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 =x
y
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 =x2–y2
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 =5 to x =9.
(B) Graph F(x) and use geometric formulas to calculate the area between the graph of F(x)
and the x–axis from x =5 to x =9.
(C) What guarantees that your answers to (A) and (B) are equal?
F(x) = x( 3
2x +4)
5)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Evaluate the integral.
6)
2
1
4 x –5
xdx
(Round to three decimal places.)
6)
A)
3.001
B)
12.846
C)
7.505
D)
1.410
Find the integral.
7)
t5
31+t6
dt
7)
A)
1
12(1 +t6)2+ C
B)
1
4(1 +t6)2/3 + C
C)
1
8(1 +t6)2/3 + C
D)
1
4t6(1 +t6)2/3 + C
Evaluate the integral.
8)
8
1
x2–9
x –3dx
8)
A)
84
B)
21
2
C)
42
D)
105
2
Find the integral.
9)
5ex–1
x dx
9)
A)
5ex– ln x+ C
B)
5ex–1
2x2+ C
C)
5xex– ln x+ C
D)
5ex–2
x2+ C
Solve the problem.
10)
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.)
10)
A)
$18 million
B)
$2 million
C)
$49 million
D)
$1 million
Approximate the area under the graph of f(x) and above the x–axis using n rectangles.
11)
f(x) = 3x2– 2 from x = 1 to x = 5; n = 4; compute R4
11)
A)
150
B)
154
C)
144
D)
140
Solve the problem.
12)
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.
12)
A)
p(x) = 80e–0.058x
B)
p(x) = 4.38e0.305x
C)
p(x) = 80e1.037x
D)
p(x) = 4.38e–0.058x
Evaluate the integral.
13)
1
–1
(3x2– 8x) dx
13)
A)
2
B)
–7
C)
7
D)
12
Find the integral.
14)
11x-6 dx
14)
A)
–11
5x–5+ C
B)
11
5x
7
+ C
C)
55
x
5
+ C
D)
–66x–7+ C
Solve the problem.
15)
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.
15)
A)
$36,800
B)
$368
C)
$86
D)
$8600
16)
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.
16)
A)
p = 190x – 8x2
B)
p = 190x – 8
C)
p = 190 – 4x
D)
p = 190x – 4x2
Provide an appropriate response.
17)
Find the general solution for the differential equation y’=15x2
17)
A)
15x3+ C
B)
5x3+ C
C)
x3
3+ C
D)
x3+ C
18)
Given
5
3
f(x) dx = 7 and
5
3
g(x) dx = 1, find
5
3
[4f(x) – 2g(x)] dx.
18)
A)
30
B)
26
C)
2
D)
28
Find the integral.
19)
t e– 7t2 dt
19)
A)
1
14 e– 7t2+ C
B)
–1
14 e– 7t2+ C
C)
–1
7e– 7t2+ C
D)
1
7e– 7t2+ C
Approximate the area under the graph of f(x) and above the x–axis using n rectangles.
20)
f(x) = 2x + 3 from x = 0 to x = 2; n = 4; compute R4
20)
A)
13
B)
15
C)
17
D)
11
Evaluate the integral.
21)
9
0
2 x dx
21)
A)
36
B)
9
C)
54
D)
81
Find the integral.
22)
3 dx
22)
A)
3 + C
B)
3x + C
C)
3
2x2+ C
D)
0
Provide an appropriate response.
23)
Graph the following example of exponential decay: y =200e–0.036t , 0 t 45, 0 y 900.
23)
A)
B)
12
C)
D)
Identify the rectangles shown in the graph as left rectangles, right rectangles, or neither.
24)
24)
A)
right rectangles
B)
neither
C)
left rectangles
Solve the problem.
25)
A company finds that consumer demand quantity changes with respect to price at a rate given by
D'(p) = – 2500
p2. Find the demand function if the company knows that 828 units of the product are
demanded when the price is $5 per unit.
25)
A)
D(p) =5000
p+828
B)
D(p) =2500
p3+328
C)
D(p) =2500
p+328
D)
D(p) =2500
p+828
Find the integral.
26)
x2x3+10 dx
26)
A)
2
9(x3+10)3/2 + C
B)
2(x3+10)3/2 + C
C)
–
2
3(x3+10)–1/2 + C
D)
2
3(x3+10)3/2 + C
27)
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.
27)
A)
38
B)
40
C)
– 40
D)
–38
Solve the problem.
28)
Find the cost function if the marginal cost function is C'(x) =20x –10 and the fixed cost is $8.
28)
A)
C(x) =10x2–10x +7
B)
C(x) =20x2–10x +8
C)
C(x) =10x2–10x +8
D)
C(x) =20x2–10x +7
29)
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.
29)
A)
C(x) =ln (x + e)
9+ 11
B)
C(x) = 9 ln (x + e) + 11
C)
C(x) =ln (x + e)
9+ 20
D)
C(x) = 9 ln (x + e) + 20
30)
(3x8– 7x3+ 7) dx
30)
A)
9x9–7
4x4+ 7x + C
B)
1
3x9–7
3x4+ 7x + C
C)
9x9–7
3x4+ 7x + C
D)
1
3x9–7
4x4+ 7x + C
31)
(3x5–8x3+ 2) dx
31)
A)
3
4x4– 4x2+ C
B)
1
6x6–1
4x4+ 2x + C
C)
3
5x5–8
3x3+ 2x + C
D)
1
2x6– 2x4+ 2x + C
Provide an appropriate response.
15
32)
Graph the following example of unlimited growth: y =500e0.18t, 0 t 12, 0 y 4500.
32)
A)
B)
C)
D)
Solve the problem.
33)
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.)
33)
A)
32.67 tons
B)
75.71 tons
C)
50.67 tons
D)
33.65 tons
34)
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?
34)
A)
0.060 milligrams per cubic centimeter
B)
0.344 milligrams per cubic centimeter
C)
0.121 milligrams per cubic centimeter
D)
0.241 milligrams per cubic centimeter
Provide an appropriate response.
35)
Given that
7
2
x dx =45
2,
5
2
x2 dx =117
3,
7
2
x2 dx =335
3,find the definite integral
7
2
(4x2– 2x) dx.
35)
A)
1205
3
B)
2725
2
C)
– 6
D)
1340
3
36)
0.4
0.1
5e2x dx
(Round to three decimal places.)
36)
A)
0.425
B)
5.021
C)
0.967
D)
2.510
Solve the problem.
37)
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).
37)
A)
v(t) = 35t2+ 16
B)
v(t) = 8t2+ 35t
C)
v(t) = 16t2+ 35
D)
v(t) = 8t2+ 35
38)
Find the average value of the function y = 5 –x2 over the interval [– 3, 2].
38)
A)
8
3
B)
94
15
C)
–2
3
D)
4
15
Evaluate the integral.
39)
–1
–3
(x2+ x + 5) dx
39)
A)
24.33
B)
14.67
C)
14.5
D)
15.17
18
Evaluate the integral.
Find the integral.
40)
–3
3x
– 4 x dx
40)
A)
x2/3 – 2x3/2 + C
B)
–2x2/3 – 6x3/2 + C
C)
–9
2x2/3 –8
3x3/2 + C
D)
–3 ln x1/3 – 2x3/2 + C
41)
(–x8+4)6x7 dx
41)
A)
–8
7(–x8+ 4)7+ C
B)
–56(–x8+ 4)7+ C
C)
–1
7(–x8+ 4)7+1
8x8+ C
D)
–1
56 (–x8+4)7+ C
42)
e7x +4 dx
42)
A)
(7x + 4) e7x + 4 + C
B)
7e7x + 4 + C
C)
e7x + C
D)
1
7e7x + 4 + C
43)
12x3x dx
43)
A)
11
5x9/2 + C
B)
24
7x9/2 + C
C)
2
9x9/2 + C
D)
8
3x9/2 + C