Math 257 Final Exam, UBC, April 18, 2008, Exams of Elementary Mathematics

The final examination for mathematics 257 section 201 at the university of british columbia, held on april 18, 2008. The examination covers topics such as fourier cosine series, laplace equation, wave equation, and sturm-liouville problem. Students were required to solve various problems related to these topics and submit their answers within 2.5 hours. The examination was closed-book and no calculators or notes were allowed.

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The University of British Columbia
Final Examination - April 18, 2008
Mathematics 257 Section 201
Instructor: Dr. Alexei F. Cheviakov
Closed book examination Time: 2.5 hours
Last name First name
Student Number Signature
Special Instructions:
- Be sure that this examination has 8 pages (5 problems). Write your last name on top of each page.
- Submit only this booklet, with solution written in space provided (you may use adjacent page(s)).
Clearly outline answers. Solutions on scratch paper will not be graded.
- A 2-sided self-prepared Letter-size formula sheet is allowed. No calculators or notes are permitted.
- In case of an exam disruption such as a fire alarm, leave the exam papers in the room and exit quickly
and quietly to a pre-designated location.
Rules governing examinations
Each candidate should be prepared to produce her/his li-
brary/AMS card upon request.
No candidate shall be permitted to enter the examination room
after the expiration of one half hour, or to leave during the first
half hour of examination.
Candidates are not permitted to ask questions of the invigilators,
except in cases of supposed errors or ambiguities in examination
questions.
CAUTION - Candidates guilty of any of the following or similar
practices shall be immediately dismissed from the examination
and shall be liable to disciplinary action.
(a) Making use of any books, papers, or memoranda, 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.
Food and smoking are not permitted during examinations.
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2 20
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5 20
Total 100
Page 1 of 8
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The University of British Columbia Final Examination - April 18, 2008 Mathematics 257 Section 201 Instructor: Dr. Alexei F. Cheviakov

Closed book examination Time: 2.5 hours

Last name First name

Student Number Signature

Special Instructions:

  • Be sure that this examination has 8 pages (5 problems). Write your last name on top of each page.
  • Submit only this booklet, with solution written in space provided (you may use adjacent page(s)). Clearly outline answers. Solutions on scratch paper will not be graded.
  • A 2-sided self-prepared Letter-size formula sheet is allowed. No calculators or notes are permitted.
  • In case of an exam disruption such as a fire alarm, leave the exam papers in the room and exit quickly and quietly to a pre-designated location.

Rules governing examinations

  • Each candidate should be prepared to produce her/his li- brary/AMS card upon request.
  • No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of examination.
  • Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
  • CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Making use of any books, papers, or memoranda, 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.
  • Food and smoking are not permitted during examinations.

Total 100

Page 1 of 8

[20] Problem 1.

a) Find the coefficients of Fourier cosine series of the function f (x) =

1 , 0 ≤ x ≤ 2 , 4 − x, 2 < x ≤ 4.

b) Write the series in the form where coefficients do not contain trigonometric functions.

c) What is the period of the series you found? Sketch the graph of the function to which the series converges over at least two periods.

d) Does the series converge uniformly or pointwise? At which points x does Gibbs phenomenon occur?

[20] Problem 3. Consider the wave equation utt = uxx for an infinite string x ∈ (−∞, ∞) with initial conditions u(x, 0) = f (x), ut(x, 0) = g(x).

a) Give the formula for u(x, t) for all t > 0, −∞ < x < +∞.

b) Draw the solution u(x, t) for t = 2, if f (x) and g(x) are functions given below:

f ( x )

-5 -4 -3 -2 -1 0 1 x

2

4

g ( x )

-3 -2 -1 0 1 2 3 4^ x

1

2

-

[Hint: consider separately a problem with f = 0 and a problem with g = 0.]

c) Now suppose the string is finite: − 10 ≤ x ≤ 10. Specify the maximum time T until which your solution found in a) for the infinite string, is correct for this finite string.

[20] Problem 4. Consider a Sturm-Liouville problem

  

X′′^ + λX = 0, 0 < x < 1 , X(0) = 0, X(1) − X′(1) = 0.

a) Is the problem regular or singular? Why? (Use the definition.)

b) Solve the problem to find all its eigenfunctions and eigenvalues.

c) Write down the orthogonality condition satisfied by the eigenfunctions. Find norms of all eigenfunctions.

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