UBC Mathematics 101 Final Examination, April 2009, Exams of Calculus

The final examination for mathematics 101 at the university of british columbia, held on april 24, 2009. The examination consists of short-answer and full-solution problems covering various topics in calculus and integration. Students were not allowed to use books, notes, or calculators during the exam, which lasted 2.5 hours.

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The University of British Columbia
Final Examination - April 24, 2009
Mathematics 101
All Sections
Closed book examination Time: 2.5 hours
Last Name First Signature
Student Number
Section :
Instructor :
Special Instructions:
No books, notes, or calculators are allowed. Unless it is otherwise specified, answers may be left
in “calculator-ready” form, where calculator means basic scientific calculator. Show all your work,
little or no credit will be given for a numerical answer without the correct accompanying work. If
you need more space than the space provided, use the back of the previous page. Where boxes are
provided for answers, put your final answers in them.
Rules governing examinations
Each candidate must be prepared to produce, upon request, a UBC-
card for identification.
Candidates are not permitted to ask questions of the invigilators,
except in cases of supposed errors or ambiguities in examination ques-
tions.
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, dishonest
practices shall be immediately dismissed from the examination and
shall be liable to disciplinary action.
(a) Having at the place of writing any books, papers or memoranda,
calculators, computers, sound or image players/recorders/transmitters
(including telephones), or other memory aid devices, other than those
authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candidates
or imaging devices. The plea of accident or forgetfulness shall not be
received.
Candidates must not destroy or mutilate any examination material;
must hand in all examination papers; and must not take any exami-
nation material from the examination room without permission of the
invigilator.
Candidates must follow any additional examination rules or directions
communicated by the instructor or invigilator.
1 30
2 6
3 10
4 12
5 10
6 12
7 10
8 10
Total 100
Page 1 of 13 pages
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The University of British Columbia Final Examination - April 24, 2009

Mathematics 101

All Sections

Closed book examination Time: 2.5 hours

Last Name First Signature

Student Number

Section :

Instructor :

Special Instructions:

No books, notes, or calculators are allowed. Unless it is otherwise specified, answers may be left in “calculator-ready” form, where calculator means basic scientific calculator. Show all your work, little or no credit will be given for a numerical answer without the correct accompanying work. If you need more space than the space provided, use the back of the previous page. Where boxes are provided for answers, put your final answers in them.

Rules governing examinations

  • Each candidate must be prepared to produce, upon request, a UBC- card for identification.
  • Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination ques- tions.
  • 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, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness shall not be received.
  • Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any exami- nation material from the examination room without permission of the invigilator.
  • Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.

Total 100

Page 1 of 13 pages

  1. Short-Answer Questions. Put your answers in the boxes provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. At most one mark will be given for an incorrect answer. Unless otherwise stated, simplify your Marks answers as much as possible.

(a) Evaluate

3 + x^5 √ x

[3] dx

Answer

(b) What integral is defined by the following expression? lim n→∞

∑^ n

i=

π 4 n

tan

iπ 4 n [3] Do not evaluate the integral.

[3] (e) Find the length of the curve^ x^ = 100 + 2y

3 / (^2) , 0 ≤ y ≤ 11.

Answer

[3] (f) What is the average value of | sin θ − cos θ| over the interval 0 ≤ θ ≤ π/2?

[3]^ (g) Give the first three nonzero terms of the Maclaurin series (power series in^ x) for ∫ e−x

2 − 1 x

dx

Answer

[3]

(h) For what values of r does the function y = xr^ satisfy the following differential equation (for x > 0)? x^2 y′′^ + 4xy′^ + 2y = 0

Full-Solution Problems. In the remaining questions, justify your answers and show all your work. If a box is provided, write your final answer there. If you need more space, use the back of the previous page. Unless otherwise indicated, simplification of answers is not required.

  1. For what values of p does

e

dx x(ln x)p^

[6] converge?

[10] 3. Let R be the finite region in the xy-plane bounded by x = 0, y = 0 and y = cos x.

(a) Calculate the centroid of R.

Answer

(b) Express the volume of the solid obtained by rotating R about the line x = −2 as an integral. Do not evaluate the integral.

[10] (^) 5. Let X be a random variable with probability density function

f (x) =

kx(1 − x^4 ) if 0 ≤ x ≤ 1 , 0 if x < 0 or x > 1.

(a) Find the value of k.

Answer

(b) Find the mean μ.

Answer

(c) Find an algebraic equation satisfied by the median m.

  1. Solve these differential equations:

[6] (a) y′^ = xy^2 with initial conditions y(0) = 1.

Answer

[6] (b) y′^ − 2 y = 4 + e^3 t. (Give the general solution)

[10] (^) 8. Consider the chemical reaction

A + B → C + D.

Suppose at time t = 0 sec the concentration of chemical A is 0.1 mol/L, the concentration of chemical B is 0.2 mol/L, and the concentrations of chemicals C and D are both 0. For t ≥ 0, let x(t) be the concentration of chemical D in mol/L. It can be shown that x(t) is the solution to the initial-value problem

dx dt

= k(0. 1 − x)(0. 2 − x), x(0) = 0,

where k is a positive constant whose value can be determined by experiment.

(a) Solve the initial-value problem to find x(t) explicitly.

Answer

(b) What value does the concentration of chemical D approach as t approaches ∞?