Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use or where appropriate to describe the behavior at each zero of the denominator and identify all vertical
asymptotes.
1)
g(x) =x
6 – x
1)
A)
lim
x
6–
f(x) = ; lim
x
6+
f(x) =; x = 6 is a vertical asymptote
B)
lim
x
6–
f(x) =; lim
x
6+
f(x) = ; x = 6 is a vertical asymptote
C)
lim
x
6–
f(x) = ; lim
x
6+
f(x) = ; x = 6 is a vertical asymptote
D)
lim
x
6–
f(x) =; lim
x
6+
f(x) = ; x = 0 is a vertical asymptote
Sketch a possible graph of a function that satisfies the given conditions.
2)
f(0) =1 and lim
x
0 f(x) =1
2)
A)
B)
1
C)
D)
3)
f(–1) = – 7 ; lim
x
(–1)–
f(x) = – 2; lim
x
(–1)+
f(x) = – 7
3)
A)
B)
2
C)
D)
4)
f(1) = 4; lim
x
1–
f(x) = 4; lim
x
1+
f(x) = 3
4)
A)
B)
3
C)
D)
Provide an appropriate response.
5)
The revenue (in thousands of dollars) from producing x units of an item is modeled by
R(x) = 5x – 0.0005x2. Find the marginal revenue at x = 1000.
5)
A)
$104.00
B)
$4.00
C)
$4.50
D)
$10,300.00
Solve the problem.
6)
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.
6)
A)
R(x) = 80x –x2
20
B)
R(x) = 80x – 20
C)
R(x) = 80x – 20x2
D)
R(x) = 20x +x2
20
Use the given graph to find the indicated limit.
7)
lim
x
7–
f(x)
7)
A)
B)
C)
0
D)
7
Provide an appropriate response.
8)
Find: d
dx
4
x4– 4 5x
8)
A)
16
x3– 20 4x
B)
–16
x3–4
5
4x
C)
–16
x5–4
55x4
D)
1
x3–4
5
4x
9)
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.
9)
A)
y = – 43x + 48
B)
y = – 43x + 60
C)
y = – 39x + 52
D)
y = – 47x + 68
5
10)
Use the four step process to find f'(x) for the function f(x) =2
x2.
10)
A)
–2(h + 2x + xh)
x2(x + h)2
B)
(h + 2x)
x2(x + h)2
C)
2(h + x)
x2(x + h)2
D)
–2(h + 2x)
x2(x + h)2
11)
Find y’ if y =5
8.
11)
A)
0
B)
5
8x
C)
1
D)
5
8
12)
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
12)
A)
65
B)
67
C)
–52
D)
–50
13)
Evaluate dy and
y for y = f(x) =x2–7x + 5, x = 7, and dx =
x = 0.5.
13)
A)
dy = 3.75;
y = 3.5
B)
dy = 3.5;
y = 3.5
C)
dy = 3.5;
y = 3.75
D)
dy = 3.75;
y = 3.75
6
14)
Use a graphing utility to find the discontinuities of the given rational function.
f(x) =x2+ 2x + 1
x3+ 2x2+ 5x – 8
14)
A)
–1
B)
3
C)
1
D)
Continuous at all values of x
Find the limit, if it exists.
15)
Let f(x) =
x2– 16
x + 4 if x > 0
x2– 16
x – 4 if x < 0
Find lim
x
0f(x).
15)
A)
0
B)
–4
C)
–
D)
Does not exist
D
16)
Find: lim
h
0
f(7 + h) – f(7)
h for f(x) = – x + 1.
16)
A)
0
B)
1
C)
–1
D)
Does not exist
C
C
Solve the problem.
17)
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?
17)
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
; Yes
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
; No
Provide an appropriate response.
18)
A spherical balloon is being inflated. Find the approximate change in volume if the radius
increases from 6.0 cm to 6.2 cm. (Recall that V =4
3r3.)
18)
A)
0.96 cm3
B)
144 cm3
C)
28.8 cm3
D)
288 cm3
Solve the problem.
19)
The cost of manufacturing a particular videotape is C(x) = 9000 + 9x, where x is the number of
tapes produced. The average cost per tape, denoted by C(x), is found by dividing C(x) by x. Find
lim
x
9000 C(x).
19)
A)
14
B)
6
C)
10
D)
Does not exist
8
Use the given graph to find the indicated limit.
20)
Find lim
x
f(x).
20)
A)
3
B)
C)
D)
4
21)
lim
x
5+
f(x)
21)
A)
B)
0
C)
5
D)
Find average rate of change for the function over the given interval.
22)
Find the average rate of change for f(x) =2x if x changes from 2 to 8.
22)
A)
–3
10
B)
2
C)
7
D)
1
3
Provide an appropriate response.
23)
Evaluate dy and
y for y = f(x) = 20 + 15x2–x3, x = 2, and dx =
x = 0.3.
23)
A)
dy = 15.183;
y = 15.183
B)
dy = 15.183;
y = 14.4
C)
dy = 14.4;
y = 14.4
D)
dy = 14.4;
y = 15.183
Solve the problem.
24)
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.
24)
A)
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.
B)
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.
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 $0.60/doz. The cost of producing 1 dozen more pens at a production level
of 70 dozen pens is approximately $0.60.
The graph of y = f(x) is shown. Use the graph to answer the question.
25)
Is f continuous at x =-1?
25)
A)
No
B)
Yes
Find the limit, if it exists.
26)
Find: lim
x– 4
x2– 16
x + 4
26)
A)
16
B)
–8
C)
8
D)
– 24
Provide an appropriate response.
27)
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.
27)
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)
–$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.
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)
–$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.
11
28)
Find the horizontal asymptote, if any, of the given function.
f(x) =(x – 3)(x + 4)
x2– 4
28)
A)
x = 2, x = – 2
B)
y = 3, y = – 4
C)
y = 1
D)
None
29)
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.
29)
A)
$45
B)
$15
C)
$35
D)
$32
The graph of a function f is given. Use the graph to answer the question.
30)
Use the graph of f given below to find f(4).
10
-10 10
-10
30)
A)
6
B)
–8
C)
0
D)
4
Find the limit, if it exists.
31)
Let f(x) =
x2– 16
x + 4 if x > 0
x2– 16
x – 4 if x < 0
Find lim
x
0
f(x).
31)
A)
B)
–4
C)
4
D)
Does not exist
Provide an appropriate response.
32)
Use a sign chart to solve the inequality. Express answers in interval notation.
x2> 16
32)
A)
(–4, 4 )
B)
(–, –4) (4, )
C)
(4, )
D)
(–4, )
D)
Find the limit, if it exists.
33)
Find: lim
x–1
6x + 5
5x – 6
33)
A)
1
11
B)
–1
11
C)
–11
D)
1
D)
Provide an appropriate response.
34)
Find the vertical asymptote(s) of the graph of the given function.
f(x) =3x – 9
5x + 30
34)
A)
y = 8
B)
x = – 6
C)
x = – 8
D)
y = – 3
D)
D)
35)
Find: d
dx
4
x4– 5 3x
35)
A)
1
4x–5– 15x2/3
B)
–16x–5–5
3x–2/3
C)
1
x3+5
3x–4/3
D)
1
4x3–5
3x–2/3
36)
Use the four step process to find f'(x) for the function f(x) =5x2– 3x.
36)
A)
10x + 5h – 3
B)
5h2– 3h
C)
5h – 3
D)
10x – 3
37)
Find the derivative of y =3x5–7x2– 4
x2.
37)
A)
y=9x2+8x3
B)
y=9x2+8x–3
C)
y=18x2+8x–3
D)
y=9x–2+8x–3
38)
Use the four step process to find f'(x) for the function f(x) =x
6 – x .
38)
A)
6
(x – 6)(x + h – 6)
B)
1
(x – 6)(x + h – 6)
C)
–6
h(x – 6)(x + h – 6)
D)
–x
(x – 6)(x + h – 6)
39)
Find the horizontal asymptote, if any, of the given function.
f(x) =2x3– 3x – 9
9x3– 5x + 3
39)
A)
y =3
5
B)
y =2
9
C)
y = 0
D)
None
40)
Use a graphing utility to find the discontinuities of the given rational function.
g(x) =x + 1
x3+ 2x2+ 10x – 13
40)
A)
3
B)
1
C)
–1
D)
Continuous at all values of x
B
41)
If the limit at infinity exists, find the limit.
lim
x
3x3+ 5x
4x4+ 10x3+ 2
41)
A)
B)
3
4
C)
0
D)
1
C
42)
If the limit at infinity exists, find the limit.
lim
x
5x2+ 7x – 9
– 6x2+ 2
42)
A)
–5
6
B)
C)
0
D)
–2
9
A
B
Use the graph to evaluate the indicated limit and function value or state that it does not exist.
43)
Find lim
x
0–
f(x) and lim
x
0+
f(x).
43)
A)
4; –1
B)
4; Does not exist
C)
Does not exist; does not exist
D)
–1; 4
Provide an appropriate response.
44)
Find the equation of the tangent line at x = – 6 for f(x) =x3
2. Write the answer in the form y = mx +
b.
44)
A)
y = 18x + 216
B)
y = 54x + 216
C)
y = 216x + 54
D)
y = 216x + 18
45)
Given f(x + h) – f(x) = 4xh + 4h + 2h2, find the slope of the tangent line at x = 4.
45)
A)
8
B)
16
C)
22
D)
20
46)
Find the slope of the secant line joining (2, f(2)) and (3, f(3)) for f(x) = – 3x2– 8.
46)
A)
15
B)
55
C)
–55
D)
–15
47)
Use a sign chart to solve the inequality. Express answers in interval notation.
– 5
–3x – 4 > 0
47)
A)
–4
3 ,
B)
(0, )
C)
–, 4
3
D)
–, –3
4
Find the limit, if it exists.
48)
Find: lim
x
3
x2– 9
x – 3 +x2+ 7
48)
A)
10
B)
3
C)
2
D)
Does not exist
A
Find the equation of the tangent line to the curve when x has the given value.
49)
f(x) =-4–x2; x =2
49)
A)
y =2x +0
B)
y = – 2x
C)
y =4x –0
D)
y =-4x +0
D
Solve the problem.
50)
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.
50)
A)
45 items per hour
B)
5 items per hour
C)
60 items per hour
D)
20 items per hour
B
A
Find the limit, if it exists.
51)
Given lim
x
5f(x) = 4 and lim
x
5 g(x) = – 5, find lim
x
5
2f(x) + 3g(x)
3f(x) .
51)
A)
7
15
B)
–7
15
C)
7
12
D)
–7
12
Use the graph to evaluate the indicated limit and function value or state that it does not exist.
52)
Find lim
x
0f(x) and f(0).
52)
A)
Does not exist; 6
B)
6; 0
C)
0; does not exist
D)
0; 6
Solve the problem.
53)
Suppose that the value V of a certain product decreases, or depreciates, with time t, in months,
where
V(t) = 100 –30t2
(t + 2) 2 .
Find lim
t
V(t).
53)
A)
100
B)
85
C)
30
D)
70
Provide an appropriate response.
54)
Use a graphing utility to approximate the partition numbers of the function to four decimal places:
f(x) =x4– 8x2– 4x + 1.
54)
A)
(–, –2.4976)
(–2.4976, –0.7203)
B)
(–, –2.4976)
(0.1832 , 3.0347)
C)
(–, –2.4976)
(–2.4976, –0.7203) (–0.7203, 0.1832 )
(0.1832 , 3.0347)
D)
(–, –2.4976)
Use the definition f'(x) =lim
h
0
f(x +
h) – f(x)
h to find the derivative at x.
55)
f(x) =10 – 5x2
55)
A)
–10x2
B)
10 – 10x
C)
10 – 5x
D)
–10x
D
Provide an appropriate response.
56)
Find: dy
dt if y =3t–4–5t–1
56)
A)
–12 t5– 5t2
B)
–12t–5–5t–2
C)
–12
t5–5
t2
D)
–12t–5+5t–2
D
Solve the problem.
57)
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 2.1 cm? (Use 3.14 for .)
57)
A)
0.3 cm3
B)
5.2 cm3
C)
5.0 cm3
D)
4.8 cm3
C
C
Provide an appropriate response.
58)
Find f'(x) if f(x) = 3x4+ 6x7.
58)
A)
7x3+ 13x6
B)
3x5+ 7x8
C)
4x3+ 7x6
D)
12x3+ 42x6
59)
Find f'(x) for f(x) = 2x5+ 6x8.
59)
A)
2x4+ 6x7
B)
10x3+ 48x2
C)
10x6+ 48x9
D)
10x4+ 48x7
Sketch a possible graph of a function that satisfies the given conditions.
60)
f(0) = 6; lim
x
0–
f(x) = 0; lim
x
0+
f(x) = 0
60)
A)
B)
20