Provide the proper response.
74)
If F’(x) = f(x), then
b
a
f(x) dx =
i) F(a) – F(b).
ii) F(b) – F(a).
iii) F(b) + F(a).
74)
A)
Only i is correct.
B)
Only iii is correct.
C)
Either i or ii could be correct.
D)
Only ii is correct.
Solve the problem.
75)
The rate of growth of the profit (in millions) from an invention is approximated by P'(x) = xe–x2,
where x represents time measured in years. The total profit in year 1 that the invention is in
operation is $25,000. Find the total profit function. Round to three decimal places where
appropriate.
75)
A)
P(x) = – 0.5e–x2– 0.209
B)
P(x) = – 0.5e–x2+ 0.209
C)
P(x) = – 0.5e–x2– 209,000
D)
P(x) = – 0.5e–x2+ 209,000
Solve the problem. Round your answer, if appropriate.
76)
The growth rate of a certain tree (in feet) is given by
y =2
t + 1 +e–t2/2 ,
where t is time in years. Estimate the total growth of the tree through the end of the second year by
using the trapezoidal rule with n = 2.
76)
A)
3.51 feet
B)
5.41 feet
C)
2.70 feet
D)
1.75 feet
Find the integral.
77)
(8x2+x–3) dx
77)
A)
8x3
3+x–2
2+ C
B)
–8x3
3+x–2
2+ C
C)
8x3
3–x–2
2+ C
D)
–8x3
3–x–2
2+ C
Find the exact value of the integral using a formula from geometry.
78)
5
–5
25 –x2 dx
78)
A)
50
B)
25
C)
25
4
D)
25
2
Use your calculator to approximate the integral using the method indicated, with n = 100. Round your answer to four
decimal places.
79)
4
0
2+1
x + 3 dx (trapezoidal rule)
79)
A)
8.8969
B)
8.9265
C)
8.8473
D)
8.9373
22
Solve the problem.
80)
A piece of tissue paper is picked up in gusty wind. The table below shows the velocity of the paper
at 2–second intervals. Estimate the distance the paper traveled using right endpoints.
Time
(sec)
Velocity
(ft/sec)
0
2
4
6
8
10
12
14
16
0
8
12
6
22
27
17
10
2
80)
A)
185 ft
B)
188 ft
C)
104 ft
D)
208 ft
Evaluate the definite integral.
81)
1
0
et
(6 +et)2
81)
A)
1
6+ e –1
7
B)
1
6+ e –1
6
C)
1
6–1
6+ e
D)
1
7–1
6+ e
Use n = 4 to approximate the value of the integral by the trapezoidal rule.
82)
2
0
8x2 dx
82)
A)
22
B)
30
C)
64
3
D)
44
Solve the problem.
83)
In town A, the birth rate is given by b'(t) =59e0.40t (births per year), where t is the number of years
since 1990. In town B, the birth rate is given by B'(t) =80e0.46t (births per year), where t is the
number of years since 1990. How many more births are there in town B than in town A during the
1990s (from t = 0 to t = 10)?
83)
A)
9248 births
B)
9222 births
C)
17,128 births
D)
4716 births
Find the area of the shaded region.
84)
y =1
x
84)
A)
ln 4
B)
ln 5.5
C)
ln 5
D)
ln 4.5
Approximate the area under the graph of f(x) and above the x–axis using n rectangles.
85)
f(x) =9– x2 from x = – 3 to x =3; n = 2; use midpoints
85)
A)
13.5
B)
20.25
C)
40.5
D)
6
Use your calculator to approximate the integral using the method indicated, with n = 100. Round your answer to four
decimal places.
86)
2
0
ex dx (trapezoidal rule)
86)
A)
6.3893
B)
7.3891
C)
6.3895
D)
6.2903
Use n = 4 to approximate the value of the integral by the trapezoidal rule.
87)
5
1
11x 2x – 1 dx
87)
A)
297.3
B)
316.3
C)
404.3
D)
28.8
Solve the problem.
88)
The rate at which an assembly line worker’s efficiency E (expressed as a percent) changes with
respect to time t is given by E'(t) =70 –4t, where t is the number of hours since the worker’s shift
began. Assuming that E(1) =88, find E(t).
88)
A)
E(t) =70t –2t2+20
B)
E(t) =70t –4t2+20
C)
E(t) =70t –2t2+156
D)
E(t) =70t –2t2+88
Find the area between the curves.
89)
y = x2– 5x + 4, y = – (x – 1)2
89)
A)
8
9
B)
7
8
C)
9
8
D)
8
7
25
Find the integral.
90)
4(2x + 5)3 dx
90)
A)
3
8(2x + 5)4+ C
B)
1
2(2x + 5)4+ C
C)
3
4(2x + 5)4+ C
D)
1
4(2x + 5)4+ C
91)
x
(7x2+ 3)5 dx
91)
A)
–7
3(7x2+ 3)4+ C
B)
–1
14(7x2+ 3)6+ C
C)
–7
3(7x2+ 3)6+ C
D)
–1
56(7x2+ 3)4+ C
Approximate the area under the graph of f(x) and above the x–axis using n rectangles.
92)
f(x) = 3x2– 2 from x = 1 to x = 5; n = 4; use right endpoints
92)
A)
140
B)
154
C)
150
D)
144
Use your calculator to approximate the integral using the method indicated, with n = 100. Round your answer to four
decimal places.
93)
3
1
x ln x dx (trapezoidal rule)
93)
A)
2.8734
B)
2.8899
C)
2.9436
D)
3.0359
Find the integral.
94)
3 x – 5
x2 dx
94)
A)
– 6
x
–5
x+ C
B)
6
x
–5
x+ C
C)
–6
x
+5
x+ C
D)
6
x
+5
x+ C
Solve the problem.
95)
Suppose the supply function of a certain item is given by S(q) = 2q + 7, and the demand function is
given by D(q) = 27 – q2/3. Find the producers’ surplus. (Hint: The equilibrium quantity q0 is a
perfect cube.)
95)
A)
32
B)
64
C)
32
5
D)
64
5
Evaluate the definite integral.
96)
4
1
x1/2 dx
96)
A)
14
5
B)
17
3
C)
17
5
D)
14
3
97)
1
0
x +9dx
97)
A)
20
310 –18
B)
10 10 –27
C)
20
310
D)
15 10 –15
27
Find the integral.
98)
t4
48+t5
dt
98)
A)
1
15(8 +t5)3+ C
B)
4
15 t5(8 +t5)3/4 + C
C)
4
25 (8 +t5)3/4 + C
D)
4
15 (8 +t5)3/4 + C
99)
(log2 (2x + 4))3
2x + 4
99)
A)
(log2 (2x + 4))4
8
B)
(ln 2)(log2 (2x + 4))4
4
C)
(ln 2)(log2 (2x + 4))4
8
D)
(log2 (2x + 4))4
8 ln 2
Find the area of the shaded region.
100)
y = 4 –x2
100)
A)
23
3
B)
3
C)
5
D)
5
3
Solve the problem.
101)
Suppose that a velocity function is given by v(t) =10t3. Find the position function s(t) if s(0) =6.
101)
A)
s(t) =30t2+6
B)
s(t) =5
2t4
C)
s(t) =10t4+6
D)
s(t) =5
2t4+6
Provide the proper response.
102)
Which integral or integrals have a value of zero?
i)
b
b
r(t) dt ii)
b
–b
t3 dt
iii)
b
0
t3 dt , where b > 0 iv)
10
b
t3 dt , where b < 0
102)
A)
Only i
B)
Both i and iv
C)
Both i and ii
D)
All of these
Find the integral.
103)
(3x + 5x–1) dx
103)
A)
3x3+ 30x –25
3x–1+C
B)
3x3+ 15x –25
3x–1+C
C)
9
4 x4+ 25 ln x2+ C
D)
3
2x2+ 5 ln x+ C
104)
5
x+2ex dx
104)
A)
5 ln x+ 2xex–1+ C
B)
10
x2+ 2ex+C
C)
10
x2+ 2xex–1+C
D)
5 ln x+ 2ex+ C
Find the area between the curves.
105)
x = 0, x = 1, y = x2+ 6, y = x2+ 2
105)
A)
4
B)
12
C)
16
D)
8
Find the integral.
106)
x2+16x
(x +8)2 dx
106)
A)
x +8
x +8+ C
B)
x +128
(x +8)3+ C
C)
64
x +8+ C
D)
x +64
x +8+ C
Find the area of the shaded region.
107)
y =(x – 3)2
107)
A)
5
3
B)
4
3
C)
2
3
D)
1
3
Evaluate the definite integral.
108)
3
–1
(x + 5) dx
108)
A)
24
B)
10
C)
15
D)
–24
Provide the proper response.
109)
If r(t) is the rate of change of revenue, then
b
a
r(t) dt is
i) the total revenue up to time b.
ii) the total revenue from time a to time b.
iii) the change in revenue at any time.
109)
A)
Only i is correct.
B)
Only iii is correct.
C)
Only ii is correct.
D)
None of the above is correct.
Solve the problem.
110)
An object is traveling with a velocity (in feet per second) given by
v(t) =3t3– 3t2+ 5t,
where t is time in seconds. Find the object’s average velocity from t = 0 to t =9 seconds.
110)
A)
4394.3 ft/sec
B)
652.5 ft/sec
C)
488.3 ft/sec
D)
221.0 ft/sec
Evaluate the definite integral.
111)
2
0
x(x2+ 1)3 dx
111)
A)
624
B)
156
C)
78
D)
31
2
32
Solve the problem.
112)
The slope of the tangent line of a curve is given by
f'(x) =x2–7x +5.
If the point (0, 7) is on the curve, find an equation of the curve.
112)
A)
f(x) =1
3x3–7
2x2+5x +7
B)
f(x) =1
3x3–7
2x2+5x + 1
C)
f(x) =1
3x3–8x2+5x + 1
D)
f(x) =1
3x3–8x2+5x +7
Evaluate the definite integral.
113)
4
1
3
x(3+ ln x) dx
113)
A)
7.731
B)
1.140
C)
0.277
D)
9.863
Solve the problem.
114)
Find the cost function if the marginal cost function is C'(x) =10x –5 and the fixed cost is $2.
114)
A)
C(x) =5x2–5x +2
B)
C(x) =5x2–5x +1
C)
C(x) =10x2–5x +2
D)
C(x) =10x2–5x +1
Evaluate the definite integral.
115)
1
–1
(x2+ 1) dx
115)
A)
8
3
B)
–2
3
C)
–8
3
D)
0
33
Find the area of the shaded region.
116)
y = 2x + 1
116)
A)
5
B)
12.5
C)
7.5
D)
10
Answer:
D
Explanation:
A)
B)
C)
D)
Use n = 4 to approximate the value of the integral by Simpson’s rule.
117)
1
0
1
1 +x2 dx
117)
A)
5323
6800
B)
8011
5100
C)
5323
3400
D)
8011
10200
Find the integral.
118)
8e4y dy
118)
A)
1
2e4y + C
B)
1
4e4y + C
C)
4e4y + C
D)
2e4y + C
Answer:
D
Explanation:
A)
B)
C)
D)
Answer:
D
Explanation:
A)
B)
C)
D)
119)
6x – 7 dx
119)
A)
1
6(6x – 7)3/2 + C
B)
1
9(6x – 7)3/2 + C
C)
1
2(6x – 7)3/2 + C
D)
1
3(6x – 7)3/2 + C
120)
9z 3z2– 7 dz
120)
A)
(3z2– 7)3/2 + C
B)
1
2z(3z2– 7)3/2 + C
C)
1
2(3z2– 7)3/2 + C
D)
z(3z2– 7)3/2 + C
Solve the problem.
121)
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.
121)
A)
235.03
B)
241.03
C)
470.06
D)
229.03
Evaluate the definite integral.
122)
e
1
19
xdx
122)
A)
0
B)
–19e2
C)
19
D)
–19
Solve the problem.
123)
A company has found that its expenditure rate per day (in hundreds of dollars) on a certain type of
job is given by E'(x) =8x + 3, where x is the number of days since the start of the job. Find the
expenditure if the job takes 8 days.
123)
A)
$28,000
B)
$6700
C)
$280
D)
$67
Use your calculator to approximate the integral using the method indicated, with n = 100. Round your answer to four
decimal places.
124)
3
1
x ln x dx (Simpson’s rule)
124)
A)
2.8829
B)
2.9438
C)
2.9394
D)
3.0250
Solve the problem.
125)
Find C(x) if C'(x) =x and C(9) = 40.
125)
A)
C(x) =2
3x3/2 + 22
B)
C(x) = 2x3/2 + 31
C)
C(x) =2
3x3/2 + 31
D)
C(x) = 2x3/2 + 22
126)
A company determines that its marginal revenue (in dollars per day) is given by MR(t) =50et. The
company’s marginal cost (in dollars per day) is given by MC(t) =120 –0.4t. Find the total profit
from t = 0 to t =6 (the first 6 days).
126)
A)
$19,394
B)
$19,416
C)
$19,409
D)
$19,459