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MATH 100 Fall 2018 midterm exam with solutions
MATH 100 Fall 2018 midterm exam with solutions
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MATH 100—Midterm Examinationversion 1
Date: October 26, 2018 Time: 90 minutes
Section EA1 EB1 EC1 ED1 EE1 EF1 EG1 EH1
Instructor Yaskin Leonard Shen Wang Yaskin Koziol Shen Devyatov
X
Instructions
1. Place your U of A Student ID card on your table or desk.
2. The exam has 11 pages including this cover page. Please ensure that you have all
pages.
3. The exam will be marked out of 60 points. There are 8 questions. The points for each
question are indicated beside the question number.
4. Answers for questions 1 through 7 must be accompanied by adequate justification.
Please note that this exam will be marked electronically. Anything written
on the back of the pages will not be marked.
5. For multiple-choice questions you are not required to show your work.
6. This is a closed book exam. No books, notes, calculators, cell phones or other
electronic aids are allowed!
, 1. [6 pts]
(a) Let a be a real number, and define a function by
(
3 1
arctan if x 6= 0,
f (x) = π x
a if x = 0.
Does there exist a value of a for which f (x) is continuous at x = 0? If yes, find it. If
no, explain why it does not exist.
f (x) is continuous at x = 0 if:
lim f (x) = lim+ f (x) = f (0) = a.
x→0− x→0
3 1 3 3 π 3
lim− f (x) = lim− arctan = lim arctan (z) = × − = − .
x→0 x→0 π x π z→−∞ π 2 2
3 1 3 3 π 3
lim+ f (x) = lim+ arctan = lim arctan (z) = × = .
x→0 x→0 π x π z→∞ π 2 2
lim f (x) 6= lim− f (x) =⇒ lim f (x) does not exist,
x→0− x→0 x→0
and therefore there is no value of a for which f (x) is continuous at x = 0.
(b) Let a be a real number, and define a function by
(
3
arctan x14
π
if x 6= 0,
f (x) =
a if x = 0.
Does there exist a value of a for which f (x) is continuous at x = 0? If yes, find it. If
no, explain why it does not exist.
In this case, we have:
3 1 3 3 π 3
lim f (x) = lim arctan = lim arctan (z) = × = .
x→0 x→0 π x4 π z→∞ π 2 2
f (x) is continuous at x = 0 if:
lim f (x) = f (0) = a
x→0
Thus,
3
a = .
2
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