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Math 116 — Practice for Exam 2
Generated October 28, 2018
Name:
Instructor: Section Number:
- This exam has 4 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 Winter 2016 3 5 pizzas 6 Winter 2016 3 10 14 Winter 2018 2 11 10 Fall 2017 2 10 gamma 9 Total 39
Recommended time (based on points): 42 minutes
Math 116 / Final (April 21, 2016) DO NOT WRITE YOUR NAME ON THIS PAGE page 6
- [6 points] O-guk is eating pizzas! All is well now, so he got hungry. He has put them next to each other, as depicted below, so that he can devour them one after another. There are infinitely many pizzas, and they have radii 1, 12 , 13 , 14 , 15 , ... The following figure shows the first five pizzas.
1 1/2 1/.^ ..
a. [4 points] Write infinite series for the total area and the total perimeter of the pizzas. You must write your series in sigma notation.
Total area:
Total perimeter:
b. [2 points] In the next two questions circle the correct answer.
Is the total area a finite number? YES NO
Is the total perimeter a finite number? YES NO
University of Michigan Department of Mathematics Winter, 2016 Math 116 Exam 3 Problem 5 (pizzas)
Math 116 / Final (April 21, 2016) DO NOT WRITE YOUR NAME ON THIS PAGE page 11
- (continued) For your convenience, the graph of f is given again. The numbers A, B, C are positive constants. The shaded region has finite area, but it extends infinitely in the positive x-direction. The line y = C is a horizontal asymptote of f (x) and f (x) > C for all x ≥ 0. The point (1, A) is a local maximum of f.
y
x
C
B
A
f (x)
b. [3 points] Circle the correct answer. The value of the integral
1
f (x)f ′(x) dx
is C − A is C (^2) −A 2 2 is^ B^ −^ A^ cannot be determined^ diverges
c. [3 points] Circle the correct answer. The value of the integral
1
f ′(x) dx
is C − A is C
(^2) −A 2 2 is^ C^ cannot be determined^ diverges
d. [3 points] Determine, with justification, whether the following series converges or diverges.
∑^ ∞
n=
(f (n) − C)
University of Michigan Department of Mathematics Winter, 2016 Math 116 Exam 3 Problem 10
Math 116 / Exam 2 (March 19, 2018) do not write your name on this exam page 11
- [10 points] You work for a temp agency. Today you fill in for Russ Weterson, doing important work for the city. On Mr. Weterson’s desk you find the following problems with a note: “Russ, the Mayor needs these problems done yesterday. -Brontel”
Suppose f (x) and g(x) are positive, continuous, decreasing functions such that
1 f^ (x)^ dx^ converges, and
- 0 ≤ g(x) ≤ 9 for all real numbers x.
Determine whether the following expressions must converge, must diverge, or whether convergence cannot be determined. No justification required.
a. [2 points]
1
f (x)
dx
Converges Diverges Cannot be determined
b. [2 points]
∑^ ∞
n=
f (n)
Converges Diverges Cannot be determined
c. [2 points]
1
f (x)g(x) dx
Converges Diverges Cannot be determined
d. [2 points]
∑^ ∞
n=
f (n)g(n)
Converges Diverges Cannot be determined
e. [2 points]
1
g(x) dx
Converges Diverges Cannot be determined
University of Michigan Department of Mathematics Winter, 2018 Math 116 Exam 2 Problem 11
