University of British Columbia - MATH 110 December Exam 2011, Exams of Mathematics

Information about the december 2011 exam for math 110 at the university of british columbia. It includes the exam date, time, rules, and questions covering various topics in mathematics such as limits, continuity, differentiation, and tangent lines.

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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Name (print):
ID number:
Section (circle): 001 002 003
University of British Columbia
DECEMBER EXAM for MATH 110
Date: December 10, 2011
Time: 12:00 noon to 2:30 p.m.
Number of pages: 14 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
Rules governing formal examinations:
Each candidate must be prepared to produce, upon request, a
UBC card for identification.
No candidate shall be permitted to enter the examination room
after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination.
Candidates suspected of any of the following, or similar, dis-
honest practices shall be immediately dismissed from the exam-
ination and shall be liable to disciplinary action:
Having at the place of writing any books, papers or memo-
randa, calculators, computers, sound or image players/record-
ers/transmitters (including telephones), or other memory aid
devices, other than those authorized by the examiners;
Speaking or communicating with other candidates;
Purposely exposing written papers to the view of other candi-
dates or imaging devices. The plea of accident or forgetfulness
shall not be received.
Candidates must not destroy or mutilate any examination ma-
terial; must hand in all examination papers; and must not take
any examination material from the examination room without
permission of the invigilator.
Candidates must follow any additional examination rules or di-
rections communicated by the instructor or invigilator.
For examiners’ use only
Question Mark Maximum mark
1 7
2 6
3 4
4 6
5 8
6 4
7 12
8 7
9 8
10 8
11 3 (bonus)
Total 70
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Name (print):

ID number:

Section (circle): 001 002 003

University of British Columbia

DECEMBER EXAM for MATH 110

Date: December 10, 2011 Time: 12:00 noon to 2:30 p.m. Number of pages: 14 (including cover page) Exam type: Closed book Aids: No calculators or other electronic aids

Rules governing formal examinations: Each candidate must be prepared to produce, upon request, aUBC card for identification. No candidate shall be permitted to enter the examination roomafter the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination. Candidates suspected of any of the following, or similar, dis-honest practices shall be immediately dismissed from the exam- ination and shall be liable to disciplinary action:

  • randa, calculators, computers, sound or image players/record- Having at the place of writing any books, papers or memo- ers/transmitters (including telephones), or other memory aiddevices, other than those authorized by the examiners;
  • Speaking or communicating with other candidates;
  • dates or imaging devices. The plea of accident or forgetfulness Purposely exposing written papers to the view of other candi- shall not be received. Candidates must not destroy or mutilate any examination ma-terial; must hand in all examination papers; and must not take any examination material from the examination room withoutpermission of the invigilator. Candidates must follow any additional examination rules or di-rections communicated by the instructor or invigilator.

For examiners’ use only Question Mark Maximum mark 1 7 2 6 3 4 4 6 5 8 6 4 7 12 8 7 9 8 10 8 11 3 (bonus) Total 70

  1. Evaluate each of the following limits.
  2. (a) [2 marks] (^) x→−∞lim^5 x^36 −x (^3 2) + 7x^2 + 1
  3. (b) [3 marks] (^) xlim→ 0

x(1 − 2 x) −^

x

  1. (c) [2 marks] (^) xlim→∞^ (e−x^ − x)
  1. [4 marks] Prove that the function f (x) = x^3 − 15 x + 1 has three roots in the interval [− 4 , 4]. Make sure to state any assumptions you are making, or theorems you are using.
  1. Let f (x) = (^) x + 1^1.
  2. (a) [4 marks] Find f ′(x) using the limit definition of derivative. (No marks will be given for using other methods in part (a).)
  3. (b) [2 marks] Confirm your answer in part (a) by finding f ′(x) using differentiation rules such as the Quotient Rule or Chain Rule.
  1. [4 marks] Let f (x) =

x^2 2 cos^ + bx ifif^ x < ax ≥ a. Find constants a and b such that f is differentiable everywhere. Justify your answer.

  1. For this question, let f (x) = x^2 − 6 x and let P be the point (4, −12).
  2. (a) [1 mark] Is P on the curve y = f (x)? Justify your answer.
  3. (b) [1 mark] Find the slope of the line between P and a point (a, a^2 − 6 a) on the curve y = f (x).
  4. (c) [2 marks] Find the slope of the line tangent to the curve y = f (x) at the point (a, a^2 − 6 a)
  1. Both parts of this question refer to the curve y = (ax + b)^7 , where a and b are constants.
  2. (a) [5 marks] Suppose that a 6 = 0. Find the equation of the line tangent to the curve at x = ba.
  3. (b) [2 marks] Suppose that a = 0 in the equation for the curve given above. Find the equation of the tangent line in this case.
  1. [8 marks] Write down an algebraic expression for a function f satisfying the following four criteria. (^) • lim x→ 0 −^ f^ (x) = 2^ •^ x→−∞lim^ f^ (x) = 0
  • (^) xlim→ 0 + f (x) = − 1 • f ′(x) = − 2 for x > 0 Then sketch the function on the axes given.

y

x

  1. [3 bonus marks] Find a piecewise algebraic expression for the nth^ derivative of f (x) = x ln x.

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