Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Match the differential equation with the appropriate slope field.
1)
dy
dx =x2–y2
1)
A)
B)
1
C)
D)
2)
dy
dx =x
y
2)
A)
B)
C)
3
D)
3)
dy
dx = y + 2
3)
A)
4
B)
C)
D)
4)
dy
dx = x – y
4)
A)
B)
C)
6
D)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
5)
(A) Calculate the change in F(x) from x =3 to x =7.
(B) Graph F(x) and use geometric formulas to calculate the area between the graph of F(x)
and the x–axis from x =3 to x =7.
(C) What guarantees that your answers to (A) and (B) are equal?
F(x) = x(3
2x + 1)
5)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
7
6)
Divide the interval [0, 8] into four equal subintervals and draw in the corresponding left
rectangles.
6)
A)
B)
C)
D)
Solve the problem.
7)
A population of bacteria grows at a rate of P'(t) = 12 et where t is time in hours. Determine how
much the population increases from t = 0 to t = 3. Round your answer to two decimal places.
7)
A)
241.03
B)
470.06
C)
229.03
D)
235.03
8)
A factory discharges pollutants into a large river at a rate that is estimated by a water quality
control agency to be P'(t) = t 1 +t2 for 0 t 5, where P(t) is the total number of tons of pollutants
discharged into the river after t years of operation. What quantity of pollutants will be discharged
into the river from the end of the third year to the end of the fifth year? (Round to two decimal
places.)
8)
A)
33.65 tons
B)
75.71 tons
C)
32.67 tons
D)
50.67 tons
Identify the rectangles shown in the graph as left rectangles, right rectangles, or neither.
9)
9)
A)
left rectangles
B)
right rectangles
C)
neither
Solve the problem.
10)
An rock’s acceleration at time t is given by a(t) = 16t, and its initial velocity is 35. Find the velocity
function v(t).
10)
A)
v(t) = 8t2+ 35t
B)
v(t) = 16t2+ 35
C)
v(t) = 8t2+ 35
D)
v(t) = 35t2+ 16
Evaluate the integral.
11)
2
14 x –5
xdx
(Round to three decimal places.)
11)
A)
1.410
B)
3.001
C)
7.505
D)
12.846
12)
16
33 x dx .
12)
A)
288
B)
192
C)
24
D)
128
Solve the problem.
13)
Test marketing for a new health–food snack product in a selected area suggests that sales (in
thousands of dollars) will increase at a rate given by S'(t) = 40 – 40e–0.16t, t months after an
aggressive national advertising campaign is begun. Find total sales during the second 12 months of
the campaign. (Round to the nearest thousand dollars.)
13)
A)
$693,000
B)
$449,000
C)
$267,000
D)
$511,000
14)
A single injection of a drug is administered to a patient. The amount Q in the body then decreases
at a rate proportional to the amount present, and for this particular drug the rate is 3% per hour.
Thus, dQ
dt = – 0.03Q with Q(0) =Q0, where t is time in hours. If the initial injection is 4 milliliters
[Q(0) = 4], about how many hours after the drug is given will there be 2 milliliters of the drug
remaining in the body? (Round answer to the nearest tenth of an hour.)
14)
A)
11.3 hours
B)
21.3 hours
C)
69.3 hours
D)
23.1 hours
Provide an appropriate response.
15)
Find the average value of the function y =e–x over the interval [ 0, 3]. Express an exact answer in
terms of e.
15)
A)
1 –e–3
B)
e– 3 – 1
3
C)
e– 1.5
D)
1 –e–3
3
16)
Find f(x) if f'(x) =3
x5 and f 1
2= 1.
16)
A)
–3
4x–4+ C
B)
–3
4x–4+5
4
C)
–4
5x–4+ 13
D)
–3
4x–4+ 13
Find the integral.
17)
7x6dx
(3 +x7)3
17)
A)
–1
4(3 +x7)4+ C
B)
1
4(3 +x7)4+ C
C)
–7x6
(3 +x7)2+ C
D)
–1
2(3 +x7)2+ C
Solve the problem.
18)
The rate of expenditure for maintenance of a particular machine is given by M'(x) = 12x x2+ 5,
where x is time measured in years. Total maintenance costs through the second year are $105. Find
the total maintenance function.
18)
A)
M(x) = 4(x2+ 5)3/2 – 93
B)
M(x) = 12(x2+ 5)3/2 – 3
C)
M(x) = 12(x2+ 5)3/2 – 93
D)
M(x) = 4(x2+ 5)3/2 – 3
D)
Find the integral.
19)
–3
3x– 4 x dx
19)
A)
–9
2x2/3 –8
3x3/2 + C
B)
–3 ln x1/3 – 2x3/2 + C
C)
x2/3 – 2x3/2 + C
D)
–2x2/3 – 6x3/2 + C
D)
20)
t2
41+t3 dt
20)
A)
1
9(1 +t3)3+ C
B)
4
15 (1 +t3)3/4 + C
C)
4
9(1 +t3)3/4 + C
D)
4
9t3(1 +t3)3/4 + C
D)
21)
9e 0.2x dx.
21)
A)
9e0.2x + C
B)
9e0.2x + 1
0.2x + 1 + C
C)
45e0.2x + C
D)
18e0.2x + C
D)
Provide an appropriate response.
22)
Find the particular solution for the differential equation y’ = 4xe2x ; y(0) = 20.
22)
A)
y = 2xe2x –e2x + 21
B)
y = 4xe2x –e2x + 22
C)
y = 2xe2x + 20
D)
y = 4xe2x –e2x + 21
23)
Find the particular solution for the differential equation dy
dx =1
2 + x; y(0) = 3
23)
A)
y = ln 2 + x – ln 2 + 3
B)
y = ln 2 + x
C)
y = ln 2 + x +ln 2
3
D)
y = ln 2 + x + ln 2 + 3
24)
Given 3
1f(x) dx = 4 and 3
1g(x) dx = 2, use properties of definite integrals to evaluate
3
1[2f(x) + 5g(x)] dx.
24)
A)
54
B)
24
C)
18
D)
13
Solve the problem.
25)
If the marginal price dp
dx at x units of demand per week is proportional to the price p, and if at $80
there is no weekly demand [p(0) = 80], and if at $50.18 there is a weekly demand of 8 units
[p(8) = 50.18], find the price–demand equation.
25)
A)
p(x) = 4.38e–0.058x
B)
p(x) = 80e1.037x
C)
p(x) = 80e–0.058x
D)
p(x) = 4.38e0.305x
Find the integral.
26)
(5 +x3)(4 –x2) dx
26)
A)
20x –5
3x3+x4–1
6+ C
B)
20x –5
3x3+5
3x4–1
6x6+ C
C)
20 –5
3x3+x4–1
6x6+ C
D)
20x –5
3x3+x4–1
6x6+ C
27)
4x + 1
4x2+ 2x + 3 dx
27)
A)
1
2(4x2+ 2x + 3)2+ C
B)
2
(4x2+ 2x + 3)2+ C
C)
1
2 ln 4x2+ 2x + 3 + C
D)
2 ln 4x2+ 2x + 3 + C
Provide an appropriate response.
28)
Find the particular solution for the differential equation y’ = 4x + 7; y(0) = –12.
28)
A)
y = 4x2+ 7x – 12
B)
y = 4x2+ 7x – 6
C)
y = 2x2+ 7x – 6
D)
y = 2x2+ 7x – 12
29)
Find f(x) if f'(x) =7
x4 and f(1) = 4.
29)
A)
–28x–5– 3
B)
–28x–5+ 32
C)
f(x) = – 7
3x–3+19
3
D)
–7
3x–3– 3
Evaluate the integral.
30)
b
05exdx
30)
A)
5eb
B)
5eb + 1
b + 1 –e
2
C)
5eb– 1
D)
5eb–5
Solve the problem.
31)
A company has found that the marginal cost of a new production line (in thousands) is
C'(x) =9
x + e , where x is the number of years the line is in use. Find the total cost function for the
production line (in thousands). The fixed cost is $20,000.
31)
A)
C(x) = 9 ln (x + e) + 20
B)
C(x) =ln (x + e)
9+ 20
C)
C(x) = 9 ln (x + e) + 11
D)
C(x) =ln (x + e)
9+ 11
32)
A company finds that consumer demand quantity changes with respect to price at a rate given by
D'(p) = – 3500
p2. Find the demand function if the company knows that 845 units of the product are
demanded when the price is $5 per unit.
32)
A)
D(p) =7000
p+845
B)
D(p) =3500
p+845
C)
D(p) =3500
p+145
D)
D(p) =3500
p3+145
Find the integral.
33)
(2 + 2x)e(4x + 2x2) dx
33)
A)
2e(4x + 2x2)+ C
B)
e[2(4x + 2x2)] + C
C)
1
2e(4x + 2x2)+ C
D)
e[(1/2)(4x + 2x2)] + C
34)
ln 8x
x dx
34)
A)
(ln 8x)2
16 + C
B)
(ln 8x)2
8+ C
C)
(ln 8x)2+ C
D)
(ln 8x)2
2+ C
Evaluate the integral.
35)
e
1
20
xdx
35)
A)
0
B)
–10e2
C)
20
D)
–20
Provide an appropriate response.
36)
Find the general solution for the differential equation y’=36x2
36)
A)
12x3+ C
B)
x3
3+ C
C)
x3+ C
D)
36x3+ C
Find the integral.
37)
3 dx
37)
A)
3
2x2+ C
B)
0
C)
3 + C
D)
3x + C
38)
x2
5x3+ 8 dx
38)
A)
15 ln 5x3+ 8 + C
B)
1
5 ln 5x3+ 8 + C
C)
15 ln (5x3– 8) + C
D)
1
15 ln 5x3+ 8 + C
Provide an appropriate response.
39)
Find the general solution for the differential equation dx
dt = –3x
39)
A)
x = Ce–3t
B)
t =e–3x + C
C)
t = Ce–3x
D)
x =e–3t + C
Find the integral.
40)
23x4–x–3 dx
40)
A)
4
3x7/3 – 3x–2+ C
B)
6
7x6/7 +1
2x2+ C
C)
6
7x7/3 +1
2x2+ C
D)
6
7x7/3 – 2x–2+ C
Evaluate the integral.
41)
e
116x –5
x dx
(Express your answer in terms for e.)
41)
A)
8e2– 8
B)
16e2– 5
C)
8e2– 13
D)
8e2– 5
D)
Solve the problem.
42)
The management of an oil company estimates that oil will be pumped from a producing field at a
rate given by R(t) =56
t + 7 for 0 t 20, where R(t) is the rate of production in thousands of barrels
per year, t years after pumping begins. How many barrels of oil, Q(t), will be produced the first five
years? (Round answer to the nearest thousand barrels.)
42)
A)
296 thousand barrels
B)
148 thousand barrels
C)
92 thousand barrels
D)
46 thousand barrels
D)
43)
The rate of change in a person’s body temperature, with respect to the dosage of x milligrams of a
drug, is given by D'(x) =2
x + 8 . One milligram raises the temperature 2.4°C. Find the function
giving the total change.
43)
A)
D(x) = ln 2
x + 8 – 2
B)
D(x) =2 ln x + 8 – 2
C)
D(x) = ln 2
x + 8 – 2.4
D)
D(x) =2 ln x + 8 + 2.4
D)
Provide an appropriate response.
44)
How large should n (n an integer) be chosen for Ln and Rn for the approximation of
4
1(ln x + 1) dx to be within 0.05 of the true value?
44)
A)
n 42
B)
n 5
C)
n 90
D)
n 28
45)
Given that 7
2x dx =45
2,5
2x2 dx =117
3,7
2x2 dx =335
3,find the definite integral
7
2(4x2– 2x) dx.
45)
A)
– 6
B)
1340
3
C)
2725
2
D)
1205
3
Find the integral.
46)
(–x8+4)6x7 dx
46)
A)
–1
56(–x8+4)7+ C
B)
–1
7(–x8+ 4)7+1
8x8+ C
C)
–8
7(–x8+ 4)7+ C
D)
–56(–x8+ 4)7+ C
47)
e7x +4 dx
47)
A)
7e7x + 4 + C
B)
(7x + 4) e7x + 4 + C
C)
e7x + C
D)
1
7e7x + 4 + C
Identify the rectangles shown in the graph as left rectangles, right rectangles, or neither.
48)
48)
A)
right rectangles
B)
neither
C)
left rectangles
Provide an appropriate response.
49)
Graph the following example of unlimited growth: y =450e0.19t, 0 t 12, 0 y 4500.
49)
A)
B)
20
C)
D)
Solve the problem.
50)
The marginal revenue from the sale of compact discs is given by R'(x) = 190 – 8x and R(0) = 0,
where R(x) is the revenue in dollars. Find the price–demand equation.
50)
A)
p = 190x – 4x2
B)
p = 190x – 8x2
C)
p = 190x – 8
D)
p = 190 – 4x
51)
A computer manufacturer has found that its expenditure rate per day (in hundreds of dollars) on a
certain type of job is given by C'(x) = 10x + 6, where x is the number of days since the start of the
job. Find the expenditure if the job takes 8 days.
51)
A)
$8600
B)
$36,800
C)
$368
D)
$86
52)
Find the cost function if the marginal cost function is C'(x) =10x –7 and the fixed cost is $4.
52)
A)
C(x) =5x2–7x +4
B)
C(x) =10x2–7x +3
C)
C(x) =10x2–7x +4
D)
C(x) =5x2–7x +3
Find the integral.
53)
5ex–1
x dx
53)
A)
5ex–2
x2+ C
B)
5ex–1
2x2+ C
C)
5xex– ln x+ C
D)
5ex– ln x+ C
Provide an appropriate response.
54)
Find the average value of the function g(x) =32e0.04x over the interval [10, 30]. Round your
answer to two decimal places.
54)
A)
45.71
B)
1462.63
C)
2.29
D)
73.13
Find the integral.
55)
12x3x dx
55)
A)
2
9x9/2 + C
B)
8
3x9/2 + C
C)
24
7x9/2 + C
D)
11
5x9/2 + C
Provide an appropriate response.
56)
Find the average value of the function y = 2x4 over the interval [– 2, 2].
56)
A)
0
B)
16
5
C)
32
5
D)
128
5
Find the integral.
57)
6 +x2
xdx
57)
A)
6 ln x+1
3x3+ C
B)
3
x2+x2+ C
C)
6 ln x+1
2x2+ C
D)
6
x2+x2+ C
Evaluate the integral.
58)
2
0
4x + 1
4x2+ 2x + 2 dx
(Round to three decimal places.)
58)
A)
1.778
B)
1.040
C)
2.398
D)
1.199
59)
0.4
0.1 5e2x dx
(Round to three decimal places.)
59)
A)
0.967
B)
0.425
C)
2.510
D)
5.021
Find the integral.
60)
x3– 7
xdx
60)
A)
1
3x3+ 7 ln x+ C
B)
1
4x4– 7x2+ C
C)
1
3x3–7
x–2+ C
D)
1
3x3– 7 ln x+ C
61)
b
09x8 dx
61)
A)
b9
B)
9b9
C)
b7
D)
1
9b9
Approximate the area under the graph of f(x) and above the x–axis using n rectangles.
62)
f(x) = 3x2– 2 from x = 1 to x = 5; n = 4; compute R4
62)
A)
154
B)
144
C)
140
D)
150
Provide an appropriate response.
63)
Divide the interval [0, 8] into four equal subintervals and draw in the corresponding right
rectangles.
63)
A)
B)
24
C)
D)
Find the integral.
64)
2
t–7et dt
64)
A)
ln t– 7 + C
B)
2– 7et+ C
C)
2 ln t–7et+ C
D)
2 ln t– 7 et+ C
Solve the problem.
65)
A manufacturing company is ready to introduce a new product with a national sales campaign.
After extensive test marketing, the market research department estimates that sales (in millions of
dollars) will increase at the monthly rate of S'(t) = 10 – 10e–0.2t for 0 t 24, t months after the
national campaign has started. What will the total sales be five months after the beginning of the
campaign if we assume zero sales at the beginning of the campaign? (Round the answer to the
nearest million.)
65)
A)
$49 million
B)
$2 million
C)
$1 million
D)
$18 million
Approximate the area under the graph of f(x) and above the x–axis using n rectangles.
66)
f(x) = 2x + 3 from x = 0 to x = 2; n = 4; compute R4
66)
A)
13
B)
15
C)
11
D)
17
Provide an appropriate response.
67)
Calculate the Riemann sum, Sn , for the function f(x) =x2– 3x – 10 on the interval [–3, 7]. Partition
[–3, 7] into five subintervals of equal length and for each subinterval [xk–1, xk], let Ck be the
midpoint.
67)
A)
–38
B)
– 40
C)
38
D)
40
Evaluate the integral.
68)
1
–1(3x2– 8x) dx
68)
A)
7
B)
2
C)
–7
D)
12
Identify the rectangles shown in the graph as left rectangles, right rectangles, or neither.
69)
69)
A)
neither
B)
right rectangles
C)
left rectangles
Solve the problem.
70)
The marginal price for a weekly demand of x bottles of cough medicine in a drug store is given by
p'(x) =–13,300
(5x +40)2. Find the price–demand equation if the weekly demand is 125 when the price of a
bottle of cough medicine is $4. What is the weekly demand (to the nearest bottle) when the price is
$3?
70)
A)
p(x) =5,320
5x + 40; 347 bottles
B)
p(x) =5,320
5x + 40 – 4; 144 bottles
C)
p(x) =2,660
5x + 40; 169 bottles
D)
p(x) = – 2,660
5x + 40 + 8; 98 bottles
Identify the rectangles shown in the graph as left rectangles, right rectangles, or neither.
71)
71)
A)
neither
B)
right rectangles
C)
left rectangles
Solve the problem.
72)
A drug is injected into the bloodstream of a patient through her right arm. The concentration of the
drug, C(t) (in milligrams per cubic centimeter), in the blood stream of the left arm t hours after the
injection is given by C(t) =0.15t
t2+ 1. What is the average concentration of the drug in the bloodstream
of the left arm during the first two hours after the injection?
72)
A)
0.344 milligrams per cubic centimeter
B)
0.241 milligrams per cubic centimeter
C)
0.060 milligrams per cubic centimeter
D)
0.121 milligrams per cubic centimeter
Evaluate the integral.
73)
4
05 x dx
73)
A)
60
B)
40
C)
10
D)
80
3
Solve the problem.
74)
At the beginning of an advertising campaign for a new product in a city of 500,000 people, no one is
aware of the product. After 10 days, 100,000 people are aware of the product. If N = N(t) is the
number of people (in thousands) who are aware of the product t days after the beginning of the
advertising campaign, solve the following differential equation for N(t):
dN
dt = k(500 – N); N(0) = 0; N(10) = 100.
74)
A)
N(t) = 500(e–0.022t – 1)
B)
N(t) = 500(1 –e–0.022t)
C)
N(t) = 500(1 –e0.022t)
D)
N(t) = 500(e0.022t – 1)
Evaluate the integral.
75)
8
2
x2–16
x –4dx
75)
A)
6
B)
54
C)
84
D)
42
Solve the problem.
76)
The number of cheeseburgers (in thousands) sold each day by a chain of restaurants t days after the
end of an advertising campaign is given by S(t) = 9 – 10e–0.3t. What is the average number of
cheeseburgers sold each day during the first 7 days after the end of the advertising campaign?
76)
A)
4770 cheeseburgers
B)
5904 cheeseburgers
C)
4821 cheeseburgers
D)
3740 cheeseburgers
Provide an appropriate response.
77)
Given 5
3f(x) dx = 7 and 5
3g(x) dx = 1, find 5
3[4f(x) – 2g(x)] dx.
77)
A)
26
B)
28
C)
2
D)
30
78)
Graph the following example of exponential decay: y =750e–0.025t , 0 t 45, 0 y 900.
78)
A)
B)
C)
D)
Evaluate the integral.
79)
–1
–3(x2+ x + 3) dx
79)
A)
16.33
B)
11.17
C)
10.5
D)
10.67
Find the integral.
80)
t e– 7t2 dt
80)
A)
1
7e– 7t2+ C
B)
–1
7e– 7t2+ C
C)
1
14 e– 7t2+ C
D)
–1
14 e– 7t2+ C
81)
(3x5–8x3+ 2) dx
81)
A)
1
6x6–1
4x4+ 2x + C
B)
1
2x6– 2x4+ 2x + C
C)
3
5x5–8
3x3+ 2x + C
D)
3
4x4– 4x2+ C
Solve the problem.
82)
A newspaper is launching a new advertising campaign in order to increase the number of daily
subscribers. The newspaper currently (t = 0) has 26,000 daily subscribers and management expects
that number, S(t), to grow at the rate of S'(t) = 80t1/2 subscribers per day, where t is the number of
days since the campaign began. How long (to the nearest day) should the campaign last if the
newspaper wants the number of daily subscribers to grow to 49,000?
82)
A)
57 days
B)
33 days
C)
69 days
D)
44 days
Find the integral.
83)
x3x4+3dx
83)
A)
–1
2(x4+3)–1/2 + C
B)
2
3(x4+3)3/2 + C
C)
8
3(x4+3)3/2 + C
D)
1
6(x4+3)3/2 + C
84)
13x-7 dx
84)
A)
–91x–8+ C
B)
13
6x 8+ C
C)
–13
6x–6+ C
D)
78
x6+ C
Approximate the area under the graph of f(x) and above the x–axis using n rectangles.
85)
f(x) =x2+ 2; interval [0, 5]; n = 5; compute L5
85)
A)
66
B)
32
C)
40
D)
65
Provide an appropriate response.
86)
Find the average value of the function y = 5 –x2 over the interval [– 3, 2].
86)
A)
–2
3
B)
94
15
C)
8
3
D)
4
15
Find the integral.
87)
(3x8– 7x3+ 7) dx
87)
A)
9x9–7
4x4+ 7x + C
B)
9x9–7
3x4+ 7x + C
C)
1
3x9–7
3x4+ 7x + C
D)
1
3x9–7
4x4+ 7x + C
Solve the problem.
88)
Find the amount A in an account (to the nearest dollar) after 5 years if dA
dt = rA, A(0) = 800, and
A(10) = 1800.
88)
A)
$1100
B)
$919
C)
$1000
D)
$1200
Find the integral.
89)
1
3 – 2x dx
89)
A)
2 ln 3 – 2x + C
B)
– 2 ln 3 – 2x + C
C)
–1
2 ln 3 – 2x + C
D)
1
2 ln 3 – 2x + C
90)
5(t2– 5t – 2) dt
90)
A)
5
3t3–5
2t2– 2t + C
B)
10t – 5 + C
C)
5t3– 5t2– 2t + C
D)
5
2t3– 5t2– 2t + C
Evaluate the integral.
91)
3
1(2x3– 4x–2) dx
91)
A)
45.83
B)
37.33
C)
56
D)
48
Find the integral.
92)
9x– 5 dx.
92)
A)
36
x4+ C
B)
– 45x– 6 + C
C)
9
4x6+ C
D)
–9
4x– 4 + C
Provide an appropriate response.
93)
Find the general solution for the differential equation y’ =7e3x
93)
A)
1
3 e3x + C
B)
7e3x + C
C)
7
3 e3x + C
D)
21e3x + C
Answer Key
Testname: C5
Answer Key
Testname: C5