Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 2, 7, 15, 17, 18, and 20 are courtesy of Heather Ramsey)
Math 142 Week-in-Review #10 (Exam 3 Review: Sections 6.1-6.6, 6.7 topic, 8.1, and 8.2)
Note: This collection of questions is intended to be a brief overview of the exam material (with emphasis on
sections 6.7, 8.1, and 8.2). 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. 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
C
c
d
Z c
Z c
f (x) dx −
Z c
4 f (x) dx +
a
b) Find
area of A:
area of B:
area of C:
area of D:
x
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) Find the average value of the function f on [b, c].
f) When finding a left Riemann sum of f on the interval [a, 0] using one rectangle, will we obtain an overestimate or underestimate?
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Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 2, 7, 15, 17, 18, and 20 are courtesy of Heather Ramsey)
2. The productivity of an automobile parts manufacturing company is given by f (x, y) = 30x0.35 y0.65 units,
where x and y represent the number of units of labor and capital utilized, respectively.
a) Find the marginal productivity of labor when 70 units of labor and 50 units of capital are utilized.
b) Find the marginal productivity of capital when 70 units of labor and 50 units of capital are utilized.
c) For the greatest increase in productivity, should this company increase the use of labor or capital (assuming
they are currently using 70 units of labor and 50 units of capital)?
3. When finding a midpoint 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.
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Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 2, 7, 15, 17, 18, and 20 are courtesy of Heather Ramsey)
4. Find fx if f (x, y) =
2x2 − 4x3 y4
. Simplify your answer.
7x2 + 8y5
5. Find the equation of the cross section of z = −4x2 − 10y2 in the xz-plane.
6. Find the level curve of the function f (x, y) = −5 − x2 − y2 for z0 = −3.
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Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 2, 7, 15, 17, 18, and 20 are courtesy of Heather Ramsey)
7. Acme Widget Company’s marginal profit is given by w(x) = 35e−0.01x dollars per widget, where x is the
number of widgets produced per day. The profit earned by selling 120 widgets is $300.
a) If the current production level is 250 widgets per day and the manager wishes to increase production to
275 widgets per day, how will this production increase affect profit?
b) Find the profit earned by selling 200 widgets.
8. Find
∂ 2z
if z = f (x, y) = 4x4 ln(3x5 − 4y2 ). Simplify your answer.
∂ y∂ x
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Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 2, 7, 15, 17, 18, and 20 are courtesy of Heather Ramsey)
9. 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
− 4e + π
t
dt, where B > 3.
4
√
dx
3x 6 + ln x
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Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 2, 7, 15, 17, 18, and 20 are courtesy of Heather Ramsey)
10. Find the domain of the following functions:
a) f (x, y) =
b) f (x, y) =
√
8x + y
+ 3y
exy
ln(x + y + 3)
2−x
√
4 3 xy
c) f (x, y) =
x − 2y
11. 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
2
3 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 142 WIR, copyright Angie Allen, Spring 2013
(Problems 2, 7, 15, 17, 18, and 20 are courtesy of Heather Ramsey)
12. Given
Z 2
0
f (x) dx = 10 and
Z −2
f (x) dx = 10, find
0
Z 2
f (x) dx.
−2
13. An appliance company manufactures two types of microwaves each week, x of type A and y of type B. The
weekly demand equations are given by p = 210 − 9x + y and q = 120 + x − 4y, where p is the price of a
type A microwave and q is the price of a type B microwave (both in dollars). If the company has fixed costs
of $200 per week, and it costs the company $80 to make a type A microwave and $30 to make a type B
microwave, find and interpret Px (8, 4), where P(x, y) is the company’s profit function.
14. Find the area between f (x) = −x2 + 4 and the x-axis on the interval [−4, 4]. Include a sketch with the
appropriately shaded area.
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Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 2, 7, 15, 17, 18, and 20 are courtesy of Heather Ramsey)
15. The price-demand equation for a certain product is given by p = D(x) = 75 − 0.1x dollars per item, and the
price-supply equation for this product is p = S(x) = 15e0.002x dollars per item. Find the producers’ surplus at
the equilibrium price level, and shade the region on an appropriately labeled graph. What does your answer
represent?
16. If f ′ (7) = 12 and
Z 7
2
f ′′ (x) dx = 16, then find f ′ (2).
17. The research department of Acme, Inc. has determined the marginal cost function for one particular item
to be m(x) = 0.12e0.04x dollars per item, where x is the number of items produced. If Acme’s fixed costs
amount to $3,000, find the total cost when 150 items are produced.
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Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 2, 7, 15, 17, 18, and 20 are courtesy of Heather Ramsey)
18. The rate at which the concentration of a particular drug in the blood stream increases when taken daily can
be modeled by r(t) = 2.2
t µ g/mL per day, where t is the number of days since the daily regimen was started,
1 ≤ t ≤ 17. Five days after the regimen was started, the concentration of this drug in the blood stream was
4.5 µ g/mL.
a) Find the average rate of change of the concentration of the drug in the blood stream from taking the second
dose through taking the eighth dose.
b) Find the average concentration of the drug in the bloodstream from taking the third dose through taking
the tenth dose.
c) Find the change in concentration of the drug in the bloodstream from taking the third dose through taking
the tenth dose.
19. Find the area between y = x2 − 2 and y = −x on the interval [−2, 2]. Include a sketch with the appropriately
shaded area.
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Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 2, 7, 15, 17, 18, and 20 are courtesy of Heather Ramsey)
20. The price-demand equation for a certain item is given by p = D(x) = −0.002(x + 100)2 + 7000 dollars per
item, where x is the number of items that can be sold at a price of $p. If the current price per item is $4,580,
find the consumers’ surplus. Shade the region on an appropriately labeled graph, and explain what your
answer represents.
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21. Find fxx and fyy if f (x, y) = e3xy (3x2 + xy2 ). Simplify your answer.
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