Midterm 1 Exam for MATH 150 at Simon Fraser University, Fall 2006, Exams of Calculus

A midterm exam for the math 150 course at simon fraser university, taught by dr. Mulholland in the fall 2006 semester. The exam covers various topics in mathematics, including limits, continuity, and functions. Students are required to answer questions related to computing limits, identifying even functions, and applying the squeeze law. The exam consists of five questions, each worth a different number of points, totaling 40 points.

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2012/2013

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SIMON FRASER UNIVERSITY
DEPARTMENT OF MATHEMATICS AND STATISTICS
Midterm 1
MATH 150 Fall 2006
Instructor: Dr. Mulholland
October 4, 2006, 8:30 โ€“ 9:20 a.m.
Name: (please print)
family name given name
SFU ID:
student number SFU-email
Signature:
Instructions:
1. Do not open this booklet until told to do so.
2. Write your name above in block letters. Write your
SFU student number and email ID on the line pro-
vided for it.
3. Write your answer in the space provided below the
question . If additional space is needed then use the
back of the previous page. Your final answer should
be simplified as far as is reasonable.
4. Make the method you are using clear in every case
unless it is explicitly stated that no explanation is
needed.
5. This exam has 5 questions on 8 pages (not includ-
ing this cover page). Once the exam begins please
check to make sure your exam is complete.
6. No calculators, books, papers, or electronic devices
shall be within the reach of a student during the
examination.
7. During the examination, communicating with,
or deliberately exposing written papers to the
view of, other examinees is forbidden.
Question Maximum Score
1 12
2 8
3 8
4 6
5 6
Total 40
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SIMON FRASER UNIVERSITY

DEPARTMENT OF MATHEMATICS AND STATISTICS

Midterm 1

MATH 150 Fall 2006 Instructor: Dr. Mulholland October 4, 2006, 8:30 โ€“ 9:20 a.m.

Name: (please print) family name given name

SFU ID:

student number SFU-email

Signature:

Instructions:

  1. Do not open this booklet until told to do so.
  2. Write your name above in block letters. Write your SFU student number and email ID on the line pro- vided for it.
  3. Write your answer in the space provided below the question. If additional space is needed then use the back of the previous page. Your final answer should be simplified as far as is reasonable.
  4. Make the method you are using clear in every case unless it is explicitly stated that no explanation is needed.
  5. This exam has 5 questions on 8 pages (not includ- ing this cover page). Once the exam begins please check to make sure your exam is complete.
  6. No calculators, books, papers, or electronic devices shall be within the reach of a student during the examination.
  7. During the examination, communicating with, or deliberately exposing written papers to the view of, other examinees is forbidden.

Question Maximum Score

Total 40

1. Compute the following limits.

[3] (a) lim xโ†’โˆ’ 1 (x^3 โˆ’ 2 x^2 + 5x + 1)^2

[3] (b) lim xโ†’ 5

x โˆ’ 1 โˆ’ 2 x โˆ’ 5

2. True or False. If True provide an explanation, if False give justification (for instance

give an example for which the statement doesnโ€™t hold).

[2] (a) If f is one-to-one, then f โˆ’^1 (x) =

f (x)

[2] (b) The function g(x) = x sin (x) is an even function.

[2] (c) Suppose that a continuous function f (x) satisfies the following table of values.

x โˆ’ 1 โˆ’ 0. 5 0 0. 5 1 1. 5 2 2. 5 3 f (x) โˆ’ 3 โˆ’ 1. 25 0. 25 0. 75 1 0. 75 โˆ’ 0. 25 โˆ’ 1. 25 โˆ’ 3

Then the function f (x) has at least 2 zeros in the interval (โˆ’ 1 , 3).

[2] (d) If f has domain [0, โˆž) and has no horizontal asymptote, then limxโ†’โˆž f (x) = โˆž or limxโ†’โˆž f (x) = โˆ’โˆž.

[2] 3. (a) State the definition of continuity for a function f (x) at x = a.

[3] (b) Suppose that

g(x) =

x^2 โˆ’ 1 if x โ‰ค 0 , bx^3 โˆ’ 2 if 0 < x โ‰ค 1 , 2 x โˆ’ b if x > 1 , where b is some constant.

Is the function g continuous at x = 0? Justify your answer.

[2] 4. (a) State the Squeeze Law, clearly identifying any hypotheses and the conclusion.

[4] (b) Use the Squeeze Law to compute lim xโ†’ 0 +

xesin (ฯ€/x). Justify your answer.

[4] 5. (a) Find

lim hโ†’ 0

f (2 + h) โˆ’ f (2) h where f (x) = x^3 โˆ’ 1.