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Math 116 — Practice for Exam 3
Generated November 25, 2018
Name: SOLUTIONS
Instructor: Section Number:
- This exam has 9 questions. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck.
- Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you hand in the exam.
- Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam.
- Show an appropriate amount of work (including appropriate explanation) for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate.
- You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3′′^ × 5 ′′^ note card.
- If you use graphs or tables to obtain an answer, be certain to include an explanation and sketch of the graph, and to write out the entries of the table that you use.
- You must use the methods learned in this course to solve all problems.
Semester Exam Problem Name Points Score
Fall 2016 2 2 5 Winter 2017 2 3 leaky boat 5 Fall 2017 3 5 dunking booth 5 Winter 2016 2 6 15 Fall 2017 3 1 foxes 6 Fall 2014 2 6 10 Fall 2015 2 5 10 Fall 2016 2 4 12 Winter 2018 3 12 6
Total 74
Recommended time (based on points): 72 minutes
Math 116 / Midterm (November 14, 2016) DO NOT WRITE YOUR NAME ON THIS PAGE page 3
- [5 points] Find constants A and B so that the function h(w), defined for w > 0 by
h(w) = Aw^3 +
w
is a solution to the differential equation
w^2
dh dw
− 3 wh + B = 0
satisfying h(1) = 32. Show all your work, and write your final answers in the spaces provided.
Solution: A = 1 /^2
B = 4
- [5 points] In a recent presidential election between candidate A and candidate B, Shamcorp’s rival company Hawk-I tried fixing the election by changing the votes on some of the ballots. For the last three hours of the election (between 5pm and 8pm), the company gained access to the huge ballot box containing 100 million ballots.
Hawk-I employees removed ballots from the ballot box continuously at a rate of 4 million ballots per hour. Those ballots were removed in proportion to the current ratio in the box. Hawk-I employees then instantly changed the the ballots voting for candidate B to vote for candidate A (leaving any votes for candidate A unchanged) before immediately returning the ballots to the box.
Assume that the ballot box always contains 100 million votes, and that the ballot box only contains votes for candidates A and B.
Write a differential equation that models a(t), the number of ballots voting for candidate A, in millions, in the ballot box t hours after Hawk-I began changing votes.
Solution:
da dt
a 25
University of Michigan Department of Mathematics Fall, 2016 Math 116 Exam 2 Problem 2 Solution
Math 116 / Final (December 14, 2017) do not write your name on this exam page 5
- [4 points] Let S be the region bounded by the x-axis, the line x = 0.5, and the line y = 4 − 4 x. This region is shown to the right. The units on both the x- and the y-axis are centimeters. A solid is obtained by rotating the region S about the y-axis. The mass density of the resulting solid at each point y centimeters above the x-axis is 16y grams per cubic centimeter.
- 5 1
y = 4 − 4 x
S
x (cm)
y (cm)
Write, but do not evaluate, an expression involving one or more integrals that gives the mass, in grams, of the resulting solid.
Answer: Mass =
- [5 points] Prior to the start of an indoor winter carnival, the water tank for a dunking booth is being filled from a water hose at a rate of 8 gallons per minute. Unfortunately, once the tank has 50 gallons of water in it, the tank begins leaking water at a rate (in gallons per minute) that is proportional to the square root of the volume of water in the tank (in gallons) with constant of proportionality k > 0. Let W = W (t) be the volume, in gallons, of water that is in the tank t minutes after the tank begins to leak. Write a differential equation that models W (t) and give an appropriate initial condition.
Answer: Differential Equation:
Initial Condition:
University of Michigan Department of Mathematics Fall, 2017 Math 116 Exam 3 Problem 5 (dunking booth) Solution
Math 116 / Exam 2 (March 21, 2016) DO NOT WRITE YOUR NAME ON THIS PAGE page 8
- [15 points] In the following questions, circle the correct answer. You do not need to show any work, but make sure your answer is clear. No points will be given for unclear answers. a. [3 points] The value of A for which the function y = ex
(^2) +A (^3) x solves the equation y′^ + 8y = 2xy is
b. [3 points] The function g is positive, decreasing and differentiable. The solution curves of the differential equation y′^ = e−xg(y) are
concave up concave down changing concavity
c. [3 points] Suppose that h(x) is an increasing differentiable function with h(0) = 0 and
xlim→∞ h(x) = 5. The value of the integral
0
(h(x))^4 h′(x) dx
diverges is 5^4 is 5^4 −
is 1 is 0
d. [3 points] Suppose a ≥ 1 is a constant, and the function h satisfies
x^1 /a^
≤ h(x) ≤
xa^
for
0 ≤ x ≤ 1. The integral
0
(h(x))^2 dx converges
always never sometimes
e. [3 points] The function f satisfies
x^3
≤ f (x) ≤
x
for x ≥ 1 and f (x) = g(x^2 ). The
integral
1
g(x) x
dx converges
always never sometimes
University of Michigan Department of Mathematics Winter, 2016 Math 116 Exam 2 Problem 6 Solution
Math 116 / Exam 2 (November 12, 2014) page 9
- [10 points] Match the following. For each blank, there is only one correct answer. a. [4 points] For each slope field on the left, write the letter corresponding to the differential equation that generates that slope field in the blank provided.
x
y
I. (E.)
x
y
II. (B.)
(A.)
dy dx
= (y + 2)(y − 1)
(B.)
dy dx
= (y − 2)(y + 1)
(C.)
dy dx
= (y + 1)(y − 2)^2
(D.)
dy dx
= (2 − x)(y + 1)
(E.)
dy dx
= (x − 2)(y + 1)
(F.)
dy dx
= (x − 1)(y − 2)
b. [6 points] Let r(θ) = k be a polar curve where k > 0 is a constant. Match the quantities on the left with their formulas (in terms of θ) on the right.
I.
dy dθ
= (A.)
II.
dx dθ
= (D.)
III.
dy dx
= (H.)
(A.) k cos(θ)
(B.) − k cos(θ)
(C.) k sin(θ)
(D.) − k sin(θ)
(E.) tan(θ)
(F.) − tan(θ)
(G.) (^) tan(^1 θ)
(H.) − (^) tan(^1 θ)
University of Michigan Department of Mathematics Fall, 2014 Math 116 Exam 2 Problem 6 Solution
Math 116 / Exam 2 (November 18, 2015) DO NOT WRITE YOUR NAME ON THIS PAGE page 5
- [10 points] The graph of a slope field corresponding to a differential equation is shown below.
x
y
a. [4 points] On the slope field, carefully sketch a solution curve passing through the point (-1,-1).
Solution: See graph above.
b. [2 points] The slope field pictured above is the slope field for one of the following differ- ential equations. Which one? Circle your answer. You do not need to show your work.
dy dx
= cos x cos(2y)
dy dx
= sin x cos(2y)
dy dx
= cos x sin(2y)
dy dx
= sin x sin(2y)
c. [4 points] Find two equilibrium solutions to the differential equation you circled.
Solution: The equilibrium solutions of dy dx = cos x sin(2y) are the values of y such that sin(2y) = 0. Solving, we see that the equilibrium solutions are y = 0, ± π 2 , ±π,...
University of Michigan Department of Mathematics Fall, 2015 Math 116 Exam 2 Problem 5 Solution
Math 116 / Final (April 19, 2018) page 14
- [6 points] The polynomial P 3 (x) = 2−3(x+e)^2 +5(x+e)^3 is the third-degree Taylor polynomial approximating the function g(x) for x near −e. Find the following values. Write “ni” if there is not enough information.
g′(−e) = 0 g(−e) = 2 g′′′(−e) = 30
P 3 (4) (−e) = 0 g(0) = ni^ P 3 (0) = 2 −^3 e^2 + 5e^3
- [6 points] Match the differential equations to their corresponding slope fields.
i. y′^ = x^2 + y^2 D
ii. y′^ = y x B
iii. y′^ = − x y A
iv. y′^ = x(y^2 − 1) E
v. y′^ = x(1 − y^2 ) F
vi. y′^ = 3 x
(^2) + 2 y C
x
y (A)
x
y (D)
x
y (B)
x
y (E)
x
y (C)
x
y (F)
University of Michigan Department of Mathematics Winter, 2018 Math 116 Exam 3 Problem 12 Solution
