Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
A manufacturer of a product has a marginal revenue function given by dr
dq =200 + 70q – 3q2. The
demand function for the product is given by
1)
A)
p= 200 + 35q–q2.
B)
p=70
q– 6.
C)
p= 70 – 6q.
D)
p= 200q+ 35q3–q4.
E)
p= 200q+ 35q2–q3.
2)
If y=x ln x, then dy =
2)
A)
1 + ln x.
B)
(x+ ln x) dx.
C)
x+ ln x.
D)
(1 + ln x) dx.
E)
none of the above
3)
(x4–x2+ 3) dx =
3)
A)
(x4–x2+3)2
2+C
B)
x5
5–x3
3+ 3x+C
C)
5x5– 3x3+ 3x+C
D)
4x3– 2x+C
E)
4x3
3–2x+C
4)
x3–1
x4+ 2 dx =
4)
A)
x4
4–3
x3+ 2x+C
B)
3x2–1
4x3+C
C)
x4
4–1
3x3+ 2x+C
D)
3x2+ 4x–5+C
E)
x4
4+1
3x3+ 2x+C
5)
3
1
t3+2
t2dt =
5)
A)
–16
3
B)
4
3
C)
2
9
D)
–4
3
E)
16
3
6)
x+ 1
(x2+ 2x)2dx =
6)
A)
(x2+ 2x)3
2+C
B)
–(x2+ 2x)–1+C
C)
–1
2(x2+ 2x)–1+C
D)
(x2+ 2x)3
3+C
E)
(x2+ 2x)3
6+C
7)
0
–1
(1 + 1) dx =
7)
A)
–1
B)
–2
C)
–1
D)
–2
E)
none of the above
8)
2
–1
(2x2– 4x+ 4) dx =
8)
A)
32
3
B)
14
C)
25
3
D)
12
E)
38
3
3
9)
A manufacturer’s marginal revenue function is dr
dq = 200 – 6q, where r is in dollars. What is the
change in revenue if the number of units sold increases from q= 10 to q= 20?
9)
A)
$900
B)
$1000
C)
$1100
D)
$1200
E)
$1300
10)
5
0
x
x2+ 1 dx =
10)
A)
1
2 ln 26
B)
ln 26
C)
1
2 ln 24
D)
ln 24
E)
none of the above
11)
The demand equation for a certain product is p= 25 – 0.005q, where p is the price per unit (in
dollars) for q units. If its supply equation is p= 1 + 0.03q, then the consumers’ surplus when market
equilibrium is established is
11)
A)
$1176.49
B)
$1250.87
C)
$1500.
D)
$1675.30
E)
$1800.
12)
The demand equation for a certain product is p= 400 – 2q, where p is the price per unit (in dollars)
for q units. If its supply equation is p=q+ 100, then the consumers’ surplus when market
equilibrium is established is
12)
A)
$9000.
B)
$9500.
C)
$10,000.
D)
$10,500.
E)
$11,000.
13)
2
0
2ex/2 dx =
13)
A)
1
2(
e
– 1)
B)
1
4(
e
– 1)
C)
e
– 1
D)
2(
e
– 1)
E)
4(
e
– 1)
14)
e3x+4dx =
14)
A)
(3x+ 4)e3x+3+C
B)
3e3x+4+C
C)
e3x+5
3x+ 5 +C
D)
e3x+4+C
E)
1
3e3x+4+C
15)
4x
x2+ 1 dx =
15)
A)
1
2 ln(x2+ 1) +C
B)
4 ln x+ 1 +C
C)
2
x+C
D)
2 ln(x2+ 1) +C
E)
ln(x2+ 1) +C
16)
3
1
1
3pdp =
16)
A)
2 –2
33
B)
6 – 2 3
C)
3 –3
D)
1 –3
3
E)
6 –2
33
17)
If
e4
e2
k
xdx = 1, then k=
17)
A)
1
4
B)
1
2
C)
ln 2
D)
2
E)
4
18)
If
1
(kx2– 2) dx = 1, then k=
18)
A)
15
28
B)
3
4
C)
9
13
D)
15
26
E)
9
14
19)
2
5/3
(3z–5)1000 dz =
19)
A)
1
3
B)
1
C)
1
3003
D)
1
1001
E)
–1
20)
If y’ =xex2 and y(0) =7
2, then y=
20)
A)
ex
2+7
2
B)
2
ex
2+7
2
C)
2
ex
2+3
2
D)
ex2
E)
1
2
ex
2+ 3
21)
Suppose the points (0, 2), (1, 3) and (2, 0) lie on the graph of the continuous function f. Using the
trapezoidal rule and all of these points, an approximation to
2
0
f(x) dx is
21)
A)
4.
B)
2.
C)
8.
D)
6.
E)
10.
22)
x2+ 3x–4
x+ 2 dx =
22)
A)
x2
2+x– 6 ln x+ 2 +C
B)
1
3 ln x+ 2 +C
C)
x2
2+ 5x+ 6 ln x+ 2 +C
D)
1
2 ln x+ 2 +C
E)
x3
3+3x2
2– 4x
x2
2+ 2x
+C
23)
2
1
2x– 3
x2– 3xdx =
23)
A)
e5
B)
7 ln 2
C)
1
2(4 – 3 ln 2)
D)
0
E)
none of the above
24)
0
–2
1
1 – 4xdx =
24)
A)
1
B)
2
C)
3
D)
4
E)
5
A
25)
The exact area of the region bounded by the graphs of y=x2+x+ 1, x= – 2, x= 1, and the x–axis is
25)
A)
5
2 sq units.
B)
7
2 sq units.
C)
13
2 sq units.
D)
9
2 sq units.
E)
11
2 sq units.
D
8
D
26)
If dy
dx = 3x2– 3 and y(0) = 8, then y(1) =
26)
A)
0.
B)
4.
C)
6.
D)
8.
E)
12.
27)
The exact area of the region bounded by the graphs of y=x and y=x2 is
27)
A)
1
3 sq unit.
B)
1
2 sq unit.
C)
2
3 sq unit.
D)
1
6 sq unit.
E)
5
6 sq unit.
28)
2
1
(2x–3)4dx =
28)
A)
0
B)
–1
5
C)
2
5
D)
3
10
E)
1
5
29)
3
2
1 – 2xdx =
29)
A)
2 ln 5
3
B)
–2 ln 5
3
C)
–8 ln 5
3
D)
–4 ln 5
3
E)
8 ln 5
3
30)
1
0
x(x2+ 5) dx =
30)
A)
11
B)
11
2
C)
11
4
D)
18
E)
9
31)
x2+ 4x– 3
x– 1 dx =
31)
A)
1
2 ln x– 1 +C
B)
7x2
2– 2x+C
C)
x2
2+ 5x + 2 ln x– 1 +C
D)
1
3 ln x– 1 +C
E)
x2
2+ 6x + 3 ln x– 1 +C
32)
1
0
32xdx =
32)
A)
432
B)
32
C)
332
4
D)
0
E)
1
33)
Use the trapezoidal rule with n= 4 to estimate the value of
4
2
(x2+ 1) dx.
33)
A)
83
4
B)
105
8
C)
62
3
D)
62
E)
83
34)
1
0
(x+ 4) 3x2+ 8x– 1 dx =
34)
A)
0
B)
33
2
C)
33
8
D)
45
8
E)
45
2
35)
A manufacturer of a product has a marginal cost function given by dc
dq = 0.1q2– 20q + 1500, where c
is the total cost (in dollars) of producing q units of a product. If fixed costs are $30,000, then the total
cost of producing 30 units is
35)
A)
$66,600.
B)
$66,700.
C)
$66,800.
D)
$66,900.
E)
$67,000.
36)
2x– 1
3dx =
36)
A)
1
3(x2–x) +C
B)
(2x– 1)2
6+C
C)
2
3+C
D)
x2–x
3x+C
E)
1
2+C
E)
37)
The exact area of the region bounded by the graphs of y=x2– 5 and y= 2x+ 3 is
37)
A)
24 sq units.
B)
28
3 sq units.
C)
36 sq units.
D)
73
3 sq units.
E)
60 sq units.
C
B)
E)
38)
32/3 dx =
38)
A)
5
3·35/3+C
B)
0 +C
C)
0
D)
32/3x+C
E)
3
5· 35/3+C
D
B)
E)
A
B)
39)
1
0
2dx =
39)
A)
2
B)
2
3(23/2)
C)
2
2
D)
3
2(23/2)
E)
1
2 2
40)
Given the marginal cost function dc
dq = 2q+ 50, where c is in dollars, how much would it cost to
increase production from q= 50 to q= 100?
40)
A)
$5000
B)
$10,000
C)
$15,000
D)
$16,000
E)
$20,000
41)
If dy
dx = 3x2– 3 – 4e2x and y(0) = 8, then y=
41)
A)
x3– 3x– 2e2x.
B)
x3– 3x– 2e2x+ 8.
C)
x3– 3x– 2e2x+ 10.
D)
x3– 3x– 4e2x.
E)
8.
42)
Suppose that the points (–1, 2), (–0.5, 1), (0, 0.5), (0.5, 0), and (1, 1) lie on the graph of the continuous
function f, where f(x) 0. Using Simpson‘s rule and all of these points, an approximation to the area
between the graph of f and the x–axis on the interval –1, 1 is
42)
A)
4
9 sq units.
B)
3
4 sq units.
C)
4
3 sq units.
D)
9
4 sq units.
E)
none of the above
43)
By using differentials, an approximation of 3123 is
43)
A)
497
100.
B)
413
15 .
C)
473
75 .
D)
423
25 .
E)
52
15 .
44)
1
–1
(x2+ 2x+ 1) dx =
44)
A)
0
B)
16
C)
8
3
D)
8
E)
16
3
14
45)
0
–1
4(x+ 1)e(x+1)2dx =
45)
A)
2(
e
– 1)
B)
e
(3
e
– 2)
C)
1
2(
e
– 1)
D)
0
E)
1 –
e
46)
By using differentials, an approximation of ln(1.03) is
46)
A)
–0.01.
B)
0.01.
C)
0.02.
D)
0.03.
E)
0.04.
47)
If y’ = 6x– 3 and y(2) = 4, then y=
47)
A)
3x2– 3x– 2.
B)
6x2– 3x + 2.
C)
3x2– 3x.
D)
6.
E)
3x2– 3x+ 2.
48)
1
0
x x2+ 1 dx =
48)
A)
2 2 + 1
3
B)
2 2 – 1
3
C)
2(2 2– 1)
3
D)
2(2 2+ 1)
3
E)
1
49)
The exact area of the region bounded by the graphs of y=x2– 4, and the x–axis from x= 0 to x= 4
is
49)
A)
12 sq units.
B)
16
3 sq units.
C)
32
3 sq units.
D)
16 sq units.
E)
64
3 sq units.
B
B
50)
If y=3x2– 4, then dy =
50)
A)
6x3x2– 4 dx.
B)
6x
3x2– 4
dx.
C)
2 3x2– 4
3xdx.
D)
3x
3x2– 4
dx.
E)
1
3x2– 4
dx.
51)
2
1
x+ 3
x2dx =
51)
A)
ln(2) +5
2
B)
ln(2) +3
2
C)
–ln(2) +3
2
D)
8
5
E)
–ln(2) +5
2
52)
The exact area of the region bounded by the graphs of y=x, y=x
2, y= 2, and y= 3 is
52)
A)
3
2 sq units.
B)
5
2 sq units.
C)
3
4 sq units.
D)
7
4 sq units.
E)
5
4 sq units.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
53)
Use differentials to approximate the change in the wattage W of a flood light with a
resistance R= 8 ohms, if the current I is increased from 4 amperes to 4.2 amperes. (Hint:
Use W=RI2.)
53)
54)
Use Simpson’s rule with n= 4 to find an approximate value of
2
0
1
3 +x2dx.
54)
55)
A yeast culture is growing at a rate of A’(t) = 0.3e0.2t2, where t is the time in hours and A(t)
is the amount in grams. Use n= 8 and the trapezoidal rule to approximate
4
0
0.3e0.2t2dt,
the amount the culture grew over the first four hours.
55)
56)
Find the exact area of the region bounded by the graphs of y= 8 – 2x–x2 and y= 3x+ 2.
Also sketch the region.
56)
57)
Determine: (3x–4)5dx
57)
58)
Evaluate:
2
1
d
dx
5
3
e x +x3 dx dx
58)
18
59)
Determine x– 8
x + 4 dx
59)
60)
Determine: (y2+ 1)(y2– 1) dy
60)
61)
A company has determined that its marginal revenue function (in dollars) is given by
R’(x) = 10,000 –x2, where x is the number of units sold. Find the total revenue for selling 4
units by evaluating
4
0
(10,000 –x2) dx .
61)
62)
Suppose the rate of change in the number of violent crimes per 100,000 people in the
United States can be modeled by dc
dt =280
2t+ 6 . Find c(t).
62)
63)
The marginal cost function for a manufacturer’s product is dc
dq = 0.0003 q2– 0.03q+ 4,
where c is in dollars. If fixed costs are $5000, determine: (a) the manufacturer’s total cost
function; (b) the manufacturer’s average cost function.
63)
64)
The rate of growth of a bacteria (in thousands) is estimated by dN
dt = 73t– 21, where t is in
hours. If N(9) = 3000, then find N(t).
64)
65)
Determine 4t5(t2– 6t4) dt
65)
66)
Determine:
4
–1
(2x+ 3) dx
66)
67)
A bacteria population is increasing at a rate of dp
dt = 2e2t5e2t. Find p as a function of t.
67)
68)
The marginal price for a weekly demand of x bottles of shampoo is given by p’(x) =
–3
(3x +50)2. Find p(x).
68)
69)
Use your graphing calculator to find the area between the x–axis and f(x) = – x2+ 4x– 3 on
the interval 1 x 3. Verify your result by finding
3
1
(–x2+ 4x– 3) dx.
69)
70)
Determine 5xdx
70)
71)
Determine 45xdx
71)
72)
Suppose the surface area of a 1.9 meter–tall person changes at the rate of ds
dx = 0.14x–0.62
square meters per kilogram, where x is the weight in kilograms. Find the decrease in
surface area for a person whose weight decreases from 75 kilograms to 70 kilograms by
evaluating
70
75
0.14x–0.62 dx.
72)