2:00 AM
6.(1 pt) setAlgebra35SystemMatrices/matrix
Write the system of equations
1.(1 pt) setAlgebra35SystemMatrices/sw7

 3 7.pg
1 -2 3
Given the matrix A = 0 1 -4  ,
0 0 0
(a) determine whether the matrix A is in echelon form;
Your answer is (input Yes or No)
(b) determine whether the matrix A is in reduced echelon form;
Your answer is (input Yes or No)
4x + 2y + 1z = −4
5x + 2y − 2z = −1
−5x + 3y − 2z = 1
as
rewrite
 a matrix equation,that
 is, 
 it in the
 form
x

  y  =
.
z
2.(1 pt) setAlgebra35SystemMatrices/sw7

 3 9.pg
1 0 2
Given the matrix A = 0 1 0  ,
0 0 0
(a) determine whether the matrix A is in echelon form;
Your answer is (input Yes or No)
(b) determine whether the matrix A is in reduced echelon form;
Your answer is (input Yes or No)
7.(1 pt) setAlgebra35SystemMatrices/3x3 test.pg
Given the augmented matrix below, determine if the associated
system of equations is independent, dependent, or inconsistent.



-1
-4 -2
-8
 -40 -48 18   -88 
5
4
-4
6
The system is
.
3.(1 pt)
setAlgebra35SystemMatrices/augmentedmatrix.pg

1 0 1 0 2
 0 1 2 0 2 

Let A =
 0 0 0 1 -4  .
0 0 0 0 0
Is the matrix in echelon form? (input Yes or No)
Is the matrix in reduced echelon form? (input Yes or No)
8.(1 pt) setAlgebra35SystemMatrices/classify 3x3.pg
Given the augmented matrix below, determine if the associated
system of equations is independent, dependent, or inconsistent.
If the system is independent, give the solution. If the system is
dependent, label each variable as ”free” or ”fixed”. If the system
is inconsistent, label each variable ”No Solution”.



15
5
-3
9
 120 -100 -17   -3 
15 -15 -2
-6
The system is
.
The solution to the system is: (
,
,
)
If this matrix were the augmented matrix for a system of linear equations, would the system be inconsistent, dependent, or
independent?
4.(1 pt) setAlgebra35SystemMatrices/ID row ops.pg
Identify the elementary row operation used below. Write your
answer with one space between every character.






 
0 -9 -8 -8 1

-7 7 2 -5 2 
 

-3 -5 -9 1 -3 
→
 
-5 1 -3 -8 -2  
-7 -9 2 7 9
The system is
.

0
-9
-8 -8 1
-7
7
2 -5 2 

-24 -40 -72 8 -24 

-5
1
-3 -8 -2 
-7 -9
2
7
9
9.(1 pt) setAlgebra35SystemMatrices/classify 4x4.pg
Given the augmented matrix below, determine if the associated
system of equations is independent, dependent, or inconsistent.



1 -3 7 -4
0
 -4 -4 -40 -4   -16 



 1 -7 4 -9   -4 
-2 4 -9 6
-5
.
The system is
5.(1 pt) setAlgebra35SystemMatrices/ID row ops 2.pg
Identify the elementary row operation used below. Write your
answer with one space between every character.
-7 -9 6 9
-7 -9 6
9
 3 -1 9 3   3 -1 9
3 

 

 -6 7 7 4  →  -24 28 28 16 
-2 5 7 1
-2
5 7
1

The row operation is


form.pg

.
1
10.(1 pt) setAlgebra35SystemMatrices/solve 3x3.pg
Solve the system associated with the augmented matrix below.
If the system is inconsistent, type ”No Solution” in each blank.
If the system is dependent, use the variable ”t” as your free variable.



-6 12 9
-6
 65 3 -59   -6 
11 -3 -11
-3
The solution to the system is: (
,
,
16.(1 pt) setAlgebra35SystemMatrices/inverse
Solve the system of equations
)
11.(1 pt) setAlgebra35SystemMatrices/solve RREF.pg
Given the matrix below, solve the associated system of equations. For your variables, use x1 , x2 , x3 , x4 , x5 , and x6 .



1 -3 8 6 3 9
-3
 0 0 1 -8 7 -8   1 
0 0 0 1 -7 4
4
The solution is (
,
,
,
,
2x − 4y + z = 3
−x + y − z = −1
x − 2y = 1
,
)by converting to a matrix equation and using the inverse of the
coefficient matrix, as in Example 4 of the text.
x=
y=
z=
12.(1 pt) setAlgebra35SystemMatrices/solve RREF 2.pg
Given the matrix below, solve the associated system of equations. For your variables, use x1 , x2 , x3 , x4 , x5 , x6 , x7 , and x8 .



1 -9 2 0 7 -8 5 8
7
 0 0 1 -8 1 4 0 -8   5 



 0 0 0 0 1 2 -2 4   -4 
0 0 0 0 0 0 0 1
1
The solution is (
)
,
,
13.(1 pt) setAlgebra35SystemMatrices/sw7
Solve the system of equations
,
,
17.(1 pt) setAlgebra35SystemMatrices/sw7
Write the system of equations
,
,
3x − 2y − 2z = −1
2x − 5y + 2z = 0
−4x + 2y + 5z = −2
Input your answer below:
a11 =
a12 =
a13 =
a21 =
a22 =
a23 =
a31 =
a32 =
a33 =
b1 =
b2 =
b3 =
by converting to a matrix equation and using the inverse of the
coefficient matrix, as in Example 4 of the text.
x=
y=
5 23.pg
2x − 4y + z = −4
−x + y − z = 1
x − 2y = −3
by converting to a matrix equation and using the inverse of the
coefficient matrix, as in Example 4 of the text.
x=
y=
z=
15.(1 pt) setAlgebra35SystemMatrices/inverse
Solve the system of equations
,
4 24.pg
as a matrix equation, that is, rewrite it in the form

  

a11 a12 a13
x
b1
 a21 a22 a23   y  =  b2 
a31 a32 a33
b3
z
5 19.pg
5x + 3y = −6
3x + 2y = −3
14.(1 pt) setAlgebra35SystemMatrices/sw7
Solve the system of equations
solve3x3.pg
18.(1 pt) setAlgebra35SystemMatrices/sw7
The system of equations
3 11.pg
x − 2y + z = 7,
y + 2z = −3,
x + y + 3z = −2
solve2x2.pg
has a unique solution. Find the solution using Gaussin elimination method or Gauss-Jordan elimination method.
x=
y=
z=
2x + 9y = 887
−7x + 5y = −148
19.(1 pt) setAlgebra35SystemMatrices/sw7
The system of equations
by converting to a matrix equation and using the inverse of the
coefficient matrix.
x=
y=
2
x + 2y − z = −4,
x + z = −2,
2x − y − z = −7.
3 15.pg
has a unique solution. Find the solution using Gaussin elimination method or Gauss-Jordan elimination method.
x=
y=
z=
20.(1 pt) setAlgebra35SystemMatrices/sw7 3 17.pg
The system of equations
(a) determine whether the system is inconsistent or dependent;
Your answer is (input inconsistent or dependent)
(b) if your answer is dependent in (a), find the complete solution. Write x and y as functions of z.
x=
y=
24.(1 pt) setAlgebra35SystemMatrices/sw7
Use Cramer’s rule to solve the system
x1 + 2x2 − x3 = 5,
2x1 − x3 = 5,
3x1 + 5x2 + 2x3 = 9
2x − y = −1
x + 2y = 2
x=
y=
has a unique solution. Find the solution using Gaussin elimination method or Gauss-Jordan elimination method.
x1 =
x2 =
x3 =
21.(1 pt) setAlgebra35SystemMatrices/sw7
The system of equations
25.(1 pt) setAlgebra35SystemMatrices/sw7
Use Cramer’s rule to solve the system
6 29.pg
x − 6y = 15
3x + 2y = 5
3 19.pg
x=
y=
2x − 3y − z = −2,
−x + 2y − 5z = −10,
5x − y − z = −2
26.(1 pt) setAlgebra35SystemMatrices/sw7
Use Cramer’s rule to solve the system
has a unique solution. Find the solution using Gaussin elimination method or Gauss-Jordan elimination method.
x=
y=
z=
22.(1 pt) setAlgebra35SystemMatrices/sw7 3 21.pg
Given the system of equations
x=
,y=
x1 =
(a) determine whether the system is inconsistent or dependent;
Your answer is (input inconsistent or dependent)
(b) if your answer is dependent, find the complete solution.
Write x and y as functions of z.
x=
y=
, x2 =
6 33.pg
x − y + 2z = −4
3x + z = −1
−x + 2y = 4
, and z =
27.(1 pt) setAlgebra35SystemMatrices/sw7
Use Cramer’s rule to solve the system
x + y + z = −2,
y − 3z = 12,
2x + y + 5z = −15,
23.(1 pt) setAlgebra35SystemMatrices/sw7
Given the system of equations
6 27.pg
6 35.pg
2x1 + 3x2 − 5x3 = 8
x1 + x 2 − x 3 = 1
2x2 + x3 = −2
, and x3 =
28.(1 pt) setAlgebra35SystemMatrices/cramer.pg
Use Cramer’s rule to find the value of z in the solution of the
following system:
3x + 3y − 2z = 4
−1x − 1y − 2z = −12
3 23.pg
−3x + 4y − 3z = −73
2x − 3y − 9z = 4,
x + 3z = −1,
−3x + y − 4z = 1,
z=
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
3
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra36SeqSeries due 2/5/10 at 2:00 AM
its third term is
;
its fourth term is
;
its 100th term is
.
8.(1 pt) setAlgebra36SeqSeries/srw10 1 7.pg
For the sequence an = 13 + (−1)n,
its first term is
;
its second term is
;
;
its third term is
its fourth term is
;
its 100th term is
.
9.(1 pt) setAlgebra36SeqSeries/srw10 1 11.pg
For the sequence an = 2(an−1 − 2) and a1 = 2,
its first term is
;
its second term is
;
its third term is
;
its fourth term is
;
its fifth term is
.
10.(1 pt) setAlgebra36SeqSeries/srw10 1 15.pg
For the sequence an = an−1 + an−2 and a1 = 5, a2 = 6,
its first term is
;
its second term is
;
its third term is
;
its fourth term is
;
its fifth term is
.
11.(1 pt) setAlgebra36SeqSeries/srw10 1 17.pg
Use a graphing calculator to find the first 10 terms of the sequence an = 5n + 1.
its 9th term is
;
its 10th term is
.
12.(1 pt) setAlgebra36SeqSeries/srw10 1 19.pg
Use a graphing calculator to find the first 10 terms of the se2
quence an = .
n
;
its 9th term is
its 10th term is
.
13.(1 pt) setAlgebra36SeqSeries/srw10 1 21.pg
Use a graphing calculator to find the first 10 terms of the se1
quence an =
and a1 = 4..
an−1
its 9th term is
;
its 10th term is
.
14.(1 pt) setAlgebra36SeqSeries/ur sq 4 1.pg
6n
Write down the first five terms of the sequence
.
n + 10
,
,
,
,
,
1.(1 pt) setAlgebra36SeqSeries/evalfact.pg
Calculate each of the following. Your answer must be a number. No arithmetic operations are allowed in your answer. Please
give 7 places after your decimal point if you use scientific notation.
940!
=
.
40!900!
250!
.
=
235!40!
537! − 532!
=
.
533!
2.(1 pt) setAlgebra36SeqSeries/factorial2.pg
Evaluate
13!
12!
13!
=
12!
3.(1 pt) setAlgebra36SeqSeries/factorial3.pg
Simplify the expression
(3n + 4)!
.
(3n − 1)!
(3n + 4)!
=
(3n − 1)!
4.(1 pt) setAlgebra36SeqSeries/factorial1.pg
Find the first five (5) terms of
7n!
an =
n+2
starting with n = 1. Write your answer as a comma separated
(e.g.: 1,2).
5.(1 pt) setAlgebra36SeqSeries/srw10
For the sequence an = n + 1,
its first term is
;
its second term is
;
its third term is
;
its fourth term is
;
its 100th term is
.
6.(1 pt) setAlgebra36SeqSeries/srw10
11
For the sequence an =
,
n+1
;
its first term is
its second term is
;
its third term is
;
its fourth term is
;
its 100th term is
.
7.(1 pt) setAlgebra36SeqSeries/srw10
(−1)n 15
For the sequence an =
,
n2
its first term is
;
its second term is
;
1 1.pg
1 3.pg
1 5.pg
1
15.(1 pt) setAlgebra36SeqSeries/ns8 1 5.pg
For each sequence, find a formula for the general term, an . For
example, answer n2 if given the sequence:
{1, 4, 9, 16, 25, 36, ...}
1 1 1 1
{ , , , , ...}
2 4 6 8
{
3 4 5 6
, , , , ...}
16 25 36 49
its first term is
;
its second term is
;
its third term is
;
its fourth term is
;
its fifth term is
;
its common difference d =
16.(1 pt) setAlgebra36SeqSeries/ns8 1 5a.pg
For each sequence, find a formula for the general term, An . For
example, answer n2 if given the sequence:
{1, 4, 9, 16, 25,36, ...}
1 1 1 1
1. , , , , ...
2 4 6 8
1 2 3 4
2. , , , , ...
2 3 4 5
25.(1 pt) setAlgebra36SeqSeries/srw10 2 5.pg
For the arithmetic sequence with given first term 6 and common
difference 5:
;
its nth term is
its 10-th term is
.
17.(1 pt) setAlgebra36SeqSeries/ns8 1 5new.pg
For each sequence, find a formula for the general term, an . For
example, answer n2 if given the sequence:
{1, 4, 9, 16, 25,36, ...}
1 1 1 1
{ , , , , ...}
3 6 9 12
1 1 1 1
{ , , , , ...}
3 9 27 81
18.(1 pt) setAlgebra36SeqSeries/srw10 1 25.pg
For the sequence 1, 4, 7, 10, .. .,
its nth term is
.
19.(1 pt) setAlgebra36SeqSeries/srw10 1 33.pg
1 1 1 1 1
For the sequence , 2 , 3 , 4 , 5 , . . . ,
5 5 5 5 5
its fifth partial sum S5 =
.
its sixth partial sum S6 =
.
26.(1 pt) setAlgebra36SeqSeries/srw10 2 7.pg
For the arithmetic sequence with given first term -4 and common
difference -4:
its nth term is
;
its 10-th term is
.
27.(1 pt) setAlgebra36SeqSeries/srw10 2 9.pg
Is the sequence 5, 8, 11, 14, · · · , arithmetic?
Your answer is (input yes or no)
;
if your answer is yes, its common difference is
.
28.(1 pt) setAlgebra36SeqSeries/srw10 2 13.pg
3
3
Is the sequence 3, , 0, − , · · · , arithmetic?
2
2
Your answer is (input yes or no)
;
if your answer is yes, its common difference is
21.(1 pt) setAlgebra36SeqSeries/closed formA.pg
For each sequence, find a closed formula for the general term,
an .
1. 18, 22, 26, 30, 34, . . ., an =
.
2. 1, 4, 9, 16, 25, . . ., an =
.
3. 3, 9, 27, 81, 243, . . ., an =
.
23.(1 pt) setAlgebra36SeqSeries/srw10 2 1.pg
For the sequence an = 6 + 6 ∗ (n − 1),
its first term is
;
its second term is
;
;
its third term is
its fourth term is
;
its fifth term is
;
its common difference d =
.
24.(1 pt) setAlgebra36SeqSeries/srw10 2 3.pg
For the sequence an = 2 − 5 ∗ (n − 1),
.
29.(1 pt) setAlgebra36SeqSeries/srw10 2 17.pg
Is the sequence an = 4 + 3n arithmetic?
Your answer is (input yes or no)
;
if your answer is yes,
its first term is
.
its common difference is
.
30.(1 pt) setAlgebra36SeqSeries/srw10 2 19.pg
1
arithmetic?
Is the sequence an =
3 + 5n
Your answer is (input yes or no)
;
if your answer is yes,
its first term is
.
its common difference is
.
20.(1 pt) setAlgebra36SeqSeries/closed form.pg
For each sequence, find a closed formula for the general term,
an .
1. −2, −8, −18, −32, −50, . .., an =
.
2. −19, −37, −55, −73, −91, . . ., an =
.
3. −57, −570, −5700, −57000, −570000, . . ., an =
.
22.(1 pt) setAlgebra36SeqSeries/find terms1.pg
List the first four terms of each sequence.
an = 10n − 5 :
,
,
,
, . . ..
bn = (−3)n :
,
,
,
, . . ..
c1 = 5, cn = 4cn−1 + 3 :
,
,
,
.
31.(1 pt) setAlgebra36SeqSeries/srw10 2 21.pg
Is the sequence an = 9n − 14 arithmetic?
Your answer is (input yes or no)
;
if your answer is yes,
.
its first term is
its common difference is
.
, . . ..
32.(1 pt) setAlgebra36SeqSeries/srw10 2 23.pg
Write the arithmetic sequence 7, 10, 13, 16, . . . in the standard
form:
an =
.
33.(1 pt) setAlgebra36SeqSeries/srw10 2 25.pg
Write the arithmetic sequence −9, −1, 7, 15, . . . in the standard
form:
an =
.
2
34.(1 pt) setAlgebra36SeqSeries/srw10 2 27.pg
Write the arithmetic sequence −6, −8, −10, −12, . . . in the standard form:
an =
.
46.(1 pt) setAlgebra36SeqSeries/ur sq 4 8.pg
Find x such that 6x + 3, 3x − 9, and 2x − 13 are consecutive
terms of an arithmetic sequence.
x=
35.(1 pt) setAlgebra36SeqSeries/srw10 2 29.pg
Write the arithmetic sequence 22, 17, 12, 7, .. . in the standard
form:
an =
.
47.(1 pt) setAlgebra36SeqSeries/sequence1.pg
Find a closed formula for an if
36.(1 pt) setAlgebra36SeqSeries/srw10 2 35.pg
If the 100th term of an arithmetic sequence is 414, and its common difference is 4, then
its first term a1 =
,
its second term a2 =
,
its third term a3 =
.
an =
37.(1 pt) setAlgebra36SeqSeries/srw10 2 37.pg
Which term of the arithmetic sequence 1, 5, 9, 13, .. . is 153?
It is the
th term.
49.(1 pt) setAlgebra36SeqSeries/type1.pg
All sequences for this problem are arithmetic. Give all answers
to the nearest thousandth.
If a1 = 79 and d = −13, then a30 =
.
If b14 = −88 and b42 = −80, then b1 =
.
If c15 = −76 and c45 = 67, then S10 =
.
n
∑ ak = 4n2 + 5n
k=1
48.(1 pt) setAlgebra36SeqSeries/sequence5.pg
For an arithmetic sequence, a20 = 83. If the common difference
is 8, find:
a1 =
the sum of the first 55 terms =
38.(1 pt) setAlgebra36SeqSeries/srw10 2 39.pg
Find the partial sum S16 for the arithmetic sequence with a =
8, d = 2.
S16 =
.
39.(1 pt) setAlgebra36SeqSeries/srw10 2 45.pg
The partial sum 1 + 6 + 11 + · · ·+ 141 equals
50.(1 pt) setAlgebra36SeqSeries/p12.pg
Find the 4th term of the geometric sequence with a8 = 1953125
and a9 = −9765625
a4 =
.
40.(1 pt) setAlgebra36SeqSeries/ur sq 4 2.pg
Find the 13th term of the arithmetic sequence
8, 11, 14, ...
Answer:
41.(1 pt) setAlgebra36SeqSeries/ur
Find the sum
1 + 7 + 13 + ... + (−5 + 6n)
Answer:
sq 4 3.pg
42.(1 pt) setAlgebra36SeqSeries/ur
Find the sum
3 + 4 + 5 + ... + 16
Answer:
sq 4 4.pg
51.(1 pt) setAlgebra36SeqSeries/ur sq 5 1.pg
Find the common ratio
and
write out the first four terms of the
4n+3
geometric sequence
2
Common ratio is
a1 = , a 2 = , a 3 = , a 4 =
52.(1 pt) setAlgebra36SeqSeries/ur sq 5 2.pg
Find the 5th term of the geometric sequence
9, 27, 81,...
Answer:
53.(1 pt) setAlgebra36SeqSeries/ur sq 5 3.pg
Find the nth term of the geometric sequence whose initial term
is 1 and common ration is 8.
(Your answer must be a function of n.)
43.(1 pt) setAlgebra36SeqSeries/ur sq 4 5.pg
Find the common difference
out the first four terms of
and write
5
1
n−
the arithmetic sequence
2
6
Common difference is
a1 =
, a2 =
, a3 =
, a4 =
,
54.(1 pt) setAlgebra36SeqSeries/geometric1.pg
Find all values of x such that x − 10, x + 10, and 7x − 10 form a
geometric sequence. Give your answers in increasing order.
x can equal
or
.
44.(1 pt) setAlgebra36SeqSeries/ur sq 4 6.pg
Find the nth term of the arithmetic sequence whose initial term
is 6 and common difference is 10.
(Your answer must be a function of n.)
55.(1 pt) setAlgebra36SeqSeries/sequence6.pg
Given the geometric sequence: 17, 2.83333333333333,0.472222222222222
Find an explicit formula for an .
an =
Find a7 =
Note: Your answer to part one should be a function in terms of
n. Your answer to part two should be a decimal, with at least 5
significant figures.
45.(1 pt) setAlgebra36SeqSeries/ur sq 4 7.pg
Find the first term and the common difference of the arithmetic
sequence whose 5th term is 25 and 13th term is 41.
First term is
,
Common difference is
3
56.(1 pt) setAlgebra36SeqSeries/sequence6A.pg
72
Given the geometric sequence: 10, 12
1 , 5 ...
Find an explicit formula for an .
an =
Find a8 =
.
64.(1 pt) setAlgebra36SeqSeries/srw10
1
.
∑k=
k=1
1 41.pg
65.(1 pt) setAlgebra36SeqSeries/srw10
1 43.pg
3
10
∑ [1 + (−1) ] =
57.(1 pt) setAlgebra36SeqSeries/sequence7.pg
Given the geometric sequence:
15, 4.6875,1.46484375, . . .
Find an explicit formula for an .
an =
Find a8 =
Find S8 =
Find S =
Note: Your answer to part one should be a function in terms of
n. Your other answers should be decimals, accurate to at least 5
places.
i
.
i=1
16
66.(1 pt) setAlgebra36SeqSeries/srw10
∑ k2 =
.
k=1
20
67.(1 pt) setAlgebra36SeqSeries/srw10
∑ (−1)n 2n =
.
68.(1 pt) setAlgebra36SeqSeries/srw10
Wirte the sum using sigma notation:
1 + 2 + 3 + 4 + · · ·+ 129 =
A
∑ B, where
n=1
∞
60.(1 pt) setAlgebra36SeqSeries/jj1.pg
Find the sum
4
∑ 4( 7 )n
n=0
8n =
The sum is
.
71.(1 pt) setAlgebra36SeqSeries/infsum2pg
Evaluate the following sum. If the sum is not finite, type DOES
NOT EXIST as your answer.
n=41
61.(1 pt) setAlgebra36SeqSeries/jj3.pg
Find the
indicated
sum.
n
11 10
∑2 3 =
n=1
∞
∑ −4 − 9n
n=4
62.(1 pt) setAlgebra36SeqSeries/p11.pg
Find the infinite sum (if it exists):
i
∞
1
−8
·
∑
5
i=0
The sum is
.
72.(1 pt) setAlgebra36SeqSeries/sequence2.pg
Evaluate
12
∑ (10k − 10)(9k + 8)
k=1
If the sum does not exists, type DNE in the answer blank.
sum =
63.(1 pt) setAlgebra36SeqSeries/srw10
8
For the sequence an = n ,
3
its fifth partial sum S5 =
.
its nth partial sum Sn =
.
1 59.pg
A=
,
B=
.
69.(1 pt) setAlgebra36SeqSeries/srw10 1 63.pg
Wirte the sum using sigma notation:
A
1
1
1
1
+
+
+
+ = ∑ B, where
1 · 2 2 · 3 3 · 4 126 · 127
n=1
A=
,
B=
.
70.(1 pt) setAlgebra36SeqSeries/infsum1.pg
Evaluate the following sum. If the sum is not finite, type DOES
NOT EXIST as your answer.
59.(1 pt) setAlgebra36SeqSeries/type2.pg
All sequences for this problem are geometric. Give all answers
to the nearest thousandth.
If a1 = 35 and r = −9, then a24 =
.
If b10 = −23 and b36 = −55, then b1 =
.
If c13 = 35 and c36 = 22, then S12 =
.
70
1 51.pg
n=0
58.(1 pt) setAlgebra36SeqSeries/sequence7A.pg
108
Given the geometric sequence: 27, 54
19 , 361 . . .
Find an explicit formula for an .
an =
Find a10 =
.
Find S10 =
.
Find S =
. If S is not finite, type DOES NOT EXIST.
∑
1 47.pg
73.(1 pt) setAlgebra36SeqSeries/sequence3.pg
Insert 5 arithmetic means between -5 and 57.
First mean =
Second mean =
Third mean =
Fourth mean =
Fifth mean =
1 35.pg
4
Note: Your answers must be in decimal form, given to at least 5
places.
row, 21 seats in the third row, 24 seats in the forth row, and so
on.
Total number of seats =
82.(1 pt) setAlgebra36SeqSeries/srw10 2 51.pg
The purchase value of an office computer is 12760 dollars. Its
annual depreciation is 1877 dollars.
The value of the compter after 5 years is
dollars.
74.(1 pt) setAlgebra36SeqSeries/sequence4.pg
Insert 5 geometric means between 66 and 84.
First mean =
Second mean =
Third mean =
Fourth mean =
Fifth mean =
Note: Your answers must be in decimal form, given to at least 5
places.
83.(1 pt) setAlgebra36SeqSeries/srw10 2 53.pg
A man gets a job with a salary of 30000 dollars a year. He is
promised a 2700 dollars raise each subsequent year.
His total earning for a 6-year period is
dollars.
75.(1 pt) setAlgebra36SeqSeries/sum1.pg
Evaluate the following sum:
84.(1 pt) setAlgebra36SeqSeries/ur sq 5 4.pg
Steve and Carrie want to purchase a house. Suppose they invest 300 dollars per month into a mutual fund. How much will
they have for a downpayment after 6 years if the per annum rate
of return of the mutual fund is assumed to be 11 percent compounded monthly?
26
∑ (−1)n−1(6n2 + 1n − 2)
n=10
The sum is
.
76.(1 pt) setAlgebra36SeqSeries/sum2.pg
Solve the following equation over the real numbers. If no solution exists, type NO SOLUTION as your answer. Give your
answers in increasing order to the nearest thousandth.
85.(1 pt) setAlgebra36SeqSeries/auditorium.pg
An auditorium has 36 rows of seats. The first row contains 70
seats. As you move to the rear of the auditorium, each row has
4 more seats than the previous row.
How many seats are in the 16 th row?
.
How many seats are in the auditorium?
.
9
∑ (−1)n−1 (−6(nx)2 − 8nx − 9) = 0
n=5
x=
or x =
.
77.(1 pt) setAlgebra36SeqSeries/sum3.pg
Evaluate the following sum without writing out all the terms:
86.(1 pt) setAlgebra36SeqSeries/auditorium2pg
In a certain auditorium, each row has 5 more seats than the row
in front of it. The first 6 rows contain 675 seats. How many
rows does the auditorium have if it holds 11125 seats?
rows.
The auditorium has
87.(1 pt) setAlgebra36SeqSeries/sequence8.pg
Solve for x:
25
∑ (1n2 − 1n + 5)
n=2
The sum is
.
78.(1 pt) setAlgebra36SeqSeries/ur sq 4 9.pg
Write down the first five terms of the following recursively defined sequence.
,
,
,
∞
∑ 7xn−1 = 40
n=1
x=
88.(1 pt) setAlgebra36SeqSeries/sequence9.pg
The hypotenuese of an isosceles right triangle is 18 inches. The
midpoints of its sides are connected to form an inscribed triangle, and this process is repeated. Find the sum of the areas of
these triangles if this process is continued infinitely.
S=
89.(1 pt) setAlgebra36SeqSeries/sequence10.pg
Solve for x:
a1 = −5; an+1 = −2an + 10
,
79.(1 pt) setAlgebra36SeqSeries/recursive1.pg
Suppose an = −1an−1 + 4an−2 − 4an−3 and a4 = 14, a5 = -14,
and a6 = 46. Find a1 , a2 , and a3 .
a1 =
.
a2 =
.
a3 =
.
∞
80.(1 pt) setAlgebra36SeqSeries/faris3.pg
Suppose you go to a company that pays 0.02 for the first day,
0.04 for the second day, 0.08 for the third day and so on. If the
daily wage keeps doubling, what will your total income be for
working 31 days ?
Total Income =
81.(1 pt) setAlgebra36SeqSeries/jj2.pg
Determine the seating capacity of an auditorium with 30 rows of
seats if there are 15 seats in the first row, 18 seats in the second
∑ 7x7n = 42
n=1
x=
90.(1 pt) setAlgebra36SeqSeries/sequence11.pg
Chucky takes his first step on January 1, 2000. Every day after
that, he takes 38 more steps than the day before. Tommy takes
his first steps on February 1, 2000. On that day, Tommy takes
10 steps. Every day after that, Tommy takes twice as many steps
as the day before.
5
Who walks farther on Valentine’s Day?
Who walks farther on Groundhog Day?
What is the last day in February that Chucky walks farther than
Tommy?
Note: Your answer to parts one and two should be names. Your
answer to part three should be the last day in February that
Chucky takes more steps than Tommy.
It will bounce
times before its rebound is less than 1
foot.
How far will the ball travel before it comes to rest on the ground?
It will travel
feet before it comes to rest on the
ground.
92.(1 pt) setAlgebra36SeqSeries/superball.pg
1
A superball that rebounds of the height from which it fell on
8
each bounce is dropped from 13 meters.
.
How high does it rebound, in meters, on the 5 th bounce?
How far does it travel, in meters, before coming to rest?
.
91.(1 pt) setAlgebra36SeqSeries/sequence12.pg
A Super Happy Fun Ball is dropped from a height of 17 feet
and rebounds 6/7 of the distance from which it fell. How many
times will it bounce before its rebound is less than 1 foot?
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
6
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra37Binomial due 2/6/10 at 2:00 AM
The coefficient of x5 y4 is
1.(1 pt) setAlgebra37Binomial/binomial.pg
Expand the expression using the Binomial Theorem:
(2x − 2)5 =
x5 +
x4 +
x3 +
x2 +
x+
4.(1 pt) setAlgebra37Binomial/findcnst.pg
14
Find the constant term in the expansion of (4x6 + −5
x ) .
The constant term is
.
2.(1 pt) setAlgebra37Binomial/findcoeff.pg
Find the coefficient of x10 y78 in the expansion of
5.(1 pt) setAlgebra37Binomial/whatterm.pg
Which term of the expansion of (x2 + 7)10 contains x14
Term number
contains x14 .
(1x5 + 2y6 )15 .
The coefficient of x10 y78 is
.
.
3.(1 pt) setAlgebra37Binomial/findcoeff1.pg
Find the coefficient of x5 y4 in the expansion of
6.(1 pt) setAlgebra37Binomial/whatterm1.pg
Which term of the expansion of (x + 1)10 contains x5
Term number
contains x5 .
(4x − 1y)9.
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
1
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra38Counting due 2/7/10 at 2:00 AM
(a) No condition is imposed.
Your answer is :
(b) No letter can be repeated in a word.
Your answer is :
(c) Each word must begin with the letter A.
Your answer is :
(d) The letter C must be at the end.
Your answer is :
(e) The second letter must be a vowel.
Your answer is :
10.(1 pt) setAlgebra38Counting/sw10 2 1.pg
Evaluate the expression P(15, 3).
Your answer is :
11.(1 pt) setAlgebra38Counting/sw10 2 5.pg
Evaluate the expression P(120, 1).
Your answer is :
12.(1 pt) setAlgebra38Counting/sw10 2 11.pg
How many three-letter “words” can be made from 7 letters
“FGHIJKL”? (Letters may not be repeated.)
Your answer is :
13.(1 pt) setAlgebra38Counting/sw10 2 14.pg
A pianist plans to play 4 pieces at a recital. In how many ways
can she arrange these pieces in the program?
Your answer is :
14.(1 pt) setAlgebra38Counting/sw10 2 17.pg
In how many ways can first, second, and third prizes be awarded
in a contest with 545 contestants?
Your answer is :
15.(1 pt) setAlgebra38Counting/sw10 2 19.pg
In how many ways can 3 students be seated in a row of 3 chairs
if Jack insists on sitting in the first chair?
Your answer is :
16.(1 pt) setAlgebra38Counting/sw10 2 21.pg
Find the number of distinguishable permutations of the given
letters “AABCD”.
Your answer is :
17.(1 pt) setAlgebra38Counting/sw10 2 22.pg
Find the number of distinguishable permutations of the given
letters “AAABBBCC”.
Your answer is :
18.(1 pt) setAlgebra38Counting/sw10 2 33.pg
Evaluate the expression C(18, 3).
Your answer is :
19.(1 pt) setAlgebra38Counting/sw10 2 37.pg
Evaluate the expression C(180, 1).
Your answer is :
20.(1 pt) setAlgebra38Counting/sw10 2 39.pg
In how many ways can 3 books be choosen from a group of
nine?
1.(1 pt) setAlgebra38Counting/sw10 1 1.pg
A vendor sells ice cream from a cart on the boardwalk. He offers vanilla, chocolate, strawberry, blueberry, and pistachio ice
cream, served on either a waffle, sugar, or plain cone. How
many different single-scoop ice-cream cones can you buy from
this vendor?
Your answer is :
2.(1 pt) setAlgebra38Counting/sw10 1 3.pg
How many three-letter “words” can be made from 7 letters
“FGHIJKL” if repetition of letters
(a) is allowed?
Your answer is :
(b) is not allowed?
Your answer is :
3.(1 pt) setAlgebra38Counting/sw10 1 7.pg
How many different ways can a race with 8 runners be completed? (Assume there is no tie.)
Your answer is :
4.(1 pt) setAlgebra38Counting/sw10 1 15.pg
3 different color dice are rolled, and the numbers showing are
recorded. How many different outcomes are possible?
Your answer is :
5.(1 pt) setAlgebra38Counting/sw10 1 17.pg
A girl has 7 skirts, 4 blouses, and 8 pairs of shoes. How many
different skirt-blouse-shoe outfits can she wear? (Assume that
each item matches all the others, so she is willing to wear any
combination.)
Your answer is :
6.(1 pt) setAlgebra38Counting/sw10 1 19.pg
A company has 4660 employees. Each employee is to be given
an ID number that consists of one letter followed by two digits.
Is it possible to give each employee a different ID number using
this scheme?
Your answer is (input Yes or No):
7.(1 pt) setAlgebra38Counting/sw10 1 21.pg
Standard automobile license plates in a country display 1 numbers, followed by 2 letters, followed by 4 numbers. How many
different standard plates are possible in this system? (Assume
repetitions of letters and numbers are allowed.)
Your answer is :
8.(1 pt) setAlgebra38Counting/sw10 1 23.pg
A true-false test contains 23 questions. In how many different
ways can this test be completed. (Assume we don’t care about
our scores.)
Your answer is :
9.(1 pt) setAlgebra38Counting/sw10 1 33.pg
3 -letter “words” are formed using the letters A, B, C, D, E, F,
G. How many such words are possible for each of the following
conditions?
1
27.(1 pt) setAlgebra38Counting/baseball.pg
There are nine different positions on a baseball team. If a team
has 12 players how many different line-ups can the team make?
The team can make
different line-ups.
Baseball games consist of nine innings. A team wants to change
its line-up every inning. If no game goes to extra innings, and
a season consists of 98 games, how many complete seasons can
the team play without repeating a line-up?
The team can play
complete seasons without repeating a
line-up.
Your answer is :
21.(1 pt) setAlgebra38Counting/sw10 2 40.pg
In how many ways can 5 pizza toppings be choosen from 12
available toppings?
Your answer is :
22.(1 pt) setAlgebra38Counting/sw10 2 43.pg
How many different 5 card hands can be dealt from a deck of 52
cards?
Your answer is :
23.(1 pt) setAlgebra38Counting/sw10 2 46.pg
A pizza parlor offers a choice of 16 different toppings. How
many 5-topping pizzas are possible?
Your answer is :
24.(1 pt) setAlgebra38Counting/sw10 2 49.pg
In how many ways can 3 students from a class of 16 be chosen
for a field trip?
Your answer is :
25.(1 pt) setAlgebra38Counting/sw10 2 52.pg
In the 6/48 lottery game, a player picks six numbers from 1 to
48. How many different choices does the player have?
Your answer is :
26.(1 pt) setAlgebra38Counting/sw10 2 60.pg
A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. If 6 freshmen, 8 sophomores,
8 juniors, and 9 seniors are eligible to be on the committee, in
how many ways can the committee be chosen?
Your answer is :
28.(1 pt) setAlgebra38Counting/outfits.pg
A boy owns 4 pairs of pants, 8 shirts, 7 ties, and 1 jackets. How
many different outfits can he wear to school if he must wear one
of each item?
He can wear
different outfits.
29.(1 pt) setAlgebra38Counting/wonka.pg
Willie Wonka gives everyone who visits his factory 11 pieces of
candy to take home. He never gives a person 2 or more pieces
of the same type of candy. If Mr. Wonka has 20 different types
of candy, in how many different ways could Mr. Wonka give a
visitor his candy?
Mr. Wonka can distribute candy in
different ways.
If 175 people visit Mr. Wonka’s factory each day, how many
days could Mr. Wonka go without giving two visitors the same
selection of candy
Mr. Wonka can go for
days without repeating candy
selections.
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
2
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra39Probability due 2/8/10 at 2:00 AM
8.(1 pt) setAlgebra39Probability/sw10 3 19.pg
An American roulette wheel has 38 slots: two slots are numbered 0 and 00, and the remainging slots are numbered from 1 to
36. Find the probability that the ball lands in an odd-numbered
slot.
Your answer is :
9.(1 pt) setAlgebra39Probability/sw10 3 21.pg
In the 3/26 lottery game, a player selects 3 numbers from 1 to
26. What is the probability of picking the 3 winning numbers?
Your answer is :
10.(1 pt) setAlgebra39Probability/sw10 3 43.pg
A jar contains 10 red marbles numbered 1 to 10 and 4 blue marbles numbered 1 to 4. A marble is drawn at random from the jar.
Find the probability of the given event.
(a) The marble is red;
Your answer is :
(b) The marble is odd-numbered;
Your answer is :
(c) The marble is red or odd-numbered;
Your answer is :
(d) The marble is blue or even-numbered;
Your answer is :
11.(1 pt) setAlgebra39Probability/sw10 3 44.pg
A coin is tossed twice. Let E be the event “the first toss shows
heads” and F the event “the second toss shows heads”.
(a) Are the events E and F independent?
Input Yes or No here:
(b) Find the probability of showing heads on both toss.
Input your answer here:
1.(1 pt) setAlgebra39Probability/sw10 3 3.pg
A die is rolled. Find the probability of the given event.
(a) The number showing is a 6;
The probability is :
(b) The number showing is an even number;
The probability is :
(c) The number showing is greater than 1;
The probability is :
2.(1 pt) setAlgebra39Probability/sw10 3 5.pg
A card is drawn randomly from a standard 52-card deck. Find
the probability of the given event.
(a) The card drawn is 5;
The probability is :
(b) The card drawn is a face card;
The probability is :
(c) The card drawn is not a face card.
The probability is :
3.(1 pt) setAlgebra39Probability/sw10 3 7.pg
A ball is drawn randomly from a jar that contains 8 red balls, 8
white balls, and 9 yellow ball. Find the probability of the given
event.
(a) A red ball is drawn;
The probability is :
(b) A white ball is drawn;
The probability is :
(c) A yellow ball is drawn;
The probability is :
4.(1 pt) setAlgebra39Probability/sw10 3 12.pg
A poker hand, consisting of 3 cards, is dealt from a standard
deck of 52 cards. Find the probability that the hand contains 3
hearts.
Your answer is :
12.(1 pt) setAlgebra39Probability/sw10 3 48.pg
A die is rolled twice. What is the probability of showing a 2 on
both rolls?
Your answer is :
13.(1 pt) setAlgebra39Probability/sw10 3 49.pg
A die is rolled twice. What is the probability of showing a(n) 1
on the first roll and an even number on the second roll?
Your answer is :
14.(1 pt) setAlgebra39Probability/batting avg.pg
A baseball player has a batting average of 0.235. What is the
probability that he has exactly 1 hits in his next 7 at bats?
The probability is
.
5.(1 pt) setAlgebra39Probability/sw10 3 13.pg
A poker hand, consisting of 6 cards, is dealt from a standard
deck of 52 cards. Find the probability that the hand contains 6
cards of the same suit.
Your answer is :
6.(1 pt) setAlgebra39Probability/sw10 3 14.pg
A poker hand, consisting of 6 cards, is dealt from a standard
deck of 52 cards. Find the probability that the hand contains 6
face cards.
Your answer is :
15.(1 pt) setAlgebra39Probability/cards1.pg
4 cards are drawn at random from a standard deck.
Find the probability that all the cards are hearts.
Find the probability that all the cards are face cards.
Note: Face cards are kings, queens, and jacks.
Find the probability that all the cards are even.
(Consider aces to be 1, jacks to be 11, queens to be 12, and kings
to be 13)
7.(1 pt) setAlgebra39Probability/sw10 3 15.pg
A poker hand, consisting of 5 cards, is dealt from a standard
deck of 52 cards. Find the probability that the hand contains an
ace, king, queen, jack, and 10 of the same suit (royal flush).
Your answer is :
1
16.(1 pt) setAlgebra39Probability/cards2.pg
A poker hand consisting of 5 cards is dealt from a standard deck
of 52 cards. Find the probability that the hand contains exactly
2 face cards.
The probability is
The probability is
.
20.(1 pt) setAlgebra39Probability/prob1.pg
The letters in the word MATHEMATICS are arranged randomly.
What is the probability that the first letter is E?
What is the probability that the first letter is M?
17.(1 pt) setAlgebra39Probability/class1.pg
An algebra class has 8 students and 8 desks. For the sake of
variety, students change the seating arrangement each day. How
many days must pass before the class must repeat a seating arrangement?
days must pass before a seating arrangement is repeated.
Suppose the desks are arranged in rows of 4. How many seating
arrangements are there that put Larry, Moe, Curly, and Shemp
in the front seats?
There are
seating arrangements that put them in the
front seats.
What is the probability that Larry, Moe, Curly and Shemp are
sitting in the front seats?
The probability is
.
21.(1 pt) setAlgebra39Probability/prob2.pg
A bag contains 6 red marbles, 6 white marbles, and 9 blue marbles. You draw 3 marbles out at random, without replacement.
What is the probability that all the marbles are red?
The probability that all the marbles are red is
.
What is the probability that exactly two of the marbles are red?
The probability that exactly two of the marbles are red is
.
What is the probability that none of the marbles are red?
The probability of picking no red marbles is
.
22.(1 pt) setAlgebra39Probability/prob3.pg
You own 18 CDs. You want to randomly arrange 6 of them
in a CD rack. What is the probability that the rack ends up in
alphabetical order?
The probability that the CDs are in alphabetical order is
.
18.(1 pt) setAlgebra39Probability/coins1.pg
You flip a fair coin 10 times. What is the probability that it lands
on heads exactly 4 times?
.
The probability of exactly 4 heads is
What is the probability that it lands on heads at least 4 times?
The probability of at least 4 heads is
.
23.(1 pt) setAlgebra39Probability/prob4.pg
Find the number of distinguishable permutations of the given
letters “AAABBBCD”.
There are:
permutations.
If a permutation is chosen at random, what is the probability that
it begins with at least 2 A’s?
The probability is
.
19.(1 pt) setAlgebra39Probability/perm1.pg
An algebra class has 12 students and 12 desks. For the sake of
variety, students change the seating arrangement each day. How
many days must pass before the class must repeat a seating arrangement?
days must pass before a seating arrangement is repeated.
Suppose the desks are arranged in rows of 4. How many seating
arrangements are there that put Larry, Moe, Curly, and Shemp
in the front seats?
There are
seating arrangements that put them in the
front seats.
What is the probability that Larry, Moe, Curly and Shemp are
sitting in the front seats?
24.(1 pt) setAlgebra39Probability/prob num.pg
Suppose a number is chosen at random from the set
0,1,2,3,...,871.
What is the probability that the number is a perfect cube?
The probability of choosing a perfect cube is
Note: Your answer must be a fraction or a decimal number.
25.(1 pt) setAlgebra39Probability/typing.pg
What is the probability that if 8 letters are typed, no letters are
repeated?
.
The probability that no letters are repeated is
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
2
ARNOLD PIZER
Rochester WeBWorK Problem Library
rochester problib from CVS June 25, 2004
WeBWorK assignment Algebra40SolveForVariables due 2/9/10 at 2:00 AM
3.(1 pt) setAlgebra40SolveForVariables/perfect square.pg
Solve for a:
(5r 2 + 10a2) = 90
There are two solutions, a1 and a2 , where a1 ≤ a2 .
a1 =
a2 =
1.(1 pt) setAlgebra40SolveForVariables/Fraction.pg
Solve for a:
a + 2b
= 1k + 2
a − 2b
a=
2.(1 pt) setAlgebra40SolveForVariables/circle.pg
Solve for a:
(x + 8a)2 + (y + 4b)2 = 81
4.(1 pt) setAlgebra40SolveForVariables/surface area.pg
Solve for a:
p
S = 1r 2 + 1r −1r 2 + 3a2
There are two solutions, a1 and a2 , where a1 ≤ a2 .
a1 =
a2 =
There are two solutions, a1 and a2 , where a1 ≤ a2 .
a1 =
a2 =
c
Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, UR
1