Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1)
Draw the region whose area is given by the definite integral.
9
5
3xdx
1)
A)
B)
C)
D)
1
2)
Draw the region whose area is given by the definite integral.
3
2
3
x3dx
2)
A)
B)
C)
D)
2
3)
Suppose the $2500 is deposited in a savings account paying 5% interest, compounded continuously.
What will be the average value of the account during the next 10 years?
3)
A)
2500(e50 – 1)
B)
50,000(e0.5 – 1)
C)
50,000(e0.05 – 1)
D)
250(e5– 1)
E)
none of these
4)
What is the area under the curve y =1
3+ 3e3x between x = – 1
3 and x =1
3?
4)
A)
e +1
e
B)
9e2+ 2e + 9
2e
C)
2
9
D)
1
e– e
E)
none of these
5)
b
a
(f(x) – g(x)) dx expresses the area between the curves y = f(x) and y = g(x) from x = a to x = b
only if:
5)
A)
neither f(x) and g(x) cross the x–axis.
B)
f(x) is greater than or equal to g(x) for all x between a and b .
C)
f(x) and g(x) do not cross each other between x = a and x = b .
D)
f(x) is greater than or equal to g(x) for all x between a and b , and neither f(x) nor g(x) crosses
the x–axis .
E)
none of these
6)
Draw the region whose area is given by the definite integral.
4
1
x2dx
6)
A)
B)
4
C)
D)
7)
A certain commodity has demand curve p =20
x + 5 – 1 at sales level x. What is the consumers’
surplus if 5 units are currently being sold?
7)
A)
20(ln 15 + ln 5) – 10
B)
20(ln 15 – ln 5) – 10
C)
20(ln 10) – 10
D)
1
20 (ln 3) – 20
E)
none of these
Solve the problem.
8)
Suppose that the marginal revenue for a retailer is 6x2–x+ x dollars at sales level x. If 4 units are
currently being sold, what is the extra revenue received from the sale of 5 additional units?
8)
A)
568 +42
17
$570.47
B)
1304 +259
6
$1347.17
C)
$152
D)
11
6
$1.83
E)
none of these
9)
Draw the region whose area is given by the definite integral.
5
4
1
xdx
9)
A)
B)
6
C)
D)
Solve the problem.
10)
If A(t) denotes the annual rate of world consumption of oil at time t (with t = 0 corresponding to
1977), which of the following expressions represents the amount of oil consumed between 1977 and
1987?
10)
A)
1987
1977
A(t) dt
B)
10
0
A'(t) dt
C)
10
0
A(t) dt
D)
A'(10)
E)
none of these
C
D
11)
A helicopter rises straight up in the air so that its velocity t seconds after take–off is
v(t) =t3/2 +1
2t1/2 + 1 feet per second. If the landing pad is 100 feet above the ground, which of the
following gives the height of the helicopter at time t ?
11)
A)
h(t) =3
2t1/2 +1
4t–1/2 + 100
B)
h(t) =2
3t5/2 +1
4t3/2 + t + C
C)
h(t) =2
5t5/2 +1
3t3/2 + t + 100
D)
h(t) =5
3t5/2 +3
4t3/2 + t – 100
E)
none of these
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
12)
Find the area of the region bounded by y = x and y =x3.
Enter a reduced fraction a
b.
12)
Calculate.
13)
3
1
(x3+ 3)x2 dx
Enter a reduced fraction of form a
b.
13)
14)
1
0
(e3x – 1) dx
Enter your answer as a(eb±ec). Any fractions in reduced form a
b.
14)
8
Find all antiderivatives of the function.
15)
f(y) =y4
Enter your answer as a polynomial in y in standard form.
15)
16)
Use the graph below to determine the area of the shaded region.
Enter your answer exactly in the form a ±eb.
16)
Calculate.
17)
4
1
x dx
Enter just a reduced fraction of form a
b.
17)
Find all antiderivatives of the function.
18)
f(x) =x3
Enter your answer as a polynomial in x in standard form.
18)
19)
Find the area of the region bounded by f(x) = {x2(0 x 1); 2x –x2(1 x 2)} and the line
y = 0.
Enter just an integer.
19)
20)
Find the volume of the solid of revolution generated by revolving the region formed by the
graphs of y =x2,y = 2, and x = 0 about the x–axis.
Enter a reduced quotient ab c
d.
20)
Calculate.
21)
5
2
e4x –1
x dx
Enter your answer as a(eb–ec) + d.
21)
22)
Find the area of the region bounded by y =e–2x, the x–axis, and the lines x = 1 and x = 3.
Enter your answer in the form a(eb–ec).
22)
23)
Determine the average value of f(x) =e3x over the interval from x = 1 to x = 3.
Enter your answer in the form a(eb±ec)
23)
Solve the problem.
24)
Suppose that at time t (0 t 2 , t in months), the sales of a certain commodity are
decreasing at a rate of 1000e–0.05t units per month. Calculate the total change in sales from
t = 0 to t = 2.
Enter your answer as just a(1 –eb) where b is a real number to one decimal place.
24)
25)
Suppose that money is deposited in a savings account at a steady rate of $150 per month.
If the account pays 2.5% interest compounded continuously, how much will be in the
account at the end of 3 years? Enter your answer in the form a(eb± c) where b is a real
number to three decimal places (no units).
25)
26)
A region is bounded above by the graph of y =x–x3 and below by the x–axis on the
interval from x = 0 to x = 1. Find the volume of the solid of revolution generated by
revolving the region about the x–axis. Enter your answer as a reduced quotient of form
ab
c.
26)
27)
Determine the average value of f(x) =x3over the interval from x = 0 to x = 4.
Enter an integer.
27)
28)
A region is bounded by the graph of y =x3, the y–axis, and the horizontal line y = 1. Find
the volume of the solid of revolution generated by revolving the region about the x–axis.
Enter your answer as a reduced quotient in form ab
c.
28)
29)
Find: (2x4+ 3x – 4) dx
Enter a polynomial in x in standard form with any fractional coefficients or powers in
reduced form a
b.
29)
30)
Use a Riemann sum to approximate the area under the graph of f(x) =x3, 1 x 3, n = 4.
Use the midpoints of the intervals.
Enter your answer as just a reduced fraction of form a
b.
30)
11
31)
Find: 4x3/2 –1
2x3/2 dx
Integrate the terms in the order they appear.
Enter your answer as a sum of power functions in standard form with any fractional
powers or coefficients reduced of form a
b.
31)
32)
Find: (3x + 2)2 dx
Enter your answer as a polynomial in x in standard form with any fractional coefficients or
powers reduced of form a
b.
32)
33)
A ball is thrown upward with initial velocity of 144 feet per second. How high will the ball
go? (Recall that from physics, it is known that the velocity at time t is 144 – 32t feet per
second.)
Enter just an integer (no units).
33)
34)
Find the area of the region bounded by the curve f(x) = 5x – 2x2 and the line y = 3.
Enter just a reduced fraction of form a
b.
34)
35)
A region is bounded above by the graph of y =x–2 and below by the x–axis on the interval
from x = 1 to x = 3. Find the volume of the solid of revolution generated by revolving the
region about the x–axis.
Enter your answer as a reduced quotient of form ab
c.
35)
36)
Use a Riemann sum to approximate the area under the graph of f(x) on the given interval.
Use the right endpoints.
Enter just an integer.
f(x) = 2x + 1; 1 x 5, n = 4
36)
37)
Find the area of the region bounded by the curve f(x) =x2 and the line y = x.
Enter just a reduced fraction of form a
b.
37)
Calculate.
38)
10
2
1
x – 1 dx
Enter just a real number (no approximations).
38)
39)
Use a Riemann sum to approximate the area under the graph of f(x) on the given interval.
Use the right endpoints.
Enter just a real number to two decimal places.
f(x) =x3;0 x 2 , n = 4
39)
40)
Determine the average value of f(x) =x3+ 3x2+ 3x + 1 over the interval from x= – 1 to x =
3.
Enter an integer.
40)
41)
Find: dx
x1/5
Enter your answer as a power function in x in standard form with any fractional
coefficients or powers in reduced form a
b and any constant at the right end.
41)
Calculate.
42)
1
0
(2x4+ 5x + 1) dx
Enter a reduced fraction of form a
b.
42)
13
43)
Find the area of the region bounded by the curves y =x2+8
5x + 1 and y =2
5x + 1.
Enter just a reduced fraction of form a
b.
43)
44)
Determine the average value of f(x) =1
x2over the interval from x = 1 to x = 100.
Enter a reduced fraction of form a
b.
44)
45)
Given f(x) = ln x on the interval 1 x 5 and with n = 2, compute the Riemann sum (a)
using the left endpoints; (b) using the right endpoints; and (c) using the midpoints of the
subintervals. Enter your answer as just a, b, c all real numbers rounded to two decimal
places and separated by commas.
Enter the numbers in the order that answers (a), (b), (c) but do not label.
45)
46)
Find the area of the region bounded by the curve y = – x2+ 3 and the line y = 2x.
Enter just a reduced fraction of form a
b.
46)
47)
Find: (3x1/3) dx .
Enter using standard power function form axb, with any fractions reduced of form a
b.
47)
48)
Find the area under the curve y =1
ex+1
x – 2 between x = 3 and x = 5.
Enter your answer in the form ea–eb+ c (no approximations).
48)
49)
This is a sketch of the region between the two curves y =x2– 15, below the x–axis and also
below the line y = – 2x. Does the following represent the area of the region?
0
–15
(x2– 15) dx +
3
0
(–x2– 2x + 15) dx
Enter just “yes” or “no”.
49)
50)
Find: 2e–2x dx
Enter your answer in standard form (no fractions).
50)
51)
This is a sketch of the region between the two curves f(x) =x2, g(x) =x3. Compute the
area.
Enter your answer as just a reduced fraction of form a
b.
51)
15
52)
Find: x – 7
xdx
Enter terms in the same order in which they appear in the integral.
52)
53)
Find: x+1
x dx
Enter your answer using standard power function form (axb), leaving the terms in the
order in which they appear in the integral.
53)
Calculate.
54)
5
3
e5x dx
Enter your answer as a(eb–ec).
54)
55)
Find the area under the curve y =1
x– 2x from x = – 3 to x = – 2.
Enter a ± ln b using reduced fractions of form a
b and integers.
55)
56)
Find the area of the region bounded by y = 4x –x2 and the x–axis.
Enter a reduced fraction of form a
b.
56)
57)
Suppose that a 1000–L water tank takes 20 min to drain and that after t minutes, the
amount of water remaining in the tank is V(t) =5
2(20 –t2) liters. What is the average
amount of water in the tank during the time it drains?
Enter a reduced fraction of form a
b (no units)
57)
16
58)
Find all functions f(x) with the following property: f'(x) = 3x2+ 2x + 1.
Enter your answer as a polynomial in x in standard form with any fractional coefficients or
powers reduced of form a
b.
58)
59)
Find the area bounded by y =1
x, the x–axis, and the lines x = 1 and x = a > 1.
Enter a real number (no approximations).
59)
60)
Find: (2x + 1)2 dx
Enter your answer as a polynomial in x in standard form with any fractional coefficients or
powers reduced of form a
b.
60)
Calculate.
61)
2
0
(3e4 – 2x) dx
Enter your answer as a(b +ec) with any fractions in reduced form e
f.
61)
62)
Find: (x3+ 1) dx
Enter your answer as a polynomial in x in standard form with any fractional coefficients or
powers reduced of form a
b.
62)
63)
A rock is dropped from a balloon hovering at 4800 ft above the ground. Its velocity at time
t seconds is v(t) = – 32t feet per second. Find how long it takes for the rock to reach the
ground.
Enter just a real number to one decimal place (no units).
63)
Find all antiderivatives of the function.
64)
f(x) =e–x/2
Enter your answer with any fractional coefficients and powers in reduced form a
b.
64)
65)
Find the area of the region bounded by y =1 – 2x –x2 and the lines x = – 1 and x = 0.
Enter a reduced fraction of form a
b.
65)
66)
Determine the area under the curve y =e4x from x = 0 to x = 1.
Enter a(eb± c).
66)
67)
Find: dx
e3x
Enter your answer in standard form (use aeb).
67)
68)
Find: x2–1
4x dx
Enter your terms in standard forms in the order in which they appear in the integral.
68)
69)
For the Riemann sum, [5 1.4 + 5 1.8 + 5 2.2 + 5 2.6 + 5 3](0.4); a = 1, determine n, b,
and f(x).
Enter your answer as just n, b, f(x) (2 integers in that order separated by commas and
followed by a power function in x).
69)
18
70)
Determine the average value of f(x) =1
x2over the interval from x = 1 to x = 2.
Enter a reduced fraction of form a
b.
70)
71)
This is a sketch of the region between the two curves y =x and y = 2 x – 1 and the
x–axis. Does the following represent the area of the region?
1
0
x dx +
4/3
1
x– 2 x – 1 dx
Enter just “yes” or “no”.
71)
Solve the problem.
72)
Suppose that at time t, a bacteria culture is increasing at the rate of 500e0.1t bacteria per
hour. Calculate the total increase in the number of bacteria from t = 0 to t = 1.
Enter your answer in the form a(eb– 1) where b is a real number to one decimal place.
72)
73)
Find the area of the region bounded by y = 6x –x2 and y =x2– 2x.
Enter a reduced fraction of form a
b.
73)
19
74)
This is a sketch of the region between the two curves y =x3– 3x + 1 and y =x2– x + 1.
Does the following represent the area of the region?
0
–1
(x3–x2– 2x) dx +
2
0
(–x3+x2+ 2x) dx
Enter just “yes” or “no.
74)
Calculate.
75)
4
1
3 x dx
Enter just an integer.
75)
Find all antiderivatives of the function.
76)
f(x) =x5
Enter your answer as a polynomial in x in standard form.
76)
20