100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
MATH 100 Fall 2018 midterm exam with solutions CA$11.60   Add to cart

Exam (elaborations)

MATH 100 Fall 2018 midterm exam with solutions

 9 views  0 purchase

MATH 100 Fall 2018 midterm exam with solutions

Preview 2 out of 11  pages

  • March 29, 2021
  • 11
  • 2018/2019
  • Exam (elaborations)
  • Questions & answers
book image

Book Title:

Author(s):

  • Edition:
  • ISBN:
  • Edition:
All documents for this subject (38)
avatar-seller
xuyian222
MATH 100—Midterm Examination
version 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

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller xuyian222. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for CA$11.60. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

75632 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling
CA$11.60
  • (0)
  Add to cart