Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives0Theory due 01/31/2006 at 02:00am EST.
2. If f 0 (7) exists, then then the limit lim f (x) is f (7)
x→7
1. (1 pt) rochesterLibrary/setDerivatives0Theory/s2 R 1-16.pg
Enter a T or an F in each answer space below to indicate whether
the corresponding statement is true or false. A statement is true
only if it is true for all possibilities. You must get all of the
answers correct to receive credit.
lim x2 + 2x − 2
x2 + 2x − 2 x→1
=
1. lim 2
x→1 x + 4x − 3
lim x2 + 4x − 3
3. If f (x) is differentiable at a, then f (x) is continuous at
a
4. If f (x) is continuous at a, then f (x) is differentiable at
a
5. If lim f (x) = 2 and lim g(x) = 0, then lim [ f (x)/g(x)]
x→5
does not exist
x→1
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
x→5
x→5
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives1 due 02/01/2006 at 02:00am EST.
(ii) f 0 (−3) =
(iii) f 0 (1) =
(iv) f 0 (3) =
To avoid calculating four separate limits, I suggest that you
evaluate the derivative at the point when x = a. Once you have
the derivative, you can just plug in those four values for ”a” to
get the answers.
1. (1 pt) rochesterLibrary/setDerivatives1/s2 1 20.pg
If f (x) = 23, find f 0 (−6).
2. (1 pt) rochesterLibrary/setDerivatives1/s2 1 19.pg
If f (x) = 17x + 24, find f 0 (−3).
3. (1 pt) rochesterLibrary/setDerivatives1/s2 1 7.pg
If f (x) = 7 + 6x − 4x2 , find f 0 (5).
7. (1 pt) rochesterLibrary/setDerivatives1/osu dr 1 11.pg
For each of the given functions f (x), find the derivative
0
f −1 (c) at the given point c, using Theorem , first finding
a = f −1 (c).
f (x) = 3x + 9x15 ; c = −12
a=
0
f −1 (c) =
f (x) = x2 − 14x + 60 on the interval [7, ∞); c = 12
a=
0
f −1 (c) =
4. (1 pt) rochesterLibrary/setDerivatives1/s2 1 26.pg
If f (x) = x32 , find f 0 (5).
5.
Let
(1 pt) rochesterLibrary/setDerivatives1/s2 1 23.pg
f (x) =
√
2 + 3x
f 0 (2) =
6.
Let
(1 pt) rochesterLibrary/setDerivatives1/ur dr 1 10.pg
1
x+1
Use the limit definition of the derivative on page 156 to find
(i) f 0 (−5) =
f (x) =
8. (1 pt) rochesterLibrary/setDerivatives1/s2 1 8.pg
The position of a cat running from a dog down a dark alley is given by the values of the table.
t(seconds) 0 1
2 3
s(feet)
0 20 33 62
4
5
81 115
A. Find the average velocity tor the time period beginning when t=2 and lasting
1. 3 s
2. 2 s
3. 1 s
B. Draw the graph of the function for yourself and estimate the instantaneous velocity when t=2
9. (1 pt) rochesterLibrary/setDerivatives1/c1s5p8.pg
This problem tests calculating new functions from old ones:
From the table below calculate the quantities asked for:
x
−40
−12
0
−2
10
−4
f (x)
1558
130 −2
0
108
10
g(x) −3280 −312 0 −12 −180 −40
f 0 (x) −79
−23
1
−3
21
−7
g0 (x)
162
50
2
10
−38
18
( f + g)0 (−4)
( f /g)0 (−4)
If h(x) = f ( f (x)), calculate h0 (−4)
( f g)(−2)
1
10. (1 pt) rochesterLibrary/setDerivatives1/c1s5p8b.pg
Constructing new functions from old ones and calculating the derivative of the new function from the derivatives of the old functions:
From the table below calculate the quantities asked for:
x
−37
−35
−12
3
−11
2
f (x) 49323 41687 1598 −37 1223 −12
g(x) 49285 41651 1585 −35 1211 −11
f 0 (x) −4034 −3606 −409 −34 −342 −17
g0 (x) −4033 −3605 −408 −33 −341 −16
( f /g)0 (2)
( f − g)0 (2)
( f + g)0 (2)
( f g)(3)
11.
(1 pt) rochesterLibrary/setDerivatives1/c1s5p9.pg
This problem tests calculating new functions from old ones:
From the table below calculate the quantities asked for:
x
1
−51
5
−3
f (x)
1 −137955
85
−51
g(x) −3 127397 −171
5
f 0 (x) 1
8009
57
41
g0 (x) −6 −7598
−94 −14
( f /g)0 (1)
( f g)0 (1)
If h(x) = g( f (x)), calculate h0 (−3).
( f g)(−3)
( f + g)0 (1)
12.
(1 pt) rochesterLibrary/setDerivatives1/c1s5p9b.pg
This problem tests calculating new functions from old ones:
From the table below calculate the quantities asked for:
x
4
39
71
−41
−4
−73
f (x)
39
57719
352727 −70521 −73 −394201
g(x) −41 −57721 −352729 70519
71
394199
f 0 (x) 38
4483
14979
5123
54
16131
g0 (x) −38 −4483 −14979 −5123 −54 −16131
( f − g)0 (−4)
( f g)(4)
f (−4)/(g(−4) + 5)
If h(x) = g( f (x)), calculate h0 (4).
If h(x) = f ( f (x)), calculate h0 (−4)
13.
(1 pt) rochesterLibrary/setDerivatives1/c1s6p1.pg
2
Given the following table:
x
0.0071
0.0072
0.0073
0.0074
0.0075
f(x) 0.50259601 0.61217801 -0.94700349 -0.046634111 0.98305479
Calculate the value of f 0 (0.0073) =
to two places of accuracy.
To obtain more precise information about the value of f near 0.0073 enter a new increment value for x here
rule1in.01in and then press the Submit Answer button.
How will you tell when your increment is small enough to give you a good answer for the problem?
14.
is the graph of the function’s first derivative
is the graph of the function’s second derivative
(1 pt) rochesterLibrary/setDerivatives1/nsc2s10p1.pg
16.
Identify the graphs A (blue), B( red) and C (green) as the graphs
of a function and its derivatives:
is the graph of the function
is the graph of the function’s first derivative
is the graph of the function’s second derivative
15.
(1 pt) rochesterLibrary/setDerivatives1/nsc2s10p3.pg
Identify the graphs A (blue), B( red) and C (green) as the
graphs of a function and its derivatives:
is the graph of the function
is the graph of the function’s first derivative
is the graph of the function’s second derivative
(1 pt) rochesterLibrary/setDerivatives1/nsc2s10p2.pg
17.
Let
(1 pt) rochesterLibrary/setDerivatives1/ns2 8 10.pg
f (x) = 3x3 − 3x + 5
Use the limit definition of the derivative on page 163 to calculate the derivative of f :
f 0 (x) =
.
Use the same formula from above to calculate the derivative
of this new function (i.e. the second derivative of f ):
f 00 (x) =
.
18.
(1 pt) rochesterLibrary/setDerivatives1/ns2 8 31.pg
The oracle function f (x) is presented below. For each x value
you enter the oracle will tell you the value f (x). Calculate the
derivative of the function at 1.9 using the Newton quotient definition.
f 0 (x) at 1.9 =
You can use a calculator
Identify the graphs A (blue), B( red) and C (green) as the
graphs of a function and its derivatives:
is the graph of the function
3
x
Enter x
Enter x
Enter x
→
f(x)
→ result: f (x)
→ result: f (x)
→ result: f (x)
If someone now told you that the derivative (slope of the tangent
line to the graph) of f (x) at x = 2 was an integer, what would
you expect it to be?
Remember the technique for finding instantaneous velocities
from average velocities? This is the same thing.
24.
19. (1 pt) rochesterLibrary/setDerivatives1/nsc2 1 5.pg
Below is an ”oracle” function. An oracle function is a function
presented interactively. When you type in a t value, and press
the –f–> button the value f (t) appears in the right hand window.
There are three lines, so you can calculate three different values
of the function at one time.
The function f(t) represents the height in feet of a ball thrown
into the air, t seconds after it has been thrown.
Calculate the average velocity 1.23 seconds after the ball has
been thrown.
You can use a
Average velocity at 1.23 =
calculator
The java Script calculator was displayed here
Remember this technique for finding velocities. Later we
will use the same method to find the derivative of functions such
as f (t).
f (x) =
26.
22. (1 pt) rochesterLibrary/setDerivatives1/ur dr 1 3.pg
1
Let f (x) be the function x+5
. Then the quotient
for:
a=
and
b=
23.
Calculate the difference quotient
(1 pt) rochesterLibrary/setDerivatives1/ur dr 1 6.pg
√
x + 1. Calculate the difference quotient
f (48+h)− f (48)
for
h
h = .1
h = .01
h = −.01
h = −.1
If someone now told you that the derivative (slope of the tangent
line to the graph) of f (x) at x = 48 was 1/n for some integer n
what would you expect n to be?
n=
21. (1 pt) rochesterLibrary/setDerivatives1/ur dr 1 2.pg
Let f (x) be the function 6x2 − 3x + 2. Then the quotient
f (7+h)− f (7)
can be simplified to ah + b for:
h
a=
and
b=
−1
ah+b
for
Let f (x) =
x2 + 6x + 9 if x ≤ 3
ax + b
if x > 3
can be simplified to
1
x−4 .
f (2+h)− f (2)
h
25.
is differentiable everywhere.
a=
b=
f (7+h)− f (7)
h
Let f (x) =
h = .1
h = .01
h = −.01
h = −.1
If someone now told you that the derivative (slope of the tangent
line to the graph) of f (x) at x = 2 was −1/n2 for some integer n
what would you expect n to be?
n=
20. (1 pt) rochesterLibrary/setDerivatives1/ur dr 1 1.pg
Find a and b such that the function
(1 pt) rochesterLibrary/setDerivatives1/ur dr 1 5.pg
(1 pt) rochesterLibrary/setDerivatives1/ur dr 1 4.pg
Let f (x) = x3 − 8x. Calculate the difference quotient
f (2+h)− f (2)
for
h
h = .1
h = .01
h = −.01
h = −.1
4
(1 pt) rochesterLibrary/setDerivatives1/ur dr 1 7.pg
Graphs A and B are approximate graphs of f and f 0 for
f (x) = −x2 − 10x − 21.
So f is decreasing (and f 0 is negative) on the interval (a, ∞)
for a =
.
27.
Graphs A and B are approximate graphs of f and f 0 for
f (x) = x2 − 4x + 7.
So f is increasing (and f 0 is positive) on the interval (a, ∞)
for a =
.
28.
(1 pt) rochesterLibrary/setDerivatives1/ur dr 1 8.pg
5
(1 pt) rochesterLibrary/setDerivatives1/ur dr 1 9.pg
(Your answer above and the next few answers below will involve
the variables t and x.)
The expression inside the limit simplifies to a simple fraction
with
numerator =
and denominator =
We can cancel the factor
appearing in the denominator against a similar factor appearing in the numerator leaving
a simpler fraction with
numerator =
and denominator =
Taking the limit of this fractional expression gives us
f 0 (x) =
30. (1 pt) rochesterLibrary/setDerivatives1/ur dr 1 11.pg
Answer the following True-False quiz. (Enter ”T” or ”F”.)
1. A continuous function on a closed interval always attains a maximum and a minimum value.
2. If a function has a local maximum at c, then f 0 (c) exists
and is equal to 0.
3. ( f (x) + g(x))0 = f 0 (x) + g0 (x).
4. If f (x) = e2 , then f 0 (x) = 2e.
5. Differentiable functions are always continuous.
6. If f 0 (c) = 0 and f 00 (c) > 0, then f (x) has a local minimum at c.
7. If f 0 (x) < 0 for all x in (0, 1), then f (x) is decreasing
on (0, 1).
Graphs A and B are approximate graphs of f and f 0 for
f (x) = x2 (x − 15).
So f is decreasing (and f 0 is negative) on the interval (0, a)
for a =
.
29.
(1 pt) rochesterLibrary/setDerivatives1/osu dr 1 12.pg
5
.
x−4
Then according to the definition of derivative
f 0 (x) = lim
Let f (x) =
t→x
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
6
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives10MaxMin due 02/10/2006 at 02:00am EST.
10. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 3 10.pg
Consider the function f (x) = 12x5 + 45x4 − 360x3 + 2. For
this function there are four important intervals: (−∞, A],
[A, B],[B,C], and [C, ∞) where A, B, and C are the critical numbers.
Find A
and B
and C
At each critical number A, B, and C does f (x) have a local min,
a local max, or neither? Type in your answer as LMIN, LMAX,
or NEITHER.
At A
At B
At C
11. (1 pt) rochesterLibrary/setDerivatives10MaxMin/sc4 2 53.pg
A University of Rochester student decided to depart from Earth
after his graduation to find work on Mars. Before building a
shuttle, he conducted careful calculations. A model for the velocity of the shuttle, from liftoff at t = 0 s until the solid rocket
boosters were jettisoned at t = 41.3 s, is given by
1. (1 pt) rochesterLibrary/setDerivatives10MaxMin/ur dr 10 2.pg
The function f (x) = (8x + 7)e5x has one critical number. Find
it.
2. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 1 25.pg
The function f (x) = 2x3 − 27x2 + 108x − 10 has two critical
numbers. The smaller one equals
and the larger one
equals
3. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 1 11.pg
Consider the function f (x) = 2 − 6x2 , −4 ≤ x ≤ 1.
The absolute maximum value is
and this occurs at x equals
The absolute minimum value is
and this occurs at x equals
4. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 1 18.pg
The function f (x) = 10 − 3x4 has an absolute maximum value
of
and this occurs at x equals
5. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 1 40.pg
Consider the function f (x) = −3x2 + 4x − 10. The absolute
maximum of f (x) is
v(t) = 0.001798667t 3 − 0.08632t 2 + 20.83t + 1.02
(in feet per second). Using this model, estimate
the absolute maximum value
and absolute minimum value
of the ACCELERATION of the shuttle between liftoff and the
jettisoning of the boosters.
6. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 1 42.pg
The function f (x) = 2x3 − 42x2 + 240x + 1 has one local minimum and one local maximum.
This function has a local minimum at x equals
with value
and a local maximum at x equals
12. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 1 39.pg
Consider the function f (x) = 6x2 − 8x + 10, 0 ≤ x ≤ 10. The
absolute maximum of f (x) (on the given interval) is
and
the absolute minimum of f (x) (on the given interval) is
with value
7. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 1 43.pg
The function f (x) = −2x3 + 36x2 − 192x + 1 has one local minimum and one local maximum.
This function has a local minimum at x equals
with value
and a local maximum at x equals
13. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 1 44.pg
Consider the function f (x) = 2x3 + 12x2 − 192x + 5, −8 ≤
x ≤ 5. This function has an absolute minimum value equal to
with value
and an absolute maximum value equal to
8. (1 pt) rochesterLibrary/setDerivatives10MaxMin/ur dr 10 1.pg
The function f (x) = 7x + 7x−1 has one local minimum and one
local maximum.
This function has a local maximum at x =
with value
14. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 1 45.pg
Consider the function f (x) = x4 − 98x2 + 1, −6 ≤ x ≤
15. This function has an absolute minimum value equal to
and a local minimum at x =
and an absolute maximum value equal to
with value
15. (1 pt) rochesterLibrary/setDerivatives10MaxMin/c3s3p1.pg
The function
9. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 3 3.pg
Consider the function f (x) = −4x2 + 10x − 1. f (x) is increasing on the interval (−∞, A] and decreasing on the interval [A, ∞)
where A is the critical number.
Find A
At x = A, does f (x) have a local min, a local max, or neither?
Type in your answer as LMIN, LMAX, or NEITHER.
f (x) = 6x3 − 27x2 − 504x + 1
,
).
is decreasing on the interval (
It is increasing on the interval ( −∞,
) and the interval (
∞ ).
The function has a local maximum at
.
1
,
[A, B):
(B,C]:
[C, ∞):
16. (1 pt) rochesterLibrary/setDerivatives10MaxMin/c3s3p2.pg
The function
f (x) = −2x3 − 5.82x2 + 211.797x + 4.16
21. (1 pt) rochesterLibrary/setDerivatives10MaxMin/sc4 3 10.pg
Consider the function f (x) = x2 e3x .
For this function there are three important intervals: (−∞, A],
[A, B], and [B, ∞) where A and B are the critical numbers.
Find A
and B
For each of the following intervals, tell whether f (x) is increasing (type in INC) or decreasing (type in DEC).
(−∞, A]:
[A, B]:
[B, ∞)
,
).
is increasing on the interval (
) and the interval (
It is decreasing on the interval ( −∞,
, ∞ ).
.
The function has a local maximum at
17. (1 pt) rochesterLibrary/setDerivatives10MaxMin/c3s3p3.pg
For x ∈ [−14, 11] the function f is defined by
f (x) = x3 (x + 6)2
On which two intervals is the function increasing (enter intervals
in ascending order)?
to
and
to
Find the region in which the function is positive:
to
22. (1 pt) rochesterLibrary/setDerivatives10MaxMin/c3s4p1.pg
Answer the following questions for the function
p
f (x) = x x2 + 16
Where does the function achieve its minimum?
18. (1 pt) rochesterLibrary/setDerivatives10MaxMin/c3s3p4.pg
For x ∈ [−14, 15] the function f is defined by
defined on the interval [−6, 7].
A.
B.
C.
D.
E.
f (x) = x4 (x − 4)3
On which two intervals is the function increasing?
to
and
to
Find the region in which the function is positive:
to
f (x) is concave down on the region
to
f (x) is concave up on the region
to
The inflection point for this function is at
The minimum for this function occurs at
The maximum for this function occurs at
23. (1 pt) rochesterLibrary/setDerivatives10MaxMin/c3s4p2.pg
Answer the following questions for the function
p
p
f (x) = x x2 − 2x + 26 − 1 x2 − 2x + 26
Where does the function achieve its minimum?
19. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 3 6.pg
Consider the function f (x) = −2x3 + 45x2 − 300x + 5. For this
function there are three important intervals: (−∞, A], [A, B], and
[B, ∞) where A and B are the critical numbers.
Find A
and B
For each of the following intervals, tell whether f (x) is increasing (type in INC) or decreasing (type in DEC).
(−∞, A]:
[A, B]:
[B, ∞):
defined on the interval [−5, 8].
A.
B.
C.
D.
E.
f (x) is concave down on the region
to
f (x) is concave up on the region
to
The inflection point for this function is at
The minimum for this function occurs at
The maximum for this function occurs at
24. (1 pt) rochesterLibrary/setDerivatives10MaxMin/c3s4p2a.pg
Answer the following questions for the function
p
p
f (x) = x x2 + 2x + 2 + 1 x2 + 2x + 2
20. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 3 6a.pg
Consider the function f (x) = 8x + 4x−1 . For this function there
are four important intervals: (−∞, A], [A, B),(B,C), and [C, ∞)
where A, and C are the critical numbers and the function is not
defined at B.
Find A
and B
and C
For each of the following intervals, tell whether f (x) is increasing (type in INC) or decreasing (type in DEC).
(−∞, A]:
defined on the interval [−8, 6].
A.
B.
C.
D.
E.
2
f (x) is concave down on the region
to
f (x) is concave up on the region
to
The inflection point for this function is at
The minimum for this function occurs at
The maximum for this function occurs at
28. (1 pt) rochesterLibrary/setDerivatives10MaxMin/c3s4p5.pg
Answer the following questions for the function
x
f (x) = sin2
3
defined on the interval [−8.9247778, 1.65619445].
Enter points, such as inflection points in ascending order, i.e.
smallest x values first.
Rememer that you can enter ”pi” for π as part of your answer.
to
A. f (x) is concave down on the region
B. A global minimum for this function occurs at
C. A local maximum for this function which is not a global
maximum occurs at
D. The function is increasing on
to
and on
to
.
25. (1 pt) rochesterLibrary/setDerivatives10MaxMin/c3s4p3.pg
Answer the following questions for the function
f (x) =
x3
x2 − 1
defined on the interval [−15, 16].
Enter points, such as inflection points in ascending order, i.e.
smallest x values first. Enter intervals in ascending order also.
and
The function f (x) has vertical asympototes at
.
f (x) is concave up on the region
to
and
to
.
The inflection point for this function is
.
26. (1 pt) rochesterLibrary/setDerivatives10MaxMin/c3s4p3graph.pg
Answer the following questions for the function
f (x) =
29. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 4 10.pg
Consider the function f (x) = 12x5 + 45x4 − 200x3 + 7.
f (x) has inflection points at (reading from left to right) x = D,
E, and F
where D is
and E is
and F is
For each of the following intervals, tell whether f (x) is concave
up (type in CU) or concave down (type in CD).
(−∞, D]:
[D, E]:
[E, F]:
[F, ∞):
x3
.
x2 − 16
Enter points, such as inflection points in ascending order, i.e.
smallest x values first. Enter ”INF” for ∞ and ”MINF” for −∞.
Enter intervals in ascending order also.
A. The function f (x) has two vertical asympototes:
x=
and x =
B. f (x) has one local maximum and one local minimum:
max =
and min =
C. For each interval, tell whether f (x) is increasing (type in
INC) or decreasing (type in DEC).
(−∞, max)
(max, −4)
(−4, 0)
(0, 4)
(4, min)
(min, +∞)
D. f (x) is concave up on the interval (
,
)
and on the inteval (
,
)
E. The inflection point for this function is
F. Sketch the graph of f (x) and bring it to class.
30. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 4 16.pg
Consider the function f (x) = 2x+6
3x+1 . For this function there are
two important intervals: (−∞, A) and (A, ∞) where the function
is not defined at A.
Find A
For each of the following intervals, tell whether f (x) is increasing (type in INC) or decreasing (type in DEC).
(−∞, A):
(A, ∞)
Note that this function has no inflection points, but we can still
consider its concavity. For each of the following intervals, tell
whether f (x) is concave up (type in CU) or concave down (type
in CD).
(−∞, A):
(A, ∞)
27. (1 pt) rochesterLibrary/setDerivatives10MaxMin/c3s4p4.pg
Answer the following questions for the function
f (x) =
x3 + 3x2 + 3x + 1
x2 + 2x + 0
31.
4x + 7
.
6x + 5
For this function there are two important intervals: (−∞, A)
and (A, ∞) where the function is not defined at A.
Find A:
Find the horizontal asymptote of f (x):
y=
Find the vertical asymptote of f (x):
x=
defined on the interval [−20, 19].
Enter points, such as inflection points in ascending order, i.e.
smallest x values first.
A. The function f (x) has vertical asympototes at
and
B. f (x) is concave down on the region
to
and
to
(1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 4 16graph.pg
Consider the function f (x) =
3
For each of the following intervals, tell whether f (x) is increasing (type in INC) or decreasing (type in DEC).
(−∞, A):
(A, ∞):
Note that this function has no inflection points, but we can
still consider its concavity. For each of the following intervals,
tell whether f (x) is concave up (type in CU) or concave down
(type in CD).
(−∞, A):
(A, ∞):
Sketch the graph of f (x) and bring it to class.
(B,C]:
[C, ∞)
Note that this function has no inflection points, but we can still
consider its concavity. For each of the following intervals, tell
whether f (x) is concave up (type in CU) or concave down (type
in CD).
(−∞, B):
(B, ∞):
35. (1 pt) rochesterLibrary/setDerivatives10MaxMin/sc4 3 10a.pg
Consider the function f (x) = x2 e16x .
f (x) has two inflection points at x = C and x = D with C ≤ D
where C is
and D is
Finally for each of the following intervals, tell whether f (x) is
concave up (type in CU) or concave down (type in CD).
(−∞,C]:
[C, D]:
[D, ∞)
32. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 4 16a.pg
Consider the function f (x) = 7(x −2)2/3 . For this function there
are two important intervals: (−∞, A) and (A, ∞) where A is a
critical number.
Find A
For each of the following intervals, tell whether f (x) is increasing (type in INC) or decreasing (type in DEC).
(−∞, A):
(A, ∞):
For each of the following intervals, tell whether f (x) is concave
up (type in CU) or concave down (type in CD).
(−∞, A):
(A, ∞):
36. (1 pt) rochesterLibrary/setDerivatives10MaxMin/osu dr 10 1.pg
Consider the function
f (x) =
33. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 4 6.pg
Consider the function f (x) = −2x3 + 45x2 − 300x + 4. For this
function there are three important intervals: (−∞, A], [A, B], and
[B, ∞) where A and B are the critical numbers.
Find A
and B
For each of the following intervals, tell whether f (x) is increasing (type in INC) or decreasing (type in DEC).
(−∞, A]:
[A, B]:
[B, ∞)
f (x) has an inflection point at x = C
where C is
Finally for each of the following intervals, tell whether f (x) is
concave up (type in CU) or concave down (type in CD).
(−∞,C]:
[C, ∞)
ex
8 + ex
Then f 0 (x) =
The following questions ask for endpoints of intervals of increase or decrease for the function f (x).
Write INF for ∞, MINF for −∞, and NA (ie. not applicable) if
there are no intervals of that type.
The interval of increase for f (x) is from
to
The interval of decrease for f (x) is from
to
f (x) has a local minimum at
. (Put NA if none.)
f (x) has a local maximum at
. (Put NA if none.)
Then f 00 (x) =
The following questions ask for endpoints of intervals of upward and downward concavity for the function f (x).
Write INF for ∞, MINF for −∞, and put NA if there are no
intervals of that type.
The interval of upward concavity for f (x) is from
to
The interval of downward concavity for f (x) is from
34. (1 pt) rochesterLibrary/setDerivatives10MaxMin/s3 4 6a.pg
Consider the function f (x) = 2x + 8x−1 . For this function there
are four important intervals: (−∞, A], [A, B),(B,C], and [C, ∞)
where A, and C are the critical numbers and the function is not
defined at B.
Find A
and B
and C
For each of the following intervals, tell whether f (x) is increasing (type in INC) or decreasing (type in DEC).
(−∞, A]:
[A, B):
to
f (x) has a point of inflection at
. (Put NA if none.)
37. (1 pt) rochesterLibrary/setDerivatives10MaxMin/k1.pg
Let f (x) = −x2 + 3x on the interval [1, 3]. Find the absolute
maximum and absolute minimum of f (x) on this interval.
The absolute max occurs at x =
.
The absolute min occurs at x =
.
4
38. (1 pt) rochesterLibrary/setDerivatives10MaxMin/k2.pg
Let f (x) = x3 − (3/2)x2 on the interval [−1, 2]. Find the absolute maximum and absolute minimum of f (x) on this interval.
.
The absolute max occurs at x =
The absolute min occurs at x =
.
39. (1 pt) rochesterLibrary/setDerivatives10MaxMin/k3.pg
Let f (x) = 3x2/3 − 2x on the interval [−1, 1]. Find the absolute
maximum and absolute minimum of f (x) on this interval.
.
The absolute max occurs at x =
The absolute min occurs at x =
.
40. (1 pt) rochesterLibrary/setDerivatives10MaxMin/k4.pg
Let f (t) = 3 − |t − 3| on the interval [−1, 5]. Find the absolute
maximum and absolute minimum of f (t) on this interval.
The absolute max occurs at t =
.
The absolute min occurs at t =
.
41. (1 pt) rochesterLibrary/setDerivatives10MaxMin/k5.pg
Let g(s) = 1/(s − 2) on the interval [0, 1]. Find the absolute
maximum and absolute minimum of g(s) on this interval.
.
The absolute max occurs at s =
The absolute min occurs at s =
.
42. (1 pt) rochesterLibrary/setDerivatives10MaxMin/k6.pg
√
Let f (t) = t 4 − t on the interval [−1, 3]. Find the absolute
maximum and absolute minimum of f (t) on this interval.
.
The absolute max occurs at t =
The absolute min occurs at t =
.
43. (1 pt) rochesterLibrary/setDerivatives10MaxMin/k7.pg
Let g(x) = (4x)/(x2 + 1) on the interval [−4, 0]. Find the absolute maximum and absolute minimum of g(x) on this interval.
The absolute max occurs at x =
.
The absolute min occurs at x =
.
c
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5
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives10 5Optim due 02/10/2006 at 02:00pm EST.
The length of the shortest ladder is
feet.
1. (1 pt) rochesterLibrary/setDerivatives10 5Optim/c3s8p1.pg
Find the point on the line −4x + 4y + 6 = 0 which is closest to
the point (4, 5).
,
)
(
7. (1 pt) rochesterLibrary/setDerivatives10 5Optim/s3 8 6.pg
A rancher wants to fence in an area of 1500000 square feet in a
rectangular field and then divide it in half with a fence down the
middle parallel to one side. What is the shortest length of fence
that the rancher can use?
2. (1 pt) rochesterLibrary/setDerivatives10 5Optim/c3s8p2.pg
A rectangle is inscribed with its base on the x-axis and its upper
corners on the parabola y = 9 − x2 . What are the dimensions of
such a rectangle with the greatest possible area?
Width =
Height =
8. (1 pt) rochesterLibrary/setDerivatives10 5Optim/dereco1.pg
√
x2
For the given cost function C(x) = 54 x + 421875
find
a) The cost at the production level 1300
b) The average cost at the production level 1300
3. (1 pt) rochesterLibrary/setDerivatives10 5Optim/c3s8p3.pg
A cylinder is inscribed in a right circular cone of height 4 and
radius (at the base) equal to 3.5. What are the dimensions of
such a cylinder which has maximum volume?
Height =
Radius =
c) The marginal cost at the production level 1300
d) The production level that will minimize the average cost.
4. (1 pt) rochesterLibrary/setDerivatives10 5Optim/nsc4 6 3.pg
If 1300 square centimeters of material is available to make a
box with a square base and an open top, find the largest possible
volume of the box.
cubic centimeters.
Volume =
5. (1 pt) rochesterLibrary/setDerivatives10 5Optim/nsc4 6 16.pg
A fence 4 feet tall runs parallel to a tall building at a distance
of 6 feet from the building. What is the length of the shortest
ladder that will reach from the ground over the fence to the wall
of the building?
e) The minimal average cost.
9. (1 pt) rochesterLibrary/setDerivatives10 5Optim/dereco2.pg
For the given cost function
C(x) = 22500 + 600x + x2 find:
a) The cost at the production level 1350
b) The average cost at the production level 1350
c) The marginal cost at the production level 1350
d) The production level that will minimize the average cost
6. (1 pt) rochesterLibrary/setDerivatives10 5Optim/nsc4 6 16a.pg
A fence 6 feet tall runs parallel to a tall building at a distance of
3 feet from the building. We want to find the the length of the
shortest ladder that will reach from the ground over the fence to
the wall of the building.
Here are some hints for finding a solution:
Use the angle that the ladder makes with the ground to define
the position of the ladder and draw a picture of the ladder leaning against the wall of the building and just touching the top of
the fence.
If the ladder makes an angle 1.34 radians with the ground,
touches the top of the fence and just reaches the wall, calculate
the distance along the ladder from the ground to the top of the
fence.
e) The minimal average cost
10.
(1 pt) rochesterLibrary/setDerivatives10 5Optim/dereco3.pg
For the given cost function
C(x) = 5050 + 590x + 0.4x2 and the demand fuction p(x) =
1770.
Find the production level that will maximaze profit.
11. (1 pt) rochesterLibrary/setDerivatives10 5Optim/dereco4.pg
A manufacture has been selling 1250 television sets a week at
390 each. A market survey indicates that for each 25 rebate offered to a buyer, the number of sets sold will increase by 250
per week.
a) Find the demand function p(x), where x is the number of the
television sets sold per week.
p(x) =
b) How large rebate should the company offer to a buyer, in order to maximize its revenue?
c) If the weekly cost function is 81250 + 130x, how should it set
the size of the rebate to maximize its profit?
The distance along the ladder from the top of the fence to the
wall is
Using these hints write a function L(x) which gives the total length of a ladder which touches the ground at an angle x,
touches the top of the fence and just reaches the wall.
L(x) =
.
Use this function to find the length of the shortest ladder
which will clear the fence.
1
To solve this problem, we need to minimize the following function of x:
f (x) =
over the closed interval [a, b] where a =
and b =
.
We find that f (x) has only one critical number in the interval at
x=
where f (x) has value
Since this is smaller than the values of f (x) at the two endpoints,
we conclude that this is the minimal sum of distances.
16. (1 pt) rochesterLibrary/setDerivatives10 5Optim/osu dr 10 5 2.pg
Centerville is the headquarters of Greedy Cablevision Inc. The
cable company is about to expand service to two nearby towns,
Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of
cable by arranging the cable in a Y-shaped configuation. Centerville is located at (9, 0) in the xy-plane, Springfield is at (0, 3),
and Shelbyville is at (0, −3). The cable runs from Centerville to
some point (x, 0) on the x-axis where it splits into two branches
going to Springfield and Shelbyville. Find the location (x, 0)
that will minimize the amount of cable between the 3 towns and
compute the amount of cable needed. Justify your answer.
To solve this problem we need to minimize the following
function of x:
f (x) =
We find that f (x) has a critical number at x =
To verify that f (x) has a minimum at this critical number we
compute the second derivative f 00 (x) and find that its value at
the critical number is
, a positive number.
Thus the minimum length of cable needed is
12. (1 pt) rochesterLibrary/setDerivatives10 5Optim/dereco5.pg
A baseball team plays in he stadium that holds 62000 spectators. With the ticket price at 9 the average attendence has been
26000. When the price dropped to 7, the averege attendence
rose to 31000.
a) Find the demand function p(x), where x is the number of the
spectators. (assume p(x) is linear) p(x) =
b) How should be set a ticket price to maximize revenue?
13. (1 pt) rochesterLibrary/setDerivatives10 5Optim/nsc4 7 16.pg
The manager of a large apartment complex knows from experience that 100 units will be occupied if the rent is 360 dollars
per month. A market survey suggests that, on the average, one
additional unit will remain vacant for each 9 dollar increase in
rent. Similarly, one additional unit will be occupied for each 9
dollar decrease in rent. What rent should the manager charge to
maximize revenue?
14. (1 pt) rochesterLibrary/setDerivatives10 5Optim/s3 8 26.pg
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of
the rectangle. What is the area of the largest possible Norman
window with a perimeter of 50 feet?
15. (1 pt) rochesterLibrary/setDerivatives10 5Optim/osu dr 10 5 1.pg
Let Q = (0, 6) and R = (11, 7) be given points in the plane. We
want to find the point P = (x, 0) on the x-axis such that the sum
of distances PQ+PR is as small as possible. (Before proceeding
with this problem, draw a picture!)
c
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Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives11Newton due 02/11/2006 at 02:00am EST.
5. (1 pt) rochesterLibrary/setDerivatives11Newton/s2 10 22.pg
Use Newton’s method to approximate a root of the equation
cos(x2 + 5) = x3 as follows.
Let x1 = 1 be the initial approximation.
The second approximation x2 is
1. (1 pt) rochesterLibrary/setDerivatives11Newton/s2 10 3.pg
Use Newton’s method to approximate a root of the equation
x3 + x + 3 = 0 as follows.
Let x1 = −1 be the initial approximation.
The second approximation x2 is
and the third approximation x3 is
6. (1 pt) rochesterLibrary/setDerivatives11Newton/s2 10 22a.pg
Use Newton’s method to approximate a root of the equation
cos(x2 + 5) = x3 as follows.
Let x1 = 1 be the initial approximation.
The second approximation x2 is
The third approximation x3 is
2. (1 pt) rochesterLibrary/setDerivatives11Newton/s2 10 4.pg
Use Newton’s method to approximate a root of the equation
4x3 + 2x + 4 = 0 as follows.
Let x1 = −1 be the initial approximation.
,
The second approximation x2 is
and the third approximation x3 is
.
7. (1 pt) rochesterLibrary/setDerivatives11Newton/c2s10p1.pg
Find the positive value of x which satisfies x = 1.400 sin(x).
Give the answer to four places of accuracy.
Remember to calculate the trig functions in radian mode.
3. (1 pt) rochesterLibrary/setDerivatives11Newton/s2 10 20.pg
Use Newton’s method to approximate a root of the equation
2x7 + 5x4 + 4 = 0 as follows.
Let x1 = 1 be the initial approximation.
The second approximation x2 is
and the third approximation x3 is
8. (1 pt) rochesterLibrary/setDerivatives11Newton/c2s10p2.pg
Find the positive value of x which satisfies x = 1.1 cos(x). Give
the answer to six places of accuracy.
Remember to calculate the trig functions in radian mode.
4. (1 pt) rochesterLibrary/setDerivatives11Newton/s2 10 11.pg
Use Newton’s method to approximate a root of the equation
4 sin(x) = x as follows.
Let x1 = 1 be the initial approximation.
The second approximation x2 is
and the third approximation x3 is
9. (1 pt) rochesterLibrary/setDerivatives11Newton/c2s10p3.pg
Find the smallest positive value of x which satisfies –
x = 3.900 cos(2.900x)
. Give the answer to four places of accuracy.
Remember to calculate the trig functions in radian mode.
c
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Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives12MVT due 02/12/2006 at 02:00am EST.
1. (1 pt) rochesterLibrary/setDerivatives12MVT/s3 2 11.pg
Consider the function f (x) = 3 − 8x2 on the interval [−6, 3].
Find the average or mean slope of the function on this interval,
i.e.
f (3) − f (−6)
=
3 − (−6)
4. (1 pt) rochesterLibrary/setDerivatives12MVT/s3 2 12.pg
Consider the function f (x) = 2x3 − 6x2 − 90x + 6 on the interval
[−5, 8]. Find the average or mean slope of the function on this
interval.
By the Mean Value Theorem, we know there exists a c in the
open interval (−5, 8) such that f 0 (c) is equal to this mean slope.
For this problem, there are two values of c that work. The
smaller one is
By the Mean Value Theorem, we know there exists a c in the
open interval (−6, 3) such that f 0 (c) is equal to this mean slope.
For this problem, there is only one c that works. Find it.
and the larger one is
2. (1 pt) rochesterLibrary/setDerivatives12MVT/c3s2p1.pg
Consider the function
5. (1 pt) rochesterLibrary/setDerivatives12MVT/s3 2 13.pg
Consider the function f (x) = 1x on the interval [3, 11]. Find
the average or mean slope of the function on this interval.
f (x) = 2x3 − 4x2 + 2x − 4
Find the average slope of this function on the interval (−3, 2).
By the Mean Value Theorem, we know there exists a c in the
open interval (3, 11) such that f 0 (c) is equal to this mean slope.
For this problem, there is only one c that works. Find it.
By the Mean Value Theorem, we know there exists a c in the
open interval (−3, 2) such that f 0 (c) is equal to this mean slope.
Find the value of c in the interval which works
3. (1 pt) rochesterLibrary/setDerivatives12MVT/s3 2 1.pg
Consider the function f (x) = 5x3 − 5x on the interval [−2, 2].
Find the average or mean slope of the function on this interval.
6. (1 pt) rochesterLibrary/setDerivatives12MVT/s3
2 14.pg
√
Consider the function f (x) = 6 x +6 on the interval [1, 8]. Find
the average or mean slope of the function on this interval.
By the Mean Value Theorem, we know there exists at least one c
in the open interval (−2, 2) such that f 0 (c) is equal to this mean
slope.
For this problem, there are two values of c that work. The
smaller one is
and the larger one is
By the Mean Value Theorem, we know there exists a c in the
open interval (1, 8) such that f 0 (c) is equal to this mean slope.
For this problem, there is only one c that works. Find it.
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives13Higher due 02/13/2006 at 02:00am EST.
1. (1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 2f.pg
If f (x) = 3x2 − 2ex , find f 0 (x).
8. (1 pt) rochesterLibrary/setDerivatives13Higher/s2 7 10.pg
Let h(t) = 3t 3.2 − 4t −3.2 .
Then h0 (t) is
h0 (2) is
,
h00 (t) is
and h00 (2) is
Find f 00 (x).
2. (1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 2.pg
If f (x) = 4x2 − 3ex , find f 0 (x).
9. (1 pt) rochesterLibrary/setDerivatives13Higher/s2 7 18.pg
0
Let f (x) = 1−4x
1+4x . Then f (4) is
00
and f (4) is
and f 000 (4) is
Find f 0 (1).
Find f 00 (x).
10.
Find f 00 (1).
(1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 5.pg
2
3.
+4x+4
Let f (x) = x 4x+8
.
0
(a) f (3) =
(b) f 00 (3) =
[NOTE: There are two ways to do this problem. The first is the
quotient rule. The second is much easier and does not use the
quotient rule.]
(1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 4.pg
Let f (x) = x7 − 4ex .
(a) f 0 (−1) =
(b) f 00 (−1) =
4. (1 pt) rochesterLibrary/setDerivatives13Higher/s2 7 3.pg
Let f (x) = x4 + 3x3 + 6x2 + 3x.
Then f 0 (x) is
and f 0 (3) is
f 00 (x) is
and f 00 (3) is
11. (1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 6.pg
If g(t) = 2t 4 − 3t 2 + 4 find
g(0) =
g0 (0) =
g00 (0) =
g000 (0) =
g(4) (0) =
g(5) (0) =
5. (1 pt) rochesterLibrary/setDerivatives13Higher/s2 7 4.pg
Let f (x) = x7 − 5x5 + 2x3 − 2x − 10.
Then f 0 (x) is
f 0 (2) is
f 00 (x) is
and f 00 (2) is
12. (1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 7.pg
Let f (x) = x sin(x). Find f 00 (0.8).
(Remember – radian mode!)
6. (1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 1.pg
If f (x) = 3x8 − 8x5 − 5x3 + 5x, find f 0 (x).
Find
13. (1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 8.pg
Let h(t) = tan(5x + 2). Then h0 (2) is
and h00 (2) is
f 0 (3).
Find f 00 (x).
14. (1 pt) rochesterLibrary/setDerivatives13Higher/s2 7 7.pg
Let g(s) = (2s − 2)9 .
Then g0 (s) is
g0 (5) is
,
g00 (s) is
and g00 (5) is
Find f 00 (3).
7. (1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 3.pg
If f (x) = 3 + 7x + x72 , find f 0 (x).
Find f 0 (1).
15. (1 pt) rochesterLibrary/setDerivatives13Higher/c2s7p1.pg
If g(t) = (10 − t 1 )2 find
g(0) =
g0 (0) =
g00 (0) =
Find f 00 (x).
Find f 00 (1).
1
20.
Let
16. (1 pt)√
rochesterLibrary/setDerivatives13Higher/s2 7 5.pg
Let f (x) = x2 + 10.
Then f 0 (x) is
f 0 (5) is
,
f 00 (x) is
and f 00 (5) is
17.
Let
f (x) =
21.
Let
(1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 11.pg
(1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 14.pg
f (x) = −5 ln[sin(x)]
f 00 (x) =
22.
Let
f (9) (1) =
(1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 12.pg
f (x) = 3 ln[sec(x) + tan(x)]
f 00 (x) =
HINT: Simplify the first derivative before you find the second
derivative.
23. (1 pt) rochesterLibrary/setDerivatives13Higher/hder1.pg
Find the 45 th derivative of the function f (x) = cos(x).
The answer is function
24. (1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 15.pg
If f (x) = 9x2 ln(4x), then
f 0 (x) =
f 00 (x) =
f 000 (x) =
f (4) (x) =
f (5) (x) =
18. (1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 9.pg
d 4 −7x4
(
)=
dx4 1−x
Note: There is a way of doing this problem without using the
quotient rule 4 times.
(1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 10.pg
f (x) =
−4x
1−x
f (4) (x) =
f (x) = −3e−x/4
19.
Let
(1 pt) rochesterLibrary/setDerivatives13Higher/ur dr 13 13.pg
9x4
1−x
f (4) (x) =
Note: There is a way of doing this problem without using the
quotient rule 4 times.
c
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Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives14Hyperbolic due 02/14/2006 at 02:00am EST.
1.
(1
pt)
6.
(1
/csuf dr 14 06.pg
rochesterLibrary/setDerivatives14Hyperbolic-
/csuf dr 14 01.pg
If
pt)
rochesterLibrary/setDerivatives14Hyperbolic-
If
f (x) = ecosh(5x)
f (x) = tanh(5x)
then
f 0 (x) =
then f 0 (x) =
.
2.
(1
/csuf dr 14 02.pg
pt)
rochesterLibrary/setDerivatives14Hyperbolic-
7.
.
(1
pt)
rochesterLibrary/setDerivatives14Hyperbolic-
/csuf dr 14 07.pg
If
If
f (x) = x cosh x + 4 sinh x
.
then f 0 (x) =
3.
(1
pt)
f (x) = x cosh−1
x
5
−
p
x2 − 25
then f 0 (7) =
rochesterLibrary/setDerivatives14Hyperbolic-
8.
/csuf dr 14 03.pg
.
(1
pt)
rochesterLibrary/setDerivatives14Hyperbolic-
/csuf dr 14 08.pg
If
If
f (x) = sinh5 x
then
f 0 (x) =
4.
(1
/csuf dr 14 04.pg
.
pt)
9.
(1
/csuf dr 14 09.pg
f (x) =
7 − cosh x
2 + cosh x
then f 0 (x) =
5.
(1
/csuf dr 14 05.pg
then f 0 (x) =
rochesterLibrary/setDerivatives14Hyperbolic-
If
then f 0 (x) =
rochesterLibrary/setDerivatives14Hyperbolic-
10.
(1
/csuf dr 14 10.pg
pt)
p
x2 + 2
.
rochesterLibrary/setDerivatives14Hyperbolic-
If
f (t) = 3t secht
then
rochesterLibrary/setDerivatives14Hyperbolic-
f (x) = coth−1
If
f 0 (t) =
pt)
If
.
pt)
f (x) = x5 sinh−1 (8x)
.
f (x) = tanh−1
.
then f 0 (x) =
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
√
4
x
.
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives1 5Tangents due 02/01/2006 at 05:00am EST.
Use this to find the equation of the tangent line to the curve
3x
y = 1+x
2 at the point (5, 0.57692). The equation of this tangent line can be written in the form y = mx + b where m is:
1. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 1.pg
If f (x) = 2x2 − 3x + 4, find f 0 (2).
Use this to find the equation of the tangent line to the parabola
y = 2x2 − 3x + 4 at the point (2, 6). The equation of this tangent
line can be written in the form y = mx+b where m is:
and where b is:
and where b is:
8. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 8.pg
The parabola y = x2 +4 has two tangents which pass through the
point (0, −4). One is tangent to the to the parabola at (A, A2 + 4)
and the other at (−A, A2 + 4). Find (the positive number) A.
2. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 2.pg
If h(x) = 5 − 3x3 , find h0 (1).
Use this to find the equation of the tangent line to the curve
y = 5 − 3x3 at the point (1, 2). The equation of this tangent line
can be written in the form y = mx + b where m is:
and where b is:
9.
(1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 8a.pg
On a separate piece of paper, sketch the graph of the parabola
y = x2 + 6. On the same graph, plot the point (0, −4). Note that
there are two tangent lines of y = x2 + 6 that pass through the
point (0, −4).
Specifically, the tangent line of the parabola y = x2 + 6 at the
point (a, a2 + 6) passes through the point (0, −4) where a > 0.
The other tangent line that passes through the point (0, −4) occurs at the point (−a, a2 + 6).
Find the number a.
3. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 3.pg
If f (x) = 4x , find f 0 (2).
Use this to find the equation of the tangent line to the hyperbola
y = 4x at the point (2, 2.000). The equation of this tangent line
can be written in the form y = mx + b where m is:
and where b is:
4. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 4.pg
If f (x) = 5x + 2x , find f 0 (4).
10. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 9.pg
The graph of f (x) = 2x3 + 6x2 − 90x + 7 has two horizontal tangents. One occurs at a negative value of x and the other at a
positive value of x. What is the negative value of x where a
horizontal tangent occurs?
What is the positive value of x where a horizontal tangent occurs?
Use this to find the equation of the tangent line to the curve
y = 5x + 2x at the point (4, 20.50000). The equation of this tangent line can be written in the form y = mx + b where m is:
and where b is:
5. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 5.pg
4
If f (x) = x−2
, find f 0 (5).
11. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 10.pg
For what values of x does the graph of
Use this to find the equation of the tangent line to the curve
4
at the point (5, 1.33333). The equation of this tany = x−2
gent line can be written in the form y = mx + b where m is:
f (x) = 4x3 + 30x2 + 48x − 48
and where b is:
have a horizontal tangent? Enter the x values in order, smallest
first, to 4 places of accuracy:
x1 =
≤ x2 =
6. (1 pt) rochesterLibrary/setDerivatives1
5Tangents/ur dr 1 5 6.pg
√
If f (x) = 3x + 3 x, find f 0 (5).
12. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 11.pg
For what values of x does the graph of
f (x) = 7x3 − 65.625x2 + 199.29x − 11.55
Use this to√find the equation of the tangent line to the curve
y = 3x + 3 x at the point (5, 21.70820). The equation of this
tangent line can be written in the form y = mx + b where m is:
have a horizontal tangent? Enter the x values in order, smallest
first, to 4 places of accuracy:
≤ x2 =
x1 =
13. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 12.pg
For what values of x is the tangent line of the graph of
and where b is:
7.
(1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 7.pg
f (x) = 4x3 + 6x2 − 142x − 48
If
3x
f (x) =
1 + x2
parallel to the line y = 2x + 1.9 ? Enter the x values in order,
smallest first, to 4 places of accuracy:
x1 =
≤ x2 =
find f 0 (5).
1
19.
14. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 13.pg
For what values of x is the tangent line of the graph of
3
(1
pt)
rochesterLibrary/setDerivatives1 5Tangents-
/ur dr 1 5 17a.pg
Let f (x) = 57 − x2
The slope of the tangent line to the graph of f (x) at the point
.
(−7, 8) is
The equation of the tangent line to the graph of f (x) at (−7, 8)
is y = mx + b for
m=
and
.
b=
Hint: the slope is given by the derivative at x = −7, ie.
f (−7 + h) − f (−7)
lim
x→−7
h
2
f (x) = 3.4x − 18.36x + 12.056x + 38.76
parallel to the line y = −1x − 1.1 ? Enter the x values in order,
smallest first, to 4 places of accuracy:
x1 =
≤ x2 =
15. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 14.pg
Given
√
f (x) = x + x
20. (1 pt) √
rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 18.pg
Let f (x) = 65 − x
The slope of the tangent line to the graph of f (x) at the point
(1, 8) is
.
The equation of the tangent line to the graph of f (x) at (1, 8) is
y = mx + b for
m=
and
b=
.
Hint: the slope is given by the derivative at x = 1, ie.
f (1 + h) − f (1)
lim
x→1
h
21. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 19.pg
Let f (x) = 17
x
The slope of the tangent line to the graph of f (x) at the point
(−4, − 17
.
4 ) is
The equation of the tangent line to the graph of f (x) at
(−4, − 17
4 ) is y = mx + b for
m=
and
b=
.
Hint: the slope is given by the derivative at x = −4, ie.
f (−4 + h) − f (−4)
lim
x→−4
h
Calculate the tangent line at the point (25, 30)
(x − 25) + 30
y=
For similar problems see p120:36–39.
16. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 15.pg
At what point does the normal to y = −1 − 3x + 3x2 at (1, −1)
intersect the parabola a second time?
,
)
(
The normal line is perpendicular to the tangent line. If two
lines are perpendicular their slopes are negative reciprocals – i.e.
if the slope of the first line is m then the slope of the second line
is −1/m
17. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 16.pg
For what values of a and b is the line −5x + y = b tangent to the
curve y = ax3 when x = −2?
a=
b=
18. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 17.pg
Let f (x) = 3x2 − 10x + 11
The slope of the tangent line to the graph of f (x) at the point
(1, 4) is
.
The equation of the tangent line to the graph of f (x) at (1, 4) is
y = mx + b for
m=
and
b=
.
Hint: the slope is given by the derivative at x = 1, ie.
f (1 + h) − f (1)
lim
x→1
h
22. (1 pt) rochesterLibrary/setDerivatives1 5Tangents/ur dr 1 5 20.pg
Find a, b, c, d such that the cubic function f (x) = ax3 + bx2 +
cx + d has horizontal tangent lines at (−1, −7) and (1, −3).
a=
b=
c=
d=
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives20Antideriv due 02/20/2006 at 02:00am EST.
11. (1 pt) rochesterLibrary/setDerivatives20Antideriv/c3s10p5.pg
Given f 00 (x) = −4 sin(2x) and f 0 (0) = −6 and f (0) = −4.
Find f (π/5) =
1. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 2func.pg
Consider the function f (x) = 20x3 − 12x2 + 10x − 10. Enter an
antiderivative of f (x)
12.
Given
2. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 2.pg
Consider the function f (x) = 9x3 − 3x2 + 6x − 5.
An antiderivative of f (x) is F(x) = Ax4 + Bx3 +Cx2 + Dx
where A is
and B is
and C is
and D is
f 00 (x) = 9x + 5
and f 0 (−2) = 6 and f (−2) = 3.
Find f 0 (x) =
and find f (1) =
3. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 3func.pg
Consider the function f (x) = 2x8 + 7x6 − 10x2 − 10.
Enter an antiderivative of f (x)
13. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 35.pg
Given that the graph of f (x) passes through the point (5, 3) and
that the slope of its tangent line at (x, f (x)) is 4x + 2, what is
f (3)?
4. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 3.pg
Consider the function f (x) = 6x10 + 9x5 − 2x2 − 9.
An antiderivative of f (x) is F(x) = Axn + Bxm + Cx p + Dxq
where
and n is
A is
and B is
and m is
and C is
and p is
and D is
and q is
5.
(1 pt) rochesterLibrary/setDerivatives20Antideriv/c3s10p1.pg
14. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 51.pg
A particle is moving with acceleration a(t) = 6t + 12. its position at time t = 0 is s(0) = 6 and its velocity at time t = 0 is
v(0) = 8. What is its position at time t = 5?
15. (1 pt) rochesterLibrary/setDerivatives20Antideriv/c3s10p2.pg
A car traveling at 40 ft/sec decelerates at a constant 4 feet per
second squared. How many feet does the car travel before coming to a complete stop?
(1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 8func.pg
8
3
− .
x3 x7
Let F(x) be the antiderivative of f (x) with F(1) = 0.
Then F(x) =
Consider the function f (x) =
6.
16. (1 pt) rochesterLibrary/setDerivatives20Antideriv/c3s10p3.pg
A ball is shot straight up into the air with initial velocity of 44
ft/sec. Assuming that the air resistance can be ignored, how high
does it go?
(1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 8.pg
7
8
− 8.
4
x
x
Let F(x) be the antiderivative of f (x) with F(1) = 0.
Then F(5) =
Consider the function f (x) =
Hint: The acceleration due to gravity is 32 ft per second
squared.
17. (1 pt) rochesterLibrary/setDerivatives20Antideriv/c3s10p4.pg
A ball is shot at an angle of 45 degrees into the air with initial
velocity of 43 ft/sec. Assuming no air resistance, how high does
it go?
7. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 13func.pg
Consider the function f (t) = 3 sec2 (t) − 8t 3 . Let F(t) be the
antiderivative of f (t) with F(0) = 0.
Then F(t) equals
How far away does it land?
8. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 13.pg
Consider the function f (t) = 8 sec2 (t) − 2t 2 . Let F(t) be the
antiderivative of f (t) with F(0) = 0.
Then F(3) =
Hint: The acceleration due to gravity is 32 ft per second
squared.
18. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 56.pg
A stone is thrown straight up from the edge of a roof, 675 feet
above the ground, at a speed of 10 feet per second.
A. Remembering that the acceleration due to gravity is -32 feet
per second squared, how high is the stone 6 seconds later?
9. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 30func.pg
Consider the function f (x) whose second derivative is f 00 (x) =
5x + 3 sin(x). If f (0) = 2 and f 0 (0) = 2, what is f (x)?
10. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 30.pg
Consider the function f (x) whose second derivative is f 00 (x) =
8x + 9 sin(x). If f (0) = 2 and f 0 (0) = 3, what is f (3)?
B. At what time does the stone hit the ground?
C. What is the velocity of the stone when it hits the ground?
1
Enter an antiderivative of f (x)
19. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 57.pg
A stone is thrown straight down from the edge of a roof, 750
feet above the ground, at a speed of 5 feet per second.
A. Remembering that the acceleration due to gravity is -32 feet
per second squared, how high is the stone 3 seconds later?
22.
(1 pt) rochesterLibrary/setDerivatives20Antideriv/ur dr 20 2.pg
9
Let f (x) = √
.
1 − x2
Enter an antiderivative of f (x).
B. At what time does the stone hit the ground?
C. What is the velocity of the stone when it hits the ground?
23.
(1 pt) rochesterLibrary/setDerivatives20Antideriv/ur dr 20 3.pg
Let f (x) =
20. (1 pt) rochesterLibrary/setDerivatives20Antideriv/s3 10 67.pg
A stone is dropped from the edge of a roof, and hits the ground
with a velocity of -200 feet per second. How high (in feet) is the
roof?
21. (1 pt) rochesterLibrary/setDerivatives20Antideriv/ur dr 20 1.pg
5
Let f (x) = − 8ex .
x
14
x2 + 1
.
Enter an antiderivative of f (x)
24. (1 pt) rochesterLibrary/setDerivatives20Antideriv/csuf
dr 20 1.pg
√
Consider the function f (x) = x5 + 5 x.
Let F(x) be the antiderivative of f (x) with F(1) = −4.
Then F(x) =
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives21LHospital due 02/21/2006 at 02:00am EST.
1. (1 pt) rochesterLibrary/setDerivatives21LHospital/sc4 5 00.pg
Evaluate the limit
p
lim x2 + 7x + 13 − x
9.
(1 pt) rochesterLibrary/setDerivatives21LHospital/sc4 5 8.pg
Evaluate the limit using L’Hospital’s rule
x→∞
3x − 13x
x→0
x
lim
2.
(1 pt) rochesterLibrary/setDerivatives21LHospital/sc4 5 00-
asympt.pg
The function
p
10. (1 pt) rochesterLibrary/setDerivatives21LHospital/osu dr 21 1.pg
Compute the following limits using l’Hôpital’s rule if appropriate. Use INF to denote ∞ and MINF to denote −∞.
4x − 4
lim 2
=
x→1 x − 1
tan−1 (x)
lim
=
x→∞ (1/x) − 4
x2 + 3x + 5 − x
has one horizontal asymptote at y =
3.
(1 pt) rochesterLibrary/setDerivatives21LHospital/ur dr 21 1.pg
Evaluate the limit using L’Hopital’s rule
4x3
x→∞ e5x
11. (1 pt) rochesterLibrary/setDerivatives21LHospital/osu dr 21 2.pg
Compute the following limits using l’Hôpital’s rule if appropriate. Use INF to denote ∞ and MINF to denote −∞.
1 − cos(7x)
lim
=
x→0 1 − cos(4x)
x
x
5 −4 −1
lim
=
x→1
x2 − 1
lim
4.
(1 pt) rochesterLibrary/setDerivatives21LHospital/ur dr 21 2.pg
Evaluate the limit using L’Hopital’s rule
lim 4 cos(−3x) sec(−5x)
x→ π2
12.
(1
pt)
rochesterLibrary/setDerivatives21LHospital-
/osu dr 21 10.pg
5. (1 pt) rochesterLibrary/setDerivatives21LHospital/osu dr 21 4.pg
Compute the following limit using l’Hôpital’s rule if appropriate. Usep
INF to denote ∞ and MINF to denote −∞.
3 3
lim ( x − 4x2 − x) =
Compute the following limits using l’Hôpital’s rule if appropriate. Use INF to denote ∞ and MINF to denote −∞.
ln(x2 − 9)
lim
=
x→∞ ln(x) cos(1/x)
e9x
lim 10x
=
x→∞ e
− e−10x
x→∞
6.
(1 pt) rochesterLibrary/setDerivatives21LHospital/sc4 5 3.pg
13. (1 pt) rochesterLibrary/setDerivatives21LHospital/osu dr 21 5.pg
Compute the following limit using l’Hopital’s rule if appropriate. Use INF to denote ∞ and MINF to denote −∞.
lim 8 sin(x) ln(x) =
Evaluate the limit using L’Hospital’s rule
ex − 1
x→0 sin(13x)
lim
x→0+
7.
(1 pt) rochesterLibrary/setDerivatives21LHospital/sc4 5 4.pg
14. (1 pt) rochesterLibrary/setDerivatives21LHospital/osu dr 21 6.pg
Find the following limits, using l’Hôpital’s rule if appropriate
arctan(x5 )
lim
=
x→∞
x8
√
8
lim x ln(x) =
Evaluate the limit using L’Hospital’s rule if necessary
lim
x→0
8.
sin(7x)
sin(14x)
x→0+
15.
(1 pt) rochesterLibrary/setDerivatives21LHospital/sc4 5 4a.pg
(1 pt) rochesterLibrary/setDerivatives21LHospital/sc4 5 23.pg
Evaluate the limit using L’Hospital’s rule
Evaluate the limit using L’Hospital’s rule
sin(8x)
x→0 tan(9x)
lim 5xe1/x − 5x
lim
x→∞
1
16.
10-13 correct = .3
14-16 correct = .5
17-19 correct = .7
Note that l’Hôpital’s rule (in some form) may ONLY be applied
to indeterminate forms.
1. π∞
1
2. −∞
3. π−∞
4. ∞∞
5. 1 · ∞
6. ∞0
7. ∞−e
8. ∞ · ∞
9. 1−∞
10. ∞1
11. 1∞
12. ∞0
13. 0−∞
14. ∞0
15. 10
16. 00
17. 0 · ∞
18. 0∞
19. ∞−∞
20. ∞ − ∞
(1 pt) rochesterLibrary/setDerivatives21LHospital/derLH1.pg
Evaluate the limit using L’Hospital’s rule if necessary
8x
16x
lim
x→∞ 16x + 8
17. (1 pt) rochesterLibrary/setDerivatives21LHospital/osu dr 21 3.pg
Compute the following limit using l’Hôpital’s rule if appropriate. UseINF todenote ∞ and MINF to denote −∞.
5 x
=
lim 1 −
x→∞
x
18.
(1 pt) rochesterLibrary/setDerivatives21LHospital/derLH2.pg
Evaluate the limit using L’Hospital’s rule if necessary
x
10 3
lim 1 +
x→∞
x
19.
(1 pt) rochesterLibrary/setDerivatives21LHospital/derLH3.pg
Evaluate the limit using L’Hospital’s rule if necessary
ln 10+1
lim (14x) ln(15x)+1
x→∞
20.
(1
pt)
rochesterLibrary/setDerivatives21LHospital-
21.
/osu dr 21 20.pg
For each of the following forms determine whether the following limit type is indeterminate, always has a fixed finite value,
or never has a fixed finite value. In the first case answer IND, in
the second case enter the numerical value, and in the third case
answer DNE. For example
IND 00
0
(1
pt)
rochesterLibrary/setDerivatives21LHospital-
/osu dr 21 21.pg
Find the following limits, using L’Hôpital’s rule, if appropriate.
Use INF to denote ∞ and MINF to denote −∞
tan−1 (x/3)
=
(a) lim
x→∞ sin−1 (1/x)
12
x cos5 (πex )
(b) lim
=
x→0 ln(1 + 7x)
0
1
22. (1 pt) rochesterLibrary/setDerivatives21LHospital/ur dr 21 3.pg
Evaluate the limit using L’Hospital’s rule
DNE 10
To discourage blind guessing, this problem is graded on the following scale
0-9 correct = 0
ex + 2x − 1
x→0
4x
lim
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives2Formulas due 02/02/2006 at 02:00am EST.
12.
1. (1 pt) rochesterLibrary/setDerivatives2Formulas/ur dr 2 11.pg
If f (x) = 5x + 9, find f 0 (x).
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 13a.pg
Let f (t) = (t 2 + 4t + 7)(7t 2 + 2).
(a) f 0 (t) =
(b) f 0 (3) =
[NOTE: Your answer to part (a) should be a function in terms
of the variable ’t’ and not a number! Your answer to part (b)
should be a number.]
2. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 1.pg
If f (x) = 6x2 − 9x − 27, find f 0 (x).
Find f 0 (3).
13. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 7.pg
If f (t) = 6t −6 , find f 0 (t).
3. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 1b.pg
If f (x) = 6x + 7, find f 0 (x).
Find f 0 (3).
4. (1 pt)√rochesterLibrary/setDerivatives2Formulas/s2 2 1c.pg
If f (x) = 5x + 8, find f 0 (x).
14.
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 8f.pg
If
√
3
f (t) = 3 ,
t
5. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 1a.pg
If f (x) = 5x2 − 11x − 3, find f 0 (x).
find f 0 (t).
6. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 4f.pg
If f (x) = 4x8 − 4x5 − 6x3 + 4x, find f 0 (x).
If
7. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 4.pg
If f (x) = 7x8 − 6x5 − 4x3 + 2x, find f 0 (x).
find f 0 (t).
Find f 0 (2).
Find f 0 (3).
8. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 6f.pg
If f (x) = (5x2 − 3)(6x + 5), find f 0 (x).
16. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 8a.pg
If f (t) = t76 , find f 0 (t).
15.
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 8.pg
√
6
f (t) = 6 ,
t
[NOTE: Your answer should be a function in terms of the variable ’t’ and not a number! ]
9. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 6.pg
If f (x) = (6x2 − 8)(3x + 4), find f 0 (x).
17.
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 11.pg
If
Find f 0 (5).
f (x) =
4x + 7
,
5x + 3
find f 0 (x).
10. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 6a.pg
If f (x) = (7x2 − 2)(4x + 2), find f 0 (x).
Find f 0 (5).
11. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 13.pg
If f (t) = (t 2 + 6t + 4)(4t 2 + 3), find f 0 (t).
18.
Find f 0 (1).
Let f (x) =
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 11a.pg
f 0 (x) =
1
6
.
4x + 4
19.
28. (1 pt)
√ rochesterLibrary/setDerivatives2Formulas/ur dr 2 12.pg
If f (x) = 3x + 8, find f 0 (x).
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 11c.pg
If
f (x) =
6x + 8
,
7x + 4
29. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 22.pg
If f (x) = 4 + 5x + x32 , find f 0 (x).
find f 0 (2).
20.
Find f 0 (4).
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 12f.pg
If
f (x) =
3 − x2
3 + x2
30. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 22b.pg
If f (x) = 4 + 6x + x42 , find f 0 (x).
find f 0 (x).
21.
31.
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 12.pg
If
√
−5
Let f (x) = 6x5 x + 2 √ .
x x
f 0 (x) =
6 − x2
f (x) =
3 + x2
find f 0 (x).
32. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2
2 29.pg
√
If f (x) = 3x x + x26√x , find f 0 (x).
Find f 0 (3).
22.
Find f 0 (2).
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 15.pg
If
f (x) =
33. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2
2 29a.pg
√
If f (x) = 6x x + x25√x , find f 0 (9).
6x2 + 7x + 3
√
,
x
find f 0 (x).
Find
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 11b.pg
34.
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 33.pg
If
f 0 (3).
f (x) =
23. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 15a.pg
2
√
If f (x) = 5x +5x+3
, find f 0 (9).
x
24.
find f 0 (x).
Find f 0 (1).
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 16.pg
If
√
x−7
f (x) = √
x+7
35. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 33a.pg
3
If f (x) = 5xx4−5 , find f 0 (x).
find f 0 (x).
36. (1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 33b.pg
5
4 −7x3
If f (x) = −7x −8x
, find f 0 (x).
x4
Find f 0 (4).
25.
5x3 − 7
x4
37. (1 pt)√rochesterLibrary/setDerivatives2Formulas/s2
2 34.pg
√
If f (x) = 2 x(x3 − 2 x + 5), find f 0 (x).
(1 pt) rochesterLibrary/setDerivatives2Formulas/s2 2 16a.pg
Let f (x) =
f 0 (16) =
√
√x−5 .
x+5
Find f 0 (2).
26. (1 pt)
√ rochesterLibrary/setDerivatives2Formulas/s2 2 17.pg
If f (x) = 9x, find f 0 (x).
38. (1 pt)√rochesterLibrary/setDerivatives2Formulas/s2
2 34a.pg
√
If f (x) = 3 x(x3 − 5 x + 3), find f 0 (16).
Find f 0 (7).
39. (1 pt) rochesterLibrary/setDerivatives2Formulas/c2s5p1.pg
Calculate G0 (−1) to 3 significant figures where
27. (1 pt)
√ rochesterLibrary/setDerivatives2Formulas/s2 2 17a.pg
If f (x) = 21x, find f 0 (x).
G(x) = (−4x − 2)10 (−3x2 + 3x + 2)12
2
47. (1 pt) rochesterLibrary/setDerivatives2Formulas/ur dr 2 2.pg
If f (x) = 6ex − 3x5 + 1, find f 0 (x).
40. (1 pt) rochesterLibrary/setDerivatives2Formulas/c2s5p3.pg
Calculate f 0 (−1) to 3 significant figures where
f (t) = (2t 2 + 4t + 3)−8
48.
Tip: You can enter an answer such as 3.14e-1 for 0.314.
Let f (x) = −16.
Then f 0 (−2) =
.
And after simplifying f 0 (x) =
41. (1 pt) rochesterLibrary/setDerivatives2Formulas/c2s5p4.pg
Find the y-intercept of the tangent line to
−0.0999999999999999
√
y=
2 + 5x
at (3, −0.0242535625036333) .
42.
49.
(1 pt) rochesterLibrary/setDerivatives2Formulas/d2.pg
50.
Let
f 0 (0) =
[NOTE: A small algebraic manipulation is needed first to get
f(x) into a form so that the derivative can be taken.]
51.
7
f (x) = x h(x)
h(−1) = 3
h0 (−1) = 6
Calculate f 0 (−1).
[HINT: Use the product rule and the power rule.]
52.
(1 pt) rochesterLibrary/setDerivatives2Formulas/ur dr 2 6.pg
(1 pt) rochesterLibrary/setDerivatives2Formulas/ur dr 2 7.pg
Let f (x) = 2x3 + 3x − 3.
Then f 0 (−1) =
.
And after simplifying f 0 (x) =
44. (1 pt) rochesterLibrary/setDerivatives2Formulas/ns3 2 4.pg
Find the derivative of the function
g(x) = (2x2 − 3x − 1)ex
53.
g0 (x) =
3
x+8
Then f 0 (−1) =
.
And after simplifying f 0 (x) =
Let f (x) =
46.
Given
Let f (x) =
54.
x
x + 1x
The derivative function is given by
f (x) =
55.
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3
.
(1 pt) rochesterLibrary/setDerivatives2Formulas/ur dr 2 10.pg
√
Let f (x) = 11 + x
Then f 0 (25) =
.
And after simplifying f 0 (x) =
x2 + x +
(x2 + )2
.
(1 pt) rochesterLibrary/setDerivatives2Formulas/ur dr 2 9.pg
5x
x−6
Then f 0 (15) =
.
And after simplifying f 0 (x) =
(1 pt) rochesterLibrary/setDerivatives2Formulas/ur dr 2 1.pg
.
(1 pt) rochesterLibrary/setDerivatives2Formulas/ur dr 2 8.pg
45. (1 pt) rochesterLibrary/setDerivatives2Formulas/ns3 2 5.pg
Find the derivative of the function
ex
g(x) =
2 − 3x
0
g (x) =
f (x) =
.
Let f (x) = −4x(x − 1).
.
Then f 0 (2) =
.
And after simplifying f 0 (x) =
Hint: You may want to expand and simplify the expression for
f (x) first.
Given that
0
.
(1 pt) rochesterLibrary/setDerivatives2Formulas/ur dr 2 5.pg
Let f (x) = x2 + 9x − 7.
.
Then f 0 (−5) =
And after simplifying f 0 (x) =
(1 pt) rochesterLibrary/setDerivatives2Formulas/d3.pg
.
(1 pt) rochesterLibrary/setDerivatives2Formulas/ur dr 2 4.pg
Let f (x) = −6x + 1.
Then f 0 (2) =
.
And after simplifying f 0 (x) =
f (x) = 4ex+3 + e−2 .
43.
(1 pt) rochesterLibrary/setDerivatives2Formulas/ur dr 2 3.pg
.
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives2 5Implicit due 02/02/2006 at 02:00pm EST.
1. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/s2 6 1.pg
If 5x2 + 5x + xy = 2 and y(2) = −14, find y0 (2) by implicit differentiation.
10. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/s2 6 25a.pg
Find the slope of the tangent line to the curve (a lemniscate)
2. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/s2 6 4.pg
√
√
If x + y = 8 and y(4) = 36, find y0 (4) by implicit differentiation.
at the point (−3, 1).
m=
2(x2 + y2 )2 = 25(x2 − y2 )
11. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/s2 6 25.pg
Find the equation of the tangent line to the curve (a lemniscate)
2(x2 + y2 )2 = 25(x2 − y2 ) at the point (3, −1). The equation of
this tangent line can be written in the form y = mx + b where m
is:
and where b is:
3. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/s2 6 2.pg
y2
x2
If 49
+ 49
= 1 and y(1) = 6.92820, find y0 (1) by implicit differentiation.
4. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/c2s6p2.pg
Find the slope of the tangent line to the curve
12. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/ur dr 25 10.pg
Use implicit differentiation to find the slope of the tangent line
to the curve
y
= x4 + 7
x + 5y
−4x2 + 4xy − 2y3 = −512
at the point (−8, 4).
8
at the point (1, −39
).
m=
5. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/c2s6p2a.pg
Use implicit differentiation to find the slope of the tangent line
to the curve
−1x2 − 1xy + 3y3 = −23
at the point (−4, −1).
m=
6. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/c2s6p3.pg
Find the slope of the tangent line to the curve xy3 −1y −0.339 =
0 at the point (1.7, 0.9).
13. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/s2 6 13.pg
Find y0 by implicit differentiation. Match the expressions defining y implicitly with the letters labeling the expressions for y0 .
1.
2.
3.
4.
A.
7. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/s2 6 19a.pg
Use implicit differentiation to find the slope of the tangent line
to the curve
xy3 + xy = 20
at the point (10, 1).
m=
8. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/s2 6 19.pg
Use implicit differentiation to find the equation of the tangent
line to the curve xy3 + xy = 12 at the point (6, 1). The equation
of this tangent line can be written in the form y = mx + b where
m is:
and where b is:
9. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/c2s6p1.pg
Find the slope of the tangent line to the curve
B.
C.
D.
4 cos(x − y) = 5y cos x
4 sin(x − y) = 5y sin x
4 cos(x − y) = 5y sin x
4 sin(x − y) = 5y cos x
4 cos(x−y)+5y sin x
4 cos(x−y)+5 cos x
−4 sin(x−y)−5y cos x
5 sin x−4 sin(x−y)
−4 sin(x−y)+5y sin x
5 cos x−4 sin(x−y)
4 cos(x−y)−5y cos x
4 cos(x−y)+5 sin x
14. (1 pt) rochesterLibrary/setDerivatives2 5Implicit/s2 6 14.pg
Find y0 by implicit differentiation. Match the expressions defining y implicitly with the letters labeling the expressions for y0 .
1.
2.
3.
4.
2x sin y + 4 cos 2y = 7 cos y
2x cos y + 4 sin 2y = 7 sin y
2x cos y + 4 cos 2y = 7 sin y
2x sin y + 4 sin 2y = 7 cos y
sin y
A. - 2x cos y+82 cos
2y+7 sin y
B.
p
p
2x + 1y + 1xy = 10.2
at the point (6, 6).
C.
D.
c
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Team, Department of Mathematics, University of Rochester
1
2 cos y
2x sin y+8 sin 2y+7 cos y
2 sin y
8 sin 2y−2x cos y−7 sin y
2 cos y
2x sin y−8 cos 2y+7 cos y
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives3WordProblems due 02/03/2006 at 02:00am EST.
5. (1 pt) rochesterLibrary/setDerivatives3WordProblems/s2 3 27.pg
The cost of producing x units of stuffed alligator toys is c(x) =
0.004x2 + 10x + 7000. Find the marginal cost at the production
level of 1000 units.
6. (1 pt) rochesterLibrary/setDerivatives3WordProblems/s2 7 41.pg
A mass attached to a vertical spring has position function given
by s(t) = 4 sin(2t) where t is measured in seconds and s in
inches.
Find the velocity at time t = 4.
Find the acceleration at time t = 4.
7. (1 pt) rochesterLibrary/setDerivatives3WordProblems/c2s3p1.pg
The mass of the part of a rod that lies between its left end and
a point x meters to the right is 2x3 kg. The linear density of the
rod at 3 meters is
kg/meter and at 5 meters the density
kg/meter
is
1. (1 pt) rochesterLibrary/setDerivatives3WordProblems/s2 3 1.pg
A particle moves along a straight line and its position at time t
is given by s(t) = 2t 3 − 21t 2 + 36t where s is measured in feet
and t in seconds.
Find the velocity (in ft/sec) of the particle at time t = 0:
The particle stops moving (i.e. is in a rest) twice, once when
t = A and again when t = B where A < B. A is
and B is
What is the position of the particle at time 14?
Finally, what is the TOTAL distance the particle travels between
time 0 and time 14?
2. (1 pt) rochesterLibrary/setDerivatives3WordProblems/s2 3 8.pg
If a ball is thrown vertically upward from the roof of 48 foot
building with a velocity of 32 ft/sec, its height after t seconds
is s(t) = 48 + 32t − 16t 2 . What is the maximum height the ball
reaches?
What is the velocity of the ball when it hits the ground (height
0)?
8. (1 pt) rochesterLibrary/setDerivatives3WordProblems/c2s3p2.pg
If f is the focal length of a convex lens and an object is placed at
a distance p from the lens, then its image will be at a distance q
from the lens, where f , p, and q are related by the lens equation
1
1 1
= +
f
p q
What is the rate of change of p with respect to q if q = 2 and
f = 1? (Make sure you have the correct sign for the rate.)
3. (1 pt) rochesterLibrary/setDerivatives3WordProblems/s2 3 10.pg
The area of a square with side s is A(s) = s2 . What is the rate
of change of the area of a square with respect to its side length
when s = 11?
9. (1 pt) rochesterLibrary/setDerivatives3WordProblems/c2s7p2.pg
A particle moves along a straight line with equation of motion
s = t 3 − 3t 2 Find the value of t (other than 0 ) at which the acceleration is equal to zero.
4. (1 pt) rochesterLibrary/setDerivatives3WordProblems/s2 3 24.pg
The population of a slowly growing bacterial colony after t
hours is given by p(t) = 3t 2 + 25t + 200. Find the growth rate
after 4 hours.
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives4Trig due 02/04/2006 at 02:00am EST.
1. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 7.pg
Evaluate the limit
sin 4x
lim
x→0 6x
10.
(1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 24f.pg
If
f (x) =
2 sin x
1 + cos x
then f 0 (x) =
2. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 8.pg
Evaluate the limit
sin 6x
lim
x→0 sin 4x
11.
(1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 24.pg
If
f (x) =
2 sin x
1 + cos x
3. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 8a.pg
Evaluate the limit
tan 5x
lim
x→0 sin 4x
find f 0 (x).
4. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 12.pg
Evaluate the limit
tan x
lim
x→0 2x
12. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 25.pg
x
0
If f (x) = 2 tan
x , find f (x).
Find f 0 (1).
Find f 0 (5).
5. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 20f.pg
If f (x) = cos x − 5 tan x, then
f 0 (x) =
13.
(1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 26.pg
If
f (x) =
6. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 20.pg
If f (x) = cos x − 3 tan x, then
f 0 (x) =
tan x − 5
sec x
find f 0 (x).
Find f 0 (1).
and
f 0 (3) =
7.
Let
14.
Let
(1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 20a.pg
(1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 26a.pg
f (x) =
f (x) = 7 cos x − 2 tan x
f 0 (x) =
f 0 (x) =
f 0 ( 11π
6 )=
f 0 (− π4 ) =
8. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 21.pg
If f (x) = 6 sin x + 3 cos x, then
f 0 (x) =
15. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 27.pg
If f (x) = 2x(sin x + cos x), find f 0 (x).
and f 0 (2).
9.
Let
10 tan x − 12
sec x
Find f 0 (4).
(1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 21a.pg
16.
Let
f (x) = 5 sin x + 9 cos x
(1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 27a.pg
f (x) = 9x(sin x + cos x)
f 0 (x) =
f 0 (− π4 ) =
[Note: When entering trigonometric functions into Webwork, you must include parentheses around the arguement. I.e.
”sinx” would not be accepted but ”sin(x)” would.]
f 0 (x) =
f 0 (− π4 ) =
1
17.
Let
24. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 34.pg
Find the equation of the tangent line to the curve y = 6 sec x −
12 cos x at the point (π/3, 6). The equation of this tangent line
can be written in the form y = mx + b where m is:
and where b is:
25. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 35.pg
(1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 27b.pg
f (x) =
12x
sin x + cos x
f 0 (π) =
18. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 30f.pg
If f (x) = 5x sin x cos x,
then f 0 (x) =
Find the equation of the tangent line to the curve y = 6x cos x
at the point (π, −6π).
The equation of this tangent line can be written in the form
y = mx + b where
m=
and b =
26. (1 pt) rochesterLibrary/setDerivatives4Trig/ur dr 4 1.pg
7 sin x
Let f (x) =
.
2 sin x + 4 cos x
0
.
Then f (x) =
The equation of the tangent line to y = f (x) at a = π/3 can be
written in the form y = mx + b where
m=
and
b=
.
27. (1 pt) rochesterLibrary/setDerivatives4Trig/c2s5p2.pg
Match the functions and their derivatives:
1. y = cos(tan(x))
2. y = cos3 (x)
3. y = sin(x) tan(x)
4. y = tan(x)
A. y0 = − sin(tan(x))/ cos2 (x)
B. y0 = sin(x) + tan(x) sec(x)
C. y0 = −3 cos3 (x) tan(x)
D. y0 = 1 + tan2 (x)
19. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 30.pg
If f (x) = 2x sin x cos x, find f 0 (x).
Find f 0 (1).
20.
Let
(1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 30a.pg
f (x) = −5x sin x cos x
f 0 ( 3π
2 )=
21. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 31.pg
2
If f (x) = 2xsectanx x , find f 0 (x).
Find f 0 (1).
22.
(1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 32.pg
Find the equation of the tangent line to the curve y = 6 sin x
at the point (π/6, 3).
The equation of this tangent line can be written in the form
y = mx + b where
m=
and b =
28. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 7 32.pg
Find the 36th derivative of sin(x) by finding the first few derivatives and observing the pattern that occurs.
(sin(x))(36) =
23. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 4 33.pg
Find the equation of the tangent line to the curve y = 3 tan x at
the point (π/4, 3). The equation of this tangent line can be written in the form y = mx + b where m is:
and where b is:
29. (1 pt) rochesterLibrary/setDerivatives4Trig/s2 7 13.pg
Let h(t) = tan(3t + 8). Then h0 (1) is
and h00 (1) is
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives5ChainRule due 02/05/2006 at 02:00am EST.
11.
Let
1. (1 pt) rochesterLibrary/setDerivatives5ChainRule/s2 5 1.pg
If f (x) = (x2 + 4x + 6)2 , find f 0 (x).
(1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 19.pg
f (x) = −6 cos4 x
Find
f 0 (3).
2.
Let
(1 pt) rochesterLibrary/setDerivatives5ChainRule/s2 5 2f.pg
f 0 (x) =
12. (1 pt) rochesterLibrary/setDerivatives5ChainRule/s2 5 7.pg
If f (x) = tan 5x, find f 0 (x).
f (x) = (x3 + 2x + 2)2
Find f 0 (2).
f 0 (x) =
3.
Let
13.
Let
(1 pt) rochesterLibrary/setDerivatives5ChainRule/s2 5 2.pg
(1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 14.pg
f (x) = sin(2x − 6)
f (x) = (x3 + 5x + 6)2
f 0 (x) =
f 0 (x) =
f 0 (1) =
14. (1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 14a.pg
If f (x) = cos(4x + 3), find f 0 (x).
4. (1 pt) rochesterLibrary/setDerivatives5ChainRule/s2 5 3.pg
If f (x) = (2x + 3)−1 , find f 0 (x).
Find f 0 (5).
Find f 0 (3).
15.
Let
5. (1 pt)√rochesterLibrary/setDerivatives5ChainRule/s2 5 6.pg
If f (x) = 2x + 6, find f 0 (x).
(1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 15.pg
f (x) = 5 sec(8x)
f 0 (x) =
Find f 0 (2).
16. (1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 15a.pg
If f (x) = 5 sec(5x), find f 0 (x).
6. (1 pt)√rochesterLibrary/setDerivatives5ChainRule/s2 5 6a.pg
If f (x) = 5x2 + 3, find f 0 (2).
Find f 0 (5).
7.
Let
17.
Let
(1 pt) rochesterLibrary/setDerivatives5ChainRule/s2 5 8.pg
f (x) =
(1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 16.pg
f (x) = 5 sin(sin x)
p
3x2 + 3x + 7
f 0 (x) =
f 0 (x) =
f 0 (2) =
18. (1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 16a.pg
If f (x) = sin(sin(x)), find f 0 (x).
8. (1 pt) rochesterLibrary/setDerivatives5ChainRule/s2 5 4.pg
If f (x) = sin(x4 ), find f 0 (x).
Find f 0 (3).
Find f 0 (4).
19.
(1 pt) rochesterLibrary/setDerivatives5ChainRule/derchr1.pg
Let F(x) = f (x9 ) and G(x) = ( f (x))9 . You also know that
a8 = 11, f (a) = 2, f 0 (a) = 13, f 0 (a9 ) = 12.
Find F 0 (a) =
and G0 (a) =
.
9. (1 pt) rochesterLibrary/setDerivatives5ChainRule/s2 5 4a.pg
If f (x) = sin(x7 ), find f 0 (x).
20.
10. (1 pt) rochesterLibrary/setDerivatives5ChainRule/s2 5 5.pg
If f (x) = sin3 x, find f 0 (x).
(1 pt) rochesterLibrary/setDerivatives5ChainRule/derchr2.pg
Let F(x) = f ( f (x)) and G(x) = (F(x))2 . You also know that
f (8) = 5, f (5) = 2, f 0 (5) = 10, f 0 (8) = 4.
Find F 0 (8) =
and G0 (8) =
.
Find f 0 (1).
1
If
21. (1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 12.pg
d
2
2
dx ( f (4x )) = 6x .
0
Calculate f (x) =
22.
Let
25. (1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 17a.pg
If f (x) = cos(sin(x2 )), find f 0 (x).
Find f 0 (4).
(1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 18.pg
f (x) = 7ex sin x
26.
Let
f 0 (x) =
23.
Let
(1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 21.pg
p
2
f (x) = cos( ex sin(x) )
(1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 20.pg
f 0 (x) =
f (x) = (4x2 + 6)6 (9x2 + 8)11
f 0 (x) =
27.
Let
24. (1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 17.pg
Let f (x) = 8 sin(cos(x8 )).
Then f 0 (x) =
(1 pt) rochesterLibrary/setDerivatives5ChainRule/ur dr 5 22.pg
f (x) =
f 0 (x) =
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
q
3
sin(ex sin(x) )
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives6InverseTrig due 02/06/2006 at 02:00am EST.
1. (1 pt) rochesterLibrary/setDerivatives6InverseTrig/sc3 6 25.pg
If f (x) = 7 arcsin(x3 ), find f 0 (x).
7. (1 pt) rochesterLibrary/setDerivatives6InverseTrig/sc3 6 27.pg
If f (x) = 2 arctan(7ex ), find f 0 (x).
2.
Let
8.
Let
(1 pt) rochesterLibrary/setDerivatives6InverseTrig/sc3 6 25a.pg
(1 pt) rochesterLibrary/setDerivatives6InverseTrig/sc3 6 27a.pg
f (x) = tan−1 (8x )
f (x) = 7 sin−1 (x2 )
f 0 (x) =
f 0 (x) =
9. (1 pt) rochesterLibrary/setDerivatives6InverseTrig/sc3 6 32.pg
If f (x) = 8 sin(3x) arcsin(x), find f 0 (x).
3. (1 pt) rochesterLibrary/setDerivatives6InverseTrig/ur dr 6 1.pg
If f (x) = 3x arcsin(x), find f 0 (x).
10.
Let
Find f 0 (0.6).
(1 pt) rochesterLibrary/setDerivatives6InverseTrig/sc3 6 32a.pg
f (x) = 3 sin(x) sin−1 (x)
4. (1 pt) rochesterLibrary/setDerivatives6InverseTrig/sc3 6 26.pg
If f (x) = 8x3 arctan(2x4 ), find f 0 (x).
f 0 (x) =
NOTE: The webwork system will accept arcsin(x) and not
sin−1 (x) as the inverse of sin(x).
5.
Let
11.
Let
(1 pt) rochesterLibrary/setDerivatives6InverseTrig/sc3 6 26a.pg
(1 pt) rochesterLibrary/setDerivatives6InverseTrig/sc3 6 33a.pg
f (x) = tan−1 (cos(8x))
f (x) = x2 tan−1 (4x)
f 0 (x) =
f 0 (x) =
NOTE: The WeBWorK system will accept arctan(x) but not
tan−1 (x) as the inverse of tan(x).
12. (1 pt) rochesterLibrary/setDerivatives6InverseTrig/sc3 6 33.pg
If f (x) = 7 arctan(4 sin(2x)), find f 0 (x).
6. (1 pt) rochesterLibrary/setDerivatives6InverseTrig/ur dr 6 2.pg
If f (x) = 3 arctan(3x), find f 0 (x).
13.
Let
(1 pt) rochesterLibrary/setDerivatives6InverseTrig/osu dr 6 3.pg
y = tan−1
Find f 0 (2).
Then
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
dy
=
dx
p
5x2 − 1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives7Log due 02/07/2006 at 02:00am EST.
1. (1 pt) rochesterLibrary/setDerivatives7Log/sc3 7 2f.pg
If f (x) = 5 ln(7 + x), find f 0 (x).
11. (1 pt) rochesterLibrary/setDerivatives7Log/sc3 7 17.pg
If f (x) = 7 log2 (x), find f 0 (4).
2. (1 pt) rochesterLibrary/setDerivatives7Log/sc3 7 2.pg
If f (x) = 2 ln(8 + x), find f 0 (x).
12.
Let
(1 pt) rochesterLibrary/setDerivatives7Log/mec5.pg
f (x) = −5 log9 (x)
Find f 0 (2).
3.
Let
f 0 (x) =
f 0 (9) =
(1 pt) rochesterLibrary/setDerivatives7Log/mec1.pg
13.
Let
f (x) = −2 ln(4x)
f 0 (x) =
f 0 (3) =
4.
Let
f (x) = 8x log4 (x)
f 0 (x) =
(1 pt) rochesterLibrary/setDerivatives7Log/mec6.pg
14. (1 pt) rochesterLibrary/setDerivatives7Log/osu dr 7 2.pg
Find the indicated
derivatives.
6
d
(a) dx
ex + log5 (π) =
√
ln(x)
d
=
(b) dx
( 7 x)
f (x) = ln(x7 )
f 0 (x) =
f 0 (e2 ) =
5.
Let
(1 pt) rochesterLibrary/setDerivatives7Log/mec12.pg
(1 pt) rochesterLibrary/setDerivatives7Log/mec4.pg
15.
Let
f (x) = [ln x]6
(1 pt) rochesterLibrary/setDerivatives7Log/mec9.pg
f (x) = ln[x3 (x + 3)6 (x2 + 3)3 ]
f 0 (x) =
f 0 (e2 ) =
f 0 (x) =
6. (1 pt) rochesterLibrary/setDerivatives7Log/sc3
7 11.pg
√
If f (x) = 5 x ln(x), find f 0 (x).
16.
Let
(1 pt) rochesterLibrary/setDerivatives7Log/mec10.pg
Find f 0 (5).
7.
Let
f (x) =
x3 (x − 6)2
(x2 + 9)8
Use logarithmic differentiation to determine the derivative.
f 0 (x) =
f 0 (6) =
(1 pt) rochesterLibrary/setDerivatives7Log/mec3.pg
f (x) = 3x5 ln x
f 0 (x) =
17. (1 pt) rochesterLibrary/setDerivatives7Log/sc3 7 25.pg
If f (x) = (5x − 4)2 ∗ (3x2 + 3)2 , find f 0 (3).
f 0 (e3 ) =
8. (1 pt) rochesterLibrary/setDerivatives7Log/sc3 7 4f.pg
If f (x) = 3 cos(2 ln(x)), find f 0 (x).
18.
Let
(1 pt) rochesterLibrary/setDerivatives7Log/mec8.pg
r
9. (1 pt) rochesterLibrary/setDerivatives7Log/sc3 7 4.pg
If f (x) = 3 cos(3 ln(x)), find f 0 (x).
f (x) = ln
6x − 4
9x + 7
f 0 (x) =
Find f 0 (5).
19.
Let
10. (1 pt) rochesterLibrary/setDerivatives7Log/sc3 7 16.pg
If f (x) = 3 ln(6x + 7 ln(x)), find f 0 (x).
(1 pt) rochesterLibrary/setDerivatives7Log/mec7f.pg
f (x) = x6x
Use logarithmic differentiation to determine the derivative.
f 0 (x) =
Find f 0 (4).
1
20.
Let
25.
Let
(1 pt) rochesterLibrary/setDerivatives7Log/mec7.pg
(1 pt) rochesterLibrary/setDerivatives7Log/osu dr 7 1.pg
y = xlog7 (x)
f (x) = x3x
Then
dy
=
dx
Note. You must express your answer in terms of natural logs,
as Webwork doesn’t understand how to evaluate logarithms to
other bases.
26. (1 pt) rochesterLibrary/setDerivatives7Log/osu dr 7 3.pg
dy
Find
for each of the following functions
dx
2x − 7
y = ln √
x 4 x2 + 1
dy
=
dx
Use logarithmic differentiation to determine the derivative.
f 0 (x) =
f 0 (1) =
21. (1 pt) rochesterLibrary/setDerivatives7Log/sc3 7 29.pg
If f (x) = 3x2x , find f 0 (2).
22. (1 pt) rochesterLibrary/setDerivatives7Log/ur dr 7 1.pg
If f (x) = 3 sin(x) + 4xx , find f 0 (2).
23. (1 pt) rochesterLibrary/setDerivatives7Log/sc3 7 32.pg
If f (x) = 4(sin(x))x , find f 0 (1).
y = xcos(x)
dy
=
dx
27. (1 pt) rochesterLibrary/setDerivatives7Log/ur dr 7 2.pg
If f (x) = e7 + ln(6),
then f 0 (x) =
24. (1 pt) rochesterLibrary/setDerivatives7Log/sc3 7 34.pg
If f (x) = 4xln(x) , find f 0 (2).
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives8RelatedRates due 02/08/2006 at 02:00am EST.
1.
(1 pt) rochesterLibrary/setDerivatives8RelatedRates/s2 8 3.pg
dy
Let xy = 3 and
= 4.
dt
dx
Find
when x = 1.
dt
2.
8.
dr
dA
Let A be the area of a circle with radius r. If
= 4, find
dt
dt
when r = 2.
rochesterLibrary/setDerivatives8RelatedRates-
Note: Draw yourself a diagram which shows where the ships are
at noon and where they are ”some time” later on. You will need
to use geometry to work out a formula which tells you how far
apart the ships are at time t, and you will need to use ”distance
= velocity * time” to work out how far the ships have travelled
after time t.
3. (1 pt) rochesterLibrary/setDerivatives8RelatedRates/s2 8 5.pg
A spherical snowball is melting in such a way that its diameter
is decreasing at rate of 0.3 cm/min. At what rate is the volume
of the snowball decreasing when the diameter is 8 cm. (Note the
answer is a positive number).
4. (1 pt) rochesterLibrary/setDerivatives8RelatedRates/c2s8p2.pg
The altitude of a triangle is increasing at a rate of 3.000 centimeters/minute while the area of the triangle is increasing at a
rate of 2.000 square centimeters/minute. At what rate is the base
of the triangle changing when the altitude is 10.000 centimeters
and the area is 97.000 square centimeters?
pt)
pt)
At noon, ship A is 30 nautical miles due west of ship B. Ship A
is sailing west at 23 knots and ship B is sailing north at 23 knots.
How fast (in knots) is the distance between the ships changing
at 3 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)
(1 pt) rochesterLibrary/setDerivatives8RelatedRates/s2 8 2.pg
5.
(1
/SRM c2s8p2.pg
(1
/SRM s2 8 12.pg
9. (1 pt) rochesterLibrary/setDerivatives8RelatedRates/s2 8 21.pg
Gravel is being dumped from a conveyor belt at a rate of 40 cubic feet per minute. It forms a pile in the shape of a right circular
cone whose base diameter and height are always the same. How
fast is the height of the pile increasing when the pile is 24 feet
high? Recall that the volume of a right circular cone with height
h and radius of the base r is given by V = 13 πr2 h.
rochesterLibrary/setDerivatives8RelatedRates-
The altitude of a triangle is increasing at a rate of 2.500 centimeters/minute while the area of the triangle is increasing at a
rate of 5.000 square centimeters/minute. At what rate is the base
of the triangle changing when the altitude is 8.500 centimeters
and the area is 99.000 square centimeters?
Note: The ”altitude” is the ”height” of the triangle in the formula
”Area=(1/2)*base*height”. Draw yourself a general ”representative” triangle and label the base one variable and the altitude
(height) another variable. Note that to solve this problem you
don’t need to know how big nor what shape the triangle really
is.
10. (1 pt) rochesterLibrary/setDerivatives8RelatedRates/s2 8 21a.pg
Gravel is being dumped from a conveyor belt at a rate of 20 cubic feet per minute. It forms a pile in the shape of a right circular
cone whose base diameter and height are always the same. How
fast is the height of the pile increasing when the pile is 21 feet
high?
Recall that the volume of a right circular cone with height h and
radius of the base r is given by
6. (1 pt) rochesterLibrary/setDerivatives8RelatedRates/c2s8p4.pg
When air expands adiabatically (without gaining or losing heat),
its pressure P and volume V are related by the equation PV 1.4 =
C where C is a constant. Suppose that at a certain instant the
volume is 570 cubic centimeters and the pressure is 75 kPa and
is decreasing at a rate of 12 kPa/minute. At what rate in cubic
centimeters per minute is the volume increasing at this instant?
1
V = πr2 h
3
Note: See number 21 on pg 258 for a picture of this.
11. (1 pt) rochesterLibrary/setDerivatives8RelatedRates/s2 8 7.pg
A street light is at the top of a 14 ft tall pole. A woman 6 ft
tall walks away from the pole with a speed of 4 ft/sec along a
straight path. How fast is the tip of her shadow moving when
she is 35 ft from the base of the pole?
(Pa stands for Pascal – it is equivalent to one Newton/(meter
squared); kPa is a kiloPascal or 1000 Pascals. )
7. (1 pt) rochesterLibrary/setDerivatives8RelatedRates/s2 8 12.pg
At noon, ship A is 30 nautical miles due west of ship B. Ship A
is sailing west at 21 knots and ship B is sailing north at 18 knots.
How fast (in knots) is the distance between the ships changing
at 6 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)
1
12.
(1
pt)
15.
rochesterLibrary/setDerivatives8RelatedRates-
(1
pt)
rochesterLibrary/setDerivatives8RelatedRates-
/SRM s2 8 7.pg
/SRM c2s8p3.pg
A street light is at the top of a 19 ft tall pole. A woman 6 ft
tall walks away from the pole with a speed of 8 ft/sec along a
straight path. How fast is the tip of her shadow moving when
she is 50 ft from the base of the pole?
Water is leaking out of an inverted conical tank at a rate of
12300.0 cubic centimeters per min at the same time that water
is being pumped into the tank at a constant rate. The tank has
height 10.0 meters and the diameter at the top is 5.5 meters. If
the water level is rising at a rate of 21.0 centimeters per minute
when the height of the water is 5.0 meters, find the rate at which
water is being pumped into the tank in cubic centimeters per
minute.
Note: Let ”R” be the unknown rate at which water is being
pumped in. Then you know that if V is volume of water,
dV
dt = R − 12300.0. Use geometry (similar triangles?) to find
the relationship between the height of the water and the volume
of the water at any given time. Recall that the volume of a cone
with base radius r and height h is given by 13 πr2 h.
Note: You should draw a picture of a right triangle with the
vertical side representing the pole, and the other end of the hypotenuse representing the tip of the woman’s shadow. Where
does the woman fit into this picture? Label her position as a
variable, and label the tip of her shadow as another variable.
You might like to use similar triangles to find a relationship between these two variables.
13. (1 pt) rochesterLibrary/setDerivatives8RelatedRates/c2s8p5.pg
A plane flying with a constant speed of 14 km/min passes over a
ground radar station at an altitude of 14 km and climbs at an angle of 45 degrees. At what rate, in km/min is the distance from
the plane to the radar station increasing 1 minutes later?
16. (1 pt) rochesterLibrary/setDerivatives8RelatedRates/ur
dr 8 1.pg
√
A particle is moving along the curve y = 3 4x + 5. As the particle passes through the point (5, 15), its x-coordinate increases
at a rate of 4 units per second. Find the rate of change of the
distance from the particle to the origin at this instant.
14. (1 pt) rochesterLibrary/setDerivatives8RelatedRates/c2s8p3.pg
Water is leaking out of an inverted conical tank at a rate of
11500.000 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank
has height 7.000 meters and the diameter at the top is 3.500 meters. If the water level is rising at a rate of 20.000 centimeters
per minute when the height of the water is 1.500 meters, find
the rate at which water is being pumped into the tank in cubic
centimeters per minute.
17.
(1
pt)
rochesterLibrary/setDerivatives8RelatedRates-
/csuf dr 8 1.pg
Air is being pumped into a spherical balloon so that its volume
increases at a rate of 20cm3 /s. How fast is the surface area of
the balloon increasing when its radius is 11cm? Recall that a
4
ball of radius r has volume V = πr3 and surface area S = 4πr2 .
3
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Derivatives9Approximations due 02/09/2006 at 02:00am EST.
10. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2 9 Z.pg
Use linear approximation, i.e. the tangent line, to approximate
15.22 as follows:
Let f (x) = x2 and find the equation of the tangent line to f (x) at
x = 15.
Using this, find your approximation for 15.22
1. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2 9 13.pg
Let y = 2x2 .
Find the change in y, ∆y when x = 3 and ∆x = 0.4
Find the differential dy when x = 3 and dx = 0.4
2. (1 pt)√rochesterLibrary/setDerivatives9Approximations/s2 9 14.pg
Let y = 4 x.
Find the change in y, ∆y when x = 5 and ∆x = 0.3
Find the differential dy when x = 5 and dx = 0.3
11. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2 9 Y.pg
Use linear approximation, i.e. the tangent line, to approximate
1.83 as follows:
Let f (x) = x3 . The equation of the tangent line to f (x) at x = 2
can be written in the form y = mx + b where m is:
and where b is:
Using this, we find our approximation for 1.83 is
3. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2 9 7.pg
Let y = 3x2 + 3x + 4.
Find the differential dy when x = 1 and dx = 0.2
Find the differential dy when x = 1 and dx = 0.4
4. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2 9 12.pg
Let y = tan(2x + 8).
Find the differential dy when x = 2 and dx = 0.3
Find the differential dy when x = 2 and dx = 0.6
12. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2 9 22.pg
Use linear approximation, i.e. the tangent line, to approximate
4.86 as follows:
Let f (x) = x6 . The equation of the tangent line to f (x) at x = 5
can be written in the form y = mx + b
where m is:
and where b is:
Using this, we find our approximation for 4.86 is
5. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2
9 35.pg
√
The linear approximation at x = 0 to 8 + 6x is A + Bx
where A =
and where B =
6. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2 9 36.pg
The linear approximation at x = 0 to sin(6x) is A + Bx where A
is:
and where B is:
13. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2 9 A.pg
Use linear approximation, i.e. the tangent line, to approximate
1
1
0.101 as follows: Let f (x) = x and find the equation of the tangent line to f (x) at a ”nice” point near 0.101. Then use this to
1
.
approximate 0.101
7. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2 9 38.pg
1
The linear approximation at x = 0 to √2−x
is A + Bx where A is:
and where B is:
14.
8. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2 9 19.pg
Use
√ linear approximation, i.e. the tangent line, to approximate
64.2 as follows:
√
Let f (x) = x. The equation of the tangent line to f (x) at x = 64
can be written in the form y = mx + b where m is:
and where b is:
√
Using this, we find our approximation for 64.2 is
NOTE: For this part, give your answer to at least 9 significant
figures or use fractions to give the exact answer.
(1
pt)
rochesterLibrary/setDerivatives9Approximations-
/ur dr 9 1.pg
Find the linear approximation of f (x) = ln x at x = 1 and use it
to estimate ln 1.37.
L(x) =
ln 1.37 ≈
15. (1 pt) rochesterLibrary/setDerivatives9Approximations/c2s9p8.pg
Suppose that you can calculate the derivative of a function using the formula f 0 (x) = 3 f (x) + 5x. If the output value of the
function at x = 3 is 1 estimate the value of the function at
3.014.
9. (1 pt) rochesterLibrary/setDerivatives9Approximations/s2 9 20.pg
Use
linear approximation, i.e. the tangent line, to approximate
√
3
125.1 as follows:
√
Let f (x) = 3 x. The equation of the tangent line to f (x) at
x = 125 can be written in the form y = mx + b where m is:
and where b is:
√
Using this, we find our approximation for 3 125.1 is
16. (1 pt) rochesterLibrary/setDerivatives9Approximations/c2s9p6.pg
The circumference of a sphere was measured to be 90.000 cm
with a possible error of 0.50000 cm. Use linear approximation
to estimate the maximum error in the calculated surface area.
Estimate the relative error in the calculated surface area.
1
17. (1 pt) rochesterLibrary/setDerivatives9Approximations/c2s9p7.pg
Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.070000 cm
thick to a hemispherical dome with a diameter of 70.000 meters.
18.
(1
pt)
rochesterLibrary/setDerivatives9Approximations-
/c2s9p10.pg
Let f (t) be the weight (in grams) of a solid sitting in a beaker
of water. Suppose that the solid dissolves in such a way that the
rate of change (in grams/minute) of the weight of the solid at any
time t can be determined from the weight using the forumula:
f 0 (t) = −2 f (t)(2 + f (t))
If there is 6 grams of solid at time t = 2 estimate the amount
of solid 1 second later.
19.
(1 pt) rochesterLibrary/setDerivatives9Approximations-
Use linear approximation to estimate f (3.2):
Is your answer a little too big or a little too small? (Enter TB or
TS):
/nsc2s9p11.pg
Suppose you have a function f (x) and all you know is that
f (3) = 24 and the graph of its derivative is:
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2