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Practice problems for Math 116 exam. It includes instructions for the exam, 4 questions with varying difficulty levels, and recommended time for completion. The questions cover topics such as infinite series, convergence, and functions. The document also specifies the allowed calculators and note cards for the exam.
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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
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
y
x
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
Suppose f (x) and g(x) are positive, continuous, decreasing functions such that
1 f^ (x)^ dx^ converges, and
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