Test #1 Review Math 1325
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Decide whether the limit exists. If it exists, find its value.
1) lim f(x)
x→1
A) 0
1)
B) 2
C) Does not exist
D) 1
Use a calculator to estimate the limit.
x4 - 1
2) lim
x→1 x - 1
A) Does not exist
2)
B) 2
C) 4
D) 0
x - 1
3) lim
x - 1
x→1
A) 0
4) lim
x→0
3)
B) Does not exist
D) -1
C) 1
x3 + 12x2 - 5x
5x
A) Does not exist
4)
B) 5
D) -1
C) 0
Use the limit properties to find the following limit.
5) If lim f(x) = 11 and lim g(x) = 10, find lim x→-1
x→-1
x→-1
A)
21
20
B)
1
20
f(x) + g(x)
.
2g(x)
C) -1
5)
D)
7
4
Use the properties of limits to evaluate the limit if it exists.
lim 3x2 + 7x - 2
6)
x→1 3x2 - 4x - 2
A) 0
7)
B) - 8
3
6)
C) - 7
4
D) Does not exist
lim x2 - 2x - 15
x→-3
x + 3
A) Does not exist
7)
B) 5
C) 0
1
D) -8
Use the given graph to determine the limit, if it exists.
8)
8)
y
8
6
4
2
-4 -3 -2 -1-2
1
2
3
4
5x
-4
-6
-8
lim f(x) and lim f(x).
x→2+
x→2A) 4; -2
9)
C) -2; 4
B) 1; 1
D) does not exist
lim f(x)
x → 1+
9)
5
f(x)
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
A) 4
B) 3
C) 3
1
2
D) does not exist
Use the properties of limits to find the indicated limit.
x2 + 25
10) lim x + 5
x→5A) 10
11)
10)
B) 5
C) 0
D) does not exist
lim f(x),
x→99
where f(x) = x2
9
A) 9
11)
(x < -3)
(-3 < x < 3)
(x > 3)
B) -9
C) 3
2
D) does not exist
Use a calculator to estimate the limit.
5
12) lim
x + 9
x→-9+
A) -∞
13)
12)
C) ∞
B) 0
D) does not exist
x - 6
lim
x + 3
x→ -3+
13)
A) ∞
B) -∞
C) 0
D) does not exist
Solve the problem.
14) Use the graph of the function f to find lim f(x).
x→-∞
14)
y
10
8
6
4
2
-10 -8
-6
-4
-2
2
4
6
8
10 x
-2
-4
-6
-8
-10
A) 2
B) -∞
C) ∞
D) does not exist
Use the properties of limits to find the indicated limit.
8x2 - 7
15) lim
x→∞ x2 - 8x
A) 0
15)
C) - B) 8
1
8
D) does not exist
x2 + 3x + 4
16) lim x→∞ x3 - 6x2 + 15
A) 1
16)
C) ∞
B) 0
D)
4
15
Use the properties of absolute value to find the indicated limit.
x - 8
17) lim
x→-∞ |x - 8|
A) 1
C) -1
B) 0
3
17)
D) does not exist
Find the average rate of change for the function over the given interval.
18) y = 5x3 - 3x2 + 7 between x = -6 and x = 4
279
A) 146
B) 365
C)
10
19) y = 18)
279
D)
4
3
between x = 1 and x = 4
x + 2
A) -28
19)
B) -2
C)
1
2
D) - 1
6
The graph shows the total sales in thousand of dollars from the distribution of x thousand catalogs. Find the
average rate of change of sales with respect to the number of catalogs distributed for the change in x.
20) 10 to 50
1
A)
4
20)
3
C)
4
B) 2
D) 1
Solve the problem.
21) The graph shows the average cost of a barrel of crude oil for the years 1981 to 1990 in constant
1996 dollars. Find the approximate average change in price from 1981 to 1984.
21)
1996 $/Barrel
60
50
40
30
20
10
81 82 83 84 85 86 87 88 89 90
Year
A) About -$16/year
C) About -$12/year
B) About -$2/year
D) About -$4/year
22) Compute the instantaneous rate of change of the function at at x = a. f(x) = -2x2 - x, a = -7.
A) 28
B) -21
C) 21
D) 27
4
22)
23) For a motorcycle traveling at speed v (in mph) when the brakes are applied, the distance
d (in feet) required to stop the motorcycle may be approximated by the formula
d = 0.05 v2 + v. Find the instantaneous rate of change of distance with respect to velocity when
the speed is 42 mph.
A) 4.2 mph
B) 43 mph
C) 10.4 mph
D) 5.2 mph
Find the equation of the tangent line to the curve when x has the given value.
24) f(x) = x2 + 5x ; x = 4
4x 8
A) y = - + 25 5
x 1
B) y = + 20 5
C) y = 13x - 16
Find fʹ(x) for the function, then find fʹ(x) for the given x.
25) g(x) = 3x2 - 4x, gʹ(3)
A) gʹ(x) = 3x; gʹ(3) = 9
C) gʹ(x) = 3x - 4; gʹ(3) = 5
24)
D) y = -39x - 80
25)
B) gʹ(x) = 6x - 4; gʹ(3) = 14
D) gʹ(x) = 6x; gʹ(3) = 18
Solve the problem.
26) The profit in dollars from the sale of x thousand compact disc players is P(x) = x3 - 3x2 + 6x
+ 8. Find the marginal profit when the value of x is 5.
A) $95
B) $59
C) $51
D) $103
Find the derivative.
27) f(x) = 5x4 + 8x3 + 5
A) 20x3 + 24x2
C) 20x3 + 24x2 - 7
26)
27)
B) 4x3 + 3x2
D) 4x3 + 3x2 - 7
1
28) f(x) = 20x1/2 - x20
2
28)
B) 10x-1/2 - 10x10
D) 10x-1/2 - 10x19
A) 10x1/2 - 10x10
C) 10x1/2 - 10x19
Find the following.
29) fʹ(-1) if f(x) = x4 + 2x3 + 2x + 2
A) -1
B) 6
29)
C) -3
D) 4
Find the equation of the tangent line to the curve when x has the given value.
4
30) f(x) = ; x = 5
x
A) y = 13x - 16 23)
B) y = x 1
+ 20 5
C) y = -39x - 80
5
30)
D) y = - 4x 8
+ 25 5
Solve the problem.
31) Let C(x) be the cost function and R(x) the revenue function. Compute the marginal cost,
marginal revenue, and the marginal profit functions.
C(x) = 0.0003x3 - 0.036x2 + 200x + 50,000
R(x) = 350x
A) Cʹ(x) = 0.0009x2 - 0.072x + 200
Rʹ(x) = 350
Pʹ(x) = -0.0009x2 + 0.072x + 150
B) Cʹ(x) = 0.0009x2 - 0.072x + 200
Rʹ(x) = 350
Pʹ(x) = 0.0009x2 - 0.072x - 150
C) Cʹ(x) = 0.0009x2 + 0.072x + 200
Rʹ(x) = 350
Pʹ(x) = 0.0009x2 + 0.072x + 150
32) A voltage source of 12 volts can supply a voltage V (in volts) through a 72-ohm line to a
12R
variable resistance R (in ohms) given by the following equation. V = . How fast does V
72 + R
change with respect to R when R is 64 ohms?
A) 0.167 V/Ω
B) 0.211 V/Ω
Find the derivative of the function.
33) f(x) = (3x - 2)(3x + 1)
A) fʹ(x) = 18x - 1.5
B) fʹ(x) = 18x - 9
34) f(x) = (6x + 5)2
A) fʹ(x) = 12x + 10
C) 0.047 V/Ω
D) 6.353 V/Ω
C) fʹ(x) = 9x - 3
D) fʹ(x) = 18x - 3
C) fʹ(x) = 72x + 60
D) fʹ(x) = 36x + 25
22t
(t - 11)2
32)
33)
34)
B) fʹ(x) = 36x + 30
Use the quotient rule to find the derivative.
t2
35) g(t) = t - 11
A) gʹ(t) = 31)
B) gʹ(t) = 35)
t2
(t - 11)2
C) gʹ(t) = t2 - 22t
(t - 11)2
Find the derivative.
(2x + 2)(6x + 3)
36)
3x + 5
D) gʹ(t) = t2 + 22t
(t - 11)2
36)
A)
108x2+ 228x + 108
(3x + 5)2
B)
36x2+ 120x + 72
(3x + 5)2
C)
108x2+ 228x + 108
3x + 5
D)
36x2+ 120x + 72
3x + 5
6
37)
(6x - 2)(6x + 2)
x + 5
37)
A)
108x2+ 360x - 4
x + 5
B)
36x2+ 360x + 4
x + 5
C)
108x2+ 360x - 4
(x + 5)2
D)
36x2+ 360x + 4
(x + 5)2
Find the derivative of the function.
38) y = 4x + 2
4
8
A) yʹ = B) yʹ = 4x + 2
4x + 2
39) f(x) = (x3 - 8) 2/3
x
A) fʹ(x) = 3
x3 - 8
38)
2
C) yʹ = 4x + 2
39)
B) fʹ(x) = 2x2
3
C) fʹ(x) = x3 - 8
40) f(x) = (4x - 4)( x + 5)
A) fʹ(x) = 6x1/2 - 4x-1/2 + 20
C) fʹ(x) = 6x1/2 - 2x-1/2 + 20
41) y = 2x
3
D) fʹ(x) = x3 - 8
x2
3
x3 - 8
40)
B) fʹ(x) = 2.67x1/2 - 4x-1/2 + 20
D) fʹ(x) = 2.67x1/2 - 2x-1/2 + 20
x2 - 4
x
A) yʹ = 1 - 42) y = 1
D) yʹ = 4x + 2
41)
4
x2
B) yʹ = 1 + 4
x2
C) yʹ = x + 4
x2
D) yʹ = 1 + 4
x
x2 + 8x + 3
x
42)
A) yʹ = 3x2 + 8x - 3
2x3/2
B) yʹ = 2x + 8
x
C) yʹ = 2x + 8
2x3/2
D) yʹ = 3x2 + 8x - 3
x
Find the derivative.
43) y = -4e2x
A) -4e-8x
43)
B) -4xe-8x
C) -8xe2x
D) -8e2x
C) 4xe2x + 1
D) 4xe2x2 + 1
44) y = e2x2 + x
A) 4xe + 1
44)
B) 4xex2 + 1
7
Find the derivative of the function.
45) y = ln (x - 9)
1
1
A) - B)
x + 9
9 - x
46) y = ln (4x3 - x2)
12x - 2
A)
4x3 - x
45)
C)
1
x + 9
D)
1
x - 9
46)
B)
12x - 2
4x2
C)
4x - 2
4x2 - x
D)
12x - 2
4x2 - x
Find the derivative.
47) f(x) = ln 6 + e4x
4e4x
A)
6 + e4x
47)
B)
1
e4x
C)
e4x
1 + e4x
D)
1
6 + e4x
Solve the problem.
48) The sales in thousands of a new type of product are given by S(t) = 80 - 60e-.8t, where t
represents time in years. Find the rate of change of sales at the time when t = 3.
A) 4.4 thousand per year
B) -4.4 thousand per year
C) 525.3 thousand per year
D) -525.3 thousand per year
Find all points where the function is discontinuous.
49)
48)
49)
A) x = -2, x = 0
C) x = -2, x = 0, x = 2
B) x = 0, x = 2
D) x = 2
50)
50)
A) None
C) x = 1, x = 5
B) x = 1, x = 4, x = 5
D) x = 4
Of the given values of x, identify those at which the function is continuous.
-2
51) f(x) = ; x = 0, 6, 2
x(x - 6)
A) 2
B) 0, 6, 2
C) None
8
51)
D) 0, 6
Write interval notation for the graph.
52)
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
A) [-5, -1)
52)
B) [1, 5)
C) (-5, -1]
Decide whether or not the function is continuous in the indicated x-interval.
53) -3 to 2
A) Not continuous
D) (1, 5]
53)
B) Continuous
54)
54) 2 to 4
A) Continuous
B) Not continuous
9
Answer Key
Testname: TEST #1 REVIEW MATH 1325
1) C
2) C
3) B
4) D
5) A
6) B
7) D
8) A
9) B
10) B
11) A
12) C
13) B
14) C
15) B
16) B
17) C
18) A
19) D
20) C
21) D
22) D
23) D
24) C
25) B
26) C
27) A
28) D
29) D
30) D
31) A
32) C
33) D
34) C
35) C
36) B
37) D
38) C
39) B
40) C
41) B
42) A
43) D
44) D
45) D
46) D
47) A
48) A
10
Answer Key
Testname: TEST #1 REVIEW MATH 1325
49)
50)
51)
52)
53)
54)
C
A
A
B
A
A
11