The graph of y = f(x) is shown. Use the graph to answer the question.
43)
Is f continuous at x = – 0.5?
A)
Yes
B)
No
Answer:
A
44)
Is f continuous at x =0?
A)
No
B)
Yes
Answer:
B
45)
Is f continuous at x =0?
A)
Yes
B)
No
Answer:
B
13
Provide an appropriate response.
46)
Determine where the function H(x) =x2+ 7
x2+ x – 6 is continuous.
A)
(–
, –3)
B)
(–
, –3) (–3, 2) (2,
)
C)
(–
, –3) (–3, 2)
D)
(–3, 2) (2,
)
Answer:
B
47)
Determine where the function f(x) =5x
2x – 3 is continuous.
A)
B)
C)
–
, 3
2
D)
Answer:
D
48)
Determine the x–values, if any, at which the function is discontinuous.
h(x) =
x2– 9 for x < – 1
0 for –1
x 1
x2+ 9 for x >1
A)
B)
C)
–1, 1
D)
Answer:
C
49)
Use a graphing utility to approximate the partition numbers of the function to four decimal places:
f(x) =x4– 8x2– 4x + 1.
A)
(–
, –2.4976) (–2.4976, –0.7203) (–0.7203, 0.1832 ) (0.1832 , 3.0347)
B)
(–
, –2.4976)
C)
(–
, –2.4976) (0.1832 , 3.0347)
D)
(–
, –2.4976) (–2.4976, –0.7203)
Answer:
A
50)
Use a graphing utility to find the discontinuities of the given rational function.
g(x) =x + 1
x3+ 2x2+ 10x – 13
A)
3
B)
1
C)
–1
D)
Continuous at all values of x
Answer:
B
51)
Use a graphing utility to find the discontinuities of the given rational function.
g(x) =x + 1
x3+ 2x2+ 10x – 13
A)
–1
B)
3
C)
1
D)
Continuous at all values of x
Answer:
C
14
52)
Use a graphing utility to find the discontinuities of the given rational function.
f(x) =x2+ 2x + 1
x3+ 2x2+ 5x – 8
A)
–1
B)
3
C)
1
D)
Continuous at all values of x
Answer:
C
53)
Solve the inequality and express the answer in interval notation: x2– 4x
x + 5 > 0.
A)
B)
C)
(–5,
)
D)
Answer:
D
54)
Use a sign chart to solve the inequality. Express answers in interval notation.
x2> 16
A)
B)
C)
(–4,
)
D)
Answer:
A
55)
Use a sign chart to solve the inequality. Express answers in interval notation.
x2+ 6 < 2x
A)
B)
C)
{2}
D)
Answer:
B
56)
Use a sign chart to solve the inequality. Express answers in interval notation.
– 5
–3x – 4 > 0
A)
B)
C)
–4
3 ,
D)
Answer:
C
Solve the problem.
57)
The cost of renting a snowblower is $20 for the first hour (or any fraction thereof) and $5 for each additional
hour (or fraction thereof) up to a maximum rental time of 5 hours. Write a piecewise definition of the cost C(x)
of renting a snowblower for x hours. Is C(x) continuous at x = 2.5?
A)
C(x) =
20 if 0 x
1
25 if 1 x
2
30 if 2 x
3
35 if 3 x
4
40 if 4 x
5
; No
B)
C(x) =
20 if 0 < x 1
25 if 1 < x 2
30 if 2 < x 3
35 if 3 < x 4
40 if 4 < x 5
; No
C)
C(x) =
25 if 0 < x 1
30 if 1 < x 2
35 if 2 < x 3
40 if 3 < x 4
45 if 4 < x 5
; No
D)
C(x) =
20 if 0 < x 1
25 if 1 < x 2
30 if 2 < x 3
35 if 3 < x 4
40 if 4 < x 5
; Yes
Answer:
D
15
Find average rate of change for the function over the given interval.
58)
y = x2+ 3x between x =3 and x =8
A)
B)
C)
88
5
D)
Answer:
A
59)
y =4x3– 6x2– 2 between x = – 4 and x =5
A)
B)
C)
78
D)
Answer:
C
60)
Find the average rate of change for f(x) =2x if x changes from 2 to 8.
A)
B)
C)
1
3
D)
Answer:
C
61)
Find the average rate of change of y with respect to x if x changes from 3 to 5 in the function y =x2+ 3x.
A)
B)
C)
9
D)
Answer:
D
Find the instantaneous rate of change for the function at the value given.
62)
Find the instantaneous rate of change for the function x2+6x at x =3.
A)
B)
C)
6
D)
Answer:
D
63)
Find the instantaneous rate of change for the function f(x) = 5x2+ x at x = – 4.
A)
B)
C)
6
D)
Answer:
D
Provide an appropriate response.
64)
Use the four step process to find f'(x) for the function f(x) =5x2– 3x.
A)
B)
C)
10x + 5h – 3
D)
Answer:
C
65)
Use the four step process to find f'(x) for the function f(x) =2
x2.
A)
B)
C)
–2(h + 2x + xh)
x2(x + h)2
D)
Answer:
A
16
66)
Use the four step process to find f'(x) for the function f(x) =x
6 – x .
A)
–x
(x – 6)(x + h – 6)
B)
6
(x – 6)(x + h – 6)
C)
1
(x – 6)(x + h – 6)
D)
–6
h(x – 6)(x + h – 6)
Answer:
B
Use the definition f'(x) =lim
h
0
f(x +
h) – f(x)
h to find the derivative at x.
67)
f(x) =6x – 8
A)
B)
C)
6x
D)
Answer:
A
68)
f(x) =10 – 3x2
A)
B)
C)
10 – 6x
D)
Answer:
B
69)
f(x) = 4x – 4x3
A)
B)
C)
4x – 12x2
D)
Answer:
B
Provide an appropriate response.
70)
Find the slope of the secant line joining (2, f(2)) and (3, f(3)) for f(x) = – 3x2– 8.
A)
B)
C)
–15
D)
Answer:
C
71)
Find the slope of the graph f(x) = – x2+ 3x at the point (1, 2).
A)
B)
C)
2
D)
Answer:
D
72)
Find the slope of the line tangent to the graph of the function at the given value of x.
y =x4+ 2x3+ 2x + 2 at x = – 3
A)
B)
C)
67
D)
Answer:
B
73)
Given f(x + h) – f(x) = 4xh + 4h + 2h2, find the slope of the tangent line at x = 4.
A)
B)
C)
22
D)
Answer:
A
Find the equation of the tangent line to the curve when x has the given value.
74)
f(x) =2–x2; x =9
A)
B)
C)
y =9x +83
D)
Answer:
D
17
75)
Find the equation of the tangent line to the graph of the function at the given value of x.
f(x) =x2+ 5x at x = 4
A)
B)
C)
y =1
20 x +1
5
D)
Answer:
B
Solve the problem.
76)
Suppose an object moves along the y–axis so that its location is y = f(x) =x2+ x at time x (y is in meters and x is
in seconds). Find the average velocity (the average rate of change of y with respect to x) for x changing from 2 to
9 seconds.
A)
B)
C)
15 m/s
D)
Answer:
B
77)
Suppose an object moves along the y–axis so that its location is y = f(x) =x2+ x at time x (y is in meters and x is
in seconds). Find the average velocity for x changing from 3 to 3 + h seconds.
A)
B)
C)
12 – h m/s
D)
Answer:
D
78)
Suppose an object moves along the y–axis so that its location is y = f(x) =x2+ x at time x (y is in meters and x is
in seconds). Find the instantaneous velocity at x = 4 seconds.
A)
B)
C)
20 m/s
D)
Answer:
B
List the x–values in the graph at which the function is not differentiable.
79)
A)
B)
C)
x = 1
D)
Answer:
B
80)
A)
B)
C)
x = – 3, x = 3
D)
Answer:
A
18
81)
A)
B)
C)
x = 0
D)
Answer:
C
Solve the problem.
82)
If an object moves along a line so that it is at y = f(x) =3x2+ 7x – 8 at time x (in seconds), find the instantaneous
velocity function v = f'(x).
A)
B)
C)
6x2+ 7
D)
Answer:
B
83)
If an object moves along a line so that it is at y = f(x) =10x2 at time x (in seconds), find the velocity at
x = 1 (y is measured in feet).
A)
B)
C)
6 ft/sec
D)
Answer:
A
84)
The electric power p (in W) as a function of the current i (in A) in a certain circuit is given by p(i) =10i2+50i.
Find the instantaneous rate of change of p with respect to i for i =0.6 A.
A)
B)
C)
62 W/A
D)
Answer:
C
Provide an appropriate response.
85)
Find f'(x) if f(x) =.
A)
B)
C)
f'(x) =
D)
Answer:
A
86)
Find y’ if y =5
8.
A)
B)
C)
5
8x
D)
Answer:
A
87)
Find y’ if y = 6x.
A)
B)
C)
x
D)
Answer:
D
88)
Find f'(x) for f(x) = 2x5+ 6x8.
A)
B)
C)
10x3+ 48x2
D)
Answer:
B
19
89)
Find the derivative of y =3x5–7x2– 4
x2.
A)
B)
C)
y=9x–2+8x–3
D)
Answer:
D
90)
Let f and g be functions that satisfy f'(4) = 2 and g'(4) = – 3. Find h'(4) for h(x) = 3f(x) – g(x) + 2.
A)
B)
C)
11
D)
Answer:
D
91)
Find f'(x) if f(x) = 3x4+ 6x7.
A)
B)
C)
12x3+ 42x6
D)
Answer:
C
92)
Find f'(x) if f(x) = 6x–2+ 8x3+ 11x.
A)
f'(x) = – 12x–3+ 24x2+ 11
B)
f(x) = – 12x–1+ 24x2
C)
f'(x) = – 12x–1+ 24x2+ 11
D)
f'(x) = – 12x–3+ 24x2
Answer:
A
93)
Find f'(x) if f(x) = 9x7/5 – 5x2+ 10000.
A)
f'(x) =63
5x2/5 – 10x
B)
f'(x) =63
5x6/5 – 10x + 4000
C)
f'(x) =63
5x2/5 – 10x + 4000
D)
f'(x) =63
5x6/5 – 10x
Answer:
A
94)
Find: d
dx
4
x4– 4 5x
A)
B)
C)
1
x3–4
5
4x
D)
Answer:
A
95)
Find: dy
dt if y =3t–4–5t–1
A)
B)
C)
–12t–5+5t–2
D)
Answer:
C
96)
Find: d
dx
4
x4– 5 3x
A)
B)
C)
1
x3+5
3x–4/3
D)
Answer:
D
20
97)
Find d
dv (6v0.7 –v5.8)
A)
B)
C)
4.2v–0.3 –5.8v4.8
D)
Answer:
C
98)
Find dy
dx for y =1
3x3+x7
10 .
A)
B)
C)
–x–4+7
10 x6
D)
Answer:
C
99)
Find the equation of the tangent line at x = 7 for f(x) = 6 –x2. Write the answer in the form y = mx + b.
A)
B)
C)
y = 14x – 55
D)
Answer:
B
100)
Find the equation of the tangent line at x = – 6 for f(x) =x3
2. Write the answer in the form y = mx + b.
A)
B)
C)
y = 18x + 216
D)
Answer:
B
101)
Find the values of x where the tangent line is horizontal for f(x) = 3x3– 2x2– 9.
A)
B)
C)
x = 0, x =2
3
D)
Answer:
A
102)
Find the equation of the tangent line at x = 2 for f(x) = 4 + x – 2x2– 3x3. Write the answer in the form
y = mx + b.
A)
B)
C)
y = – 43x + 60
D)
Answer:
C
Solve the problem.
103)
An object moves along the y–axis (marked in feet) so that its position at time t (in seconds) is given by
f(t) = 9t3– 9t2+ t + 7. Find the velocity at three seconds.
A)
B)
C)
109 feet per second
D)
Answer:
D
21
104)
A pen manufacturer determined that the total cost in dollars of producing x dozen pens in one day is given by:
C(x) = 350 + 2x – 0.01x2, 0
x 100
Find the marginal cost at a production level of 70 dozen pens and interpret the result.
A)
The marginal cost is $0.60/doz. The cost of producing 1 dozen more pens at a production level of 70 dozen
pens is approximately $0.60.
B)
The marginal cost is $0.59/doz. The cost of producing 1 dozen more pens at a production level of 70 dozen
pens is approximately $0.59.
C)
The marginal cost is $0.62/doz. The cost of producing 1 dozen more pens at a production level of 70 dozen
pens is approximately $0.62.
D)
The marginal cost is $0.58/doz. The cost of producing 1 dozen more pens at a production level of 70 dozen
pens is approximately $0.58.
Answer:
A
105)
According to one theory of learning, the number of items, w(t), that a person can learn after t hours of instruction is
given by:
w(t) = 15 3t2,0
t 64
Find the rate of learning at the end of eight hours of instruction.
A)
B)
C)
5 items per hour
D)
Answer:
C
Find
y for the given values of x1 and x2.
106)
y = 2x + 3; x = 18, x = 0.5
A)
B)
C)
1
D)
Answer:
C
Find dy.
107)
y =9x2+ 9x – 3
A)
B)
C)
18x + 18 dx
D)
Answer:
B
108)
y = x 9x – 4
A)
B)
C)
27x + 8
2 9x – 4 dx
D)
Answer:
B
Provide an appropriate response.
109)
Evaluate dy and
y for y = f(x) =x2–7x + 5, x = 7, and dx =
x = 0.5.
A)
B)
C)
dy = 3.5;
y = 3.75
D)
Answer:
C
110)
Evaluate dy and
y for y = f(x) = 20 + 15x2–x3, x = 2, and dx =
x = 0.3.
A)
dy = 14.4;
y = 14.4
B)
dy = 14.4;
y = 15.183
C)
dy = 15.183;
y = 15.183
D)
dy = 15.183;
y = 14.4
Answer:
B
22
111)
A spherical balloon is being inflated. Find the approximate change in volume if the radius increases from 6.3
cm to 6.4 cm. (Recall that V =4
3r3.)
A)
B)
C)
333.4 cm3
D)
Answer:
D
Solve the problem.
112)
A cube 4 inches on an edge is given a protective coating 0.1 inches thick. About how much coating should a
production manager order for 1200 cubes?
A)
B)
C)
About 7680 in.3
D)
Answer:
D
113)
One hour after x milligrams of a particular drug are given to a person, the change in body temperature T (in
degrees Fahrenheit) is given by T =x21 –x
9 , where 0
x 3. Approximate the changes in body temperature
produced by changing the drug dosage from 1 to 1.4 milligrams. Round to the nearest hundredth when
necessary.
A)
B)
C)
1.67°F
D)
Answer:
A
114)
V =4
3r3, where r is the radius, in centimeters. By approximately how much does the volume of a sphere
increase when the radius is increased from 2.0 cm to 2.2 cm? (Use 3.14 for .)
A)
B)
C)
10.0 cm3
D)
Answer:
C
Provide an appropriate response.
115)
Suppose that the total profit in hundreds of dollars from selling x items is given by P(x) =4x2– 5x + 10. Find the
marginal profit at x = 5.
A)
B)
C)
$35
D)
Answer:
C
116)
The revenue (in thousands of dollars) from producing x units of an item is modeled byR(x) = 5x – 0.0005x2. Find
the marginal revenue at x = 1000.
A)
B)
C)
$4.00
D)
Answer:
C
23
117)
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– 0.06x2+ 300x + 10,000
R(x) =350x
A)
C'(x) =0.0009x2– 0.12x + 300
R'(x) =350
P'(x) =0.0009x2– 0.12x – 50
B)
C'(x) =0.0009x2– 0.12x + 300
R'(x) =350
P'(x) = – 0.0009x2+ 0.12x + 50
C)
C'(x) =0.0009x2+ 0.12x + 300
R'(x) =350
P'(x) =0.0009x2+ 0.12x + 50
Answer:
B
118)
The total cost to produce x units of paint is C(x) = (5x + 3)(7x + 4). Find the marginal average cost function.
A)
C‘(x) = 35 –12
x2
B)
C‘(x) = 35x + 41 +12
x
C)
C‘(x) = 70 –41
x
D)
C‘(x) = 70x + 41
Answer:
A
119)
The total profit from selling x units of doorknobs is P(x) =(6x – 7)(9x – 8). Find the marginal average profit
function.
A)
B)
C)
P‘(x) = 54x –111
D)
Answer:
D
120)
The total cost in dollars of producing x lawn mowers is given by C(x) = 4,000 + 90x –x2
3. Find the marginal
average cost at x = 20, C‘(20) and interpret the result.
A)
–$10.33; a unit increase in production will decrease the average cost per unit by approximately $10.33 at a
production level of 20 units.
B)
–$13.33; a unit increase in production will decrease the average cost per unit by approximately $13.33 at a
production level of 20 units.
C)
–$1.33; a unit increase in production will decrease the average cost per unit by approximately $1.33 at a
production level of 20 units.
D)
–$20.33; a unit increase in production will decrease the average cost per unit by approximately $20.33 at a
production level of 20 units.
Answer:
A
24
Solve the problem.
121)
The demand equation for a certain item is p = 14 –x
1,000 and the cost equation is C(x) = 7,000 + 4x. Find the
marginal profit at a production level of 3,000 and interpret the result.
A)
$14; at the 3,000 level of production, profit will increase by approximately $14 for each unit increase in
production.
B)
$16; at the 3,000 level of production, profit will increase by approximately $16 for each unit increase in
production.
C)
$7; at the 3,000 level of production, profit will increase by approximately $7 for each unit increase in
production.
D)
$4; at the 3,000 level of production, profit will increase by approximately $4 for each unit increase in
production.
Answer:
D
122)
A company is planning to manufacture a new blender. After conducting extensive market surveys, the research
department estimates a weekly demand of 600 blenders at a price of $50 per blender and a weekly demand of
800 blenders at a price of $40 per blender. Assuming the demand equation is linear, use the research
department’s estimates to find the revenue equation in terms of the demand x.
A)
B)
C)
R(x) = 80x – 20
D)
Answer:
B
123)
Suppose the demand for a certain item is given by D(p) = – 4p2+ 3p + 5, where p represents the price of the
item. Find D'(p), the rate of change of demand with respect to price.
A)
B)
C)
D'(p) = – 4p + 3
D)
Answer:
B
25