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Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
Math 131 Week-in-Review #10 (Exam 3 Review: Sections 4.2, 4.3, 4.6, 4.8, and 5.1-5.5)
Note: This collection of questions is intended to be a brief overview of the exam material (with emphasis on sections 5.3, 5.4,
and 5.5). When studying, you should also rework your notes, the previous week-in-reviews for this material, as well as your
suggested and online homework.
1. A Martian throws a rock downward from the top of a 60-meter cliff on Mars with an initial speed of 2 m/s. If the
acceleration due to gravity on Mars is 3.8 m/s2 , how long will it take for the rock to hit the surface of the planet?
2. Estimate
Z 10
f (x) dx using a left Riemann sum and the information in the table below.
1
x
f (x)
1 4 8 10 11
5 20 16 8 3
Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
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3. The velocity function (in meters per second) for a particle moving along a line is given by v(t) = t 2 − 3t − 10.
Find the displacement and the distance traveled by the particle during the time interval 1 ≤ t ≤ 6.
4. Use calculus to find the absolute maximum and minimum values of f (x) = 4x3 + 18x2 + 64 on the following intervals
(if they exist):
a) [−1, 4]
b) [−1, 4)
Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
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ln x
is increasing/decreasing and concave upward/downward, as well as where
5. Determine where the function f (x) =
x
any relative extrema or inflection points occur.
Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
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6. Ben needs to enclose three regions as shown below using 2500 feet of fencing. Use calculus to find the dimensions of
each region so that Ben obtains the largest total area possible.
y
x
2x
7. Compute
∑ 3i2 .
4
i=1
x
5
Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
8. Find the derivative of the function y =
Z √x
cos3 t dt.
sin x
9. When a cup of hot chocolate is bought its temperature is 175◦F. The outside temperature is 50◦ F, and the rate at which
the hot chocolate is cooling is given by c(x) = −10e−0.08x degrees per minute, where x is measured in minutes and x ≥ 0.
a) What is the temperature of the hot chocolate after 1 hour?
b) Find the change in the temperature during the first 6 minutes after the hot chocolate is bought, and interpret your
answer.
c) Find the change in the temperature between 5 and 10 minutes after the hot chocolate is bought, and interpret your
answer.
Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
10. Compute the following indefinite integrals.
a)
√
Z 2
t −3 t
b)
Z
16x ln(4x2 + 3)
dx
4x2 + 3
c)
Z
sin(π x) (cos(π x) + e)23 dx
d)
Z
3 sin(x)
dx
5 cos(x)
√
7 4
t
− csc2 t
dt
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Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
11. Evaluate
√
Z A 2
2
t + 4t − t 3
3
12. Evaluate
Z A
2
t3
7
dt, where A is a constant and A > 3.
x4
dx, where A is a constant and A > 2.
x5 + 5
13. From a 24 inch by 6 inch piece of cardboard, square corners are cut out so the sides fold up to form a box without a
top. What should the length and width of each square be to maximize the volume of the box?
Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
14. Let g(x) =
Rx
0
f (t) dt, where f is the function whose graph is shown below. (Source: #20, pg. 384 of Stewart)
a) At what values of x do the local maximum and minimum values of g occur?
b) Where does g attain its absolute maximum value?
c) On what intervals is g concave downward?
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Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
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15. a) If we know f (3) = 4 and f ′ (3) = 0, and we also know that f ′′ is continuous everywhere and f ′′ (3) = −6, then
what (if anything) can we conclude about the behavior of f at x = 3?
b) If we know f (5) = 1 and f ′ (5) = −2, and we also know that f ′′ is continuous everywhere and f ′′ (5) = 8, then
what (if anything) can we conclude about the behavior of f at x = 5?
16. If the temperature of a roast in an oven is increasing by T (m) degrees Fahrenheit per minute, where m is the number of
minutes after the roast is placed in the oven,
a) What are the units for
b) What does
Z
T (m) dm?
Z 20
T (m) dm = 18 represent?
Z 20
T (m) dm = 40 represent?
10
c) What does
0
Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
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17. A honeybee population starts with 200 honeybees and increases at a rate of n(t) = 50et bees per week, where t is in
weeks and t ≥ 0.
a) Find the increase in the number of honeybees in the first 4 weeks. (Round to the nearest integer, if necessary).
b) Find a function, h(t), representing the number of honeybees after t weeks. Then, use this function to
work part a) above.
18. Estimate
Z 6
represent?
−2
(3x − x2) dx using a Riemann sum with n = 4 and the Midpoint Rule. What does the definite integral
Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
19. Find each of the following indefinite integrals:
a)
Z √
4x2 − x3 + π x − 7ex + 2 dx
b)
Z
e3x
dx
(e3x − 5)7
c)
Z
7x(x + 4)5 dx
d)
Z
e)
Z
!
√
√
3
12 + 5 x − 3x3 + 7 x2
x
− 6e dx
x4
2x
√
dx
6−x
11
12
Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
20. Find the function R(t) if
2
dR
= 3tet and R(0) = 4.
dt
21. Let G(x) be an antiderivative of g(x). If
Z 3
g(x) dx = 15 and G(3) = 7, find G(0).
0
22. Express
n
∑ (4x2i + xi sin xi )∆x
n→∞
lim
i=1
as an integral on the interval [0, 2π ].
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Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
23. Set up an expression for
Z 5
(ex + 3) dx as a limit of a right sum. What does this integral (limit of sums) represent?
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24. Use the graph of f (t) below to answer the following. Note that the domain of f (t) is [0, 7].
a) Approximate
Z 5
f (t) dt using a left sum with 3
2
rectangles of equal width.
b) If g(x) =
Z x
f (t) dt, find g(6).
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(Source: #3, pg. 372 of Single Variable Calculus: Concepts and Contexts, 4th ed., by Stewart)
c) If f (t) gives the velocity (in ft/s) of an object after t seconds, determine the distance the object travels
during the first 3 seconds.
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Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
25. Use the graph of f (x) with the indicated areas below to answer the following:
y
f(x)
D
A
b
a
a) Find
B
Z c
a
b) Find
Z c
C
c
f (x) dx −
d
Z c
4 f (x) dx +
x
area of A:
area of B:
area of C:
area of D:
2.0
1.5
2.5
9.5
2 f (x) dx.
0
Z b
f (x) dx.
d
c
c) Find the area between f (x) and the x-axis from x = 0 to x = d.
d) Find
Z a
f (x) dx.
0
e) When finding a left Riemann sum of f on the interval [a, 0] using one rectangle, will we obtain an overestimate or
underestimate?
f) Find
Z c
0
(5 − 3 f (x)) dx.
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Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
26. When finding a Riemann sum of a function f on the interval [a, b] using n rectangles, write the formula (using sigma
notation) for the sum of the areas of the first, second, and third rectangles when using
a) right endpoints.
b) left endpoints.
c) midpoints.
27. Given
Z 2
0
f (x) dx = 10 and
Z −2
0
f (x) dx = 10, find
Z 2
−2
f (x) dx.
Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
28. Find each of the following indefinite integrals:
2
a)
Z
e1/x
dx
x3
b)
Z
1
√ dx
x ln x
c)
Z
e2x − e−2x
dx
e2x + e−2x
29. If f ′ (7) = 12 and
Z 7
2
f ′′ (x) dx = 16, then find f ′ (2).
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Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
30. Compute the following integrals:
a)
Z A
2
8
dx, where A > 2.
x ln x
√
(2 + t) t − 3 dt
b)
Z
c)
√
Z B 2
3t + 4 t
3
d)
Z
5t 3
t
− 4e + π
4
√
dx
3x 6 + lnx
dt, where B > 3.
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Math 131 WIR, copyright Angie Allen (Problems 2 and 9 are courtesy of Joe Kahlig)
31. Find
Z
x3 (x2 + 3)2 dx.
32. When finding
made.
18
Z
2 +24x
(x + 4)e3x
dx, write the integral that would be obtained after the appropriate u-substitution is