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The University of British Columbia Final Examination - April, 2010 Mathematics 200
Closed book examination Time: 2.5 hours
Last Name: , First: Signature
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Special Instructions:
No books, notes or calculators are allowed. Use backs of sheets if extra space needed.
Rules governing examinations
- Each candidate must be prepared to produce, upon request, a UBCcard for identification.
- Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
- 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, dishon- est 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 play- ers/recorders/transmitters (including telephones), or other mem- ory 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 can- didates or imaging devices. The plea of accident or forgetfulness shall not be received.
- Candidates must not destroy or mutilate any examination mate- rial; must hand in all examination papers; and must not take any examination material from the examination room without permis- sion of the invigilator.
- Candidates must follow any additional examination rules or di- rections communicated by the instructor or invigilator.
Total 100
Page 1 of 12 pages
[12] 1 .(a) A surface z(x, y) is defined by zy − y + x = ln(xyz).
(i) Compute ∂z∂x , ∂z∂y in terms of x, y, z.
(ii) Evaluate ∂z∂x and ∂z∂y at (x, y, z) = (− 1 , − 2 , 1 /2).
[14] 2. (i) For the function
z = f(x, y) = x^3 + 3xy + 3y^2 − 6 x − 3 y − 6.
Find and classify [as local maxima, local minima, or saddle points] all critical points of f(x, y)
2 (ii) The images below depict level sets f(x, y) = c of the functions in the list at heights c = 0, 0. 1 , 0. 2 ,... , 1. 9 , 2. Label the pictures with the corresponding function and mark the critical points in each picture. (Note that in some cases, the critical points might not be drawn on the images already. In those cases you should add them to the picture.)
(a) f(x, y) = (x^2 + y^2 − 1)(x − y) + 1
(b) f(x, y) =
p x^2 + y^2
(c) f(x, y) = y(x + y)(x − y) + 1
(d) f(x, y) = x^2 + y^2
[10] 4.
(a) Does the function f(x, y) = x^2 +y^2 have a maximum or a minimum on the curve xy = 1? Explain.
(b) Find all maxima and minima of f(x, y) on the curve xy = 1.
[13] 5. Let G be the region in R^2 given by
x^2 + y^2 ≤ 1 , 0 ≤ x ≤ 2 y, y ≤ 2 x.
(a) Sketch the region G.
Express the integral
RR
G f^ (x, y)dA^ as
(b) a sum of iterated integrals
RR
f (x, y)dxdy,
(c) an iterated integral in polar coordinates (r, θ) where x = r cos(θ) and y = r sin(θ).
[13] 7. A thin plate of uniform density k is bounded by the positive x and y axes and the circle x^2 + y^2 = 1. Find its centre of mass.
[13] 8. Let
I =
Z Z Z
T
(x^2 + y^2 )dV,
where T is the solid region bounded below by the cone z =
p 3 x^2 + 3y^2 and above by the sphere x^2 + y^2 + z^2 = 9.
(i) Express I as a triple integral in spherical coordinates.
(ii) Express I as a triple integral in cylindrical coordinates.
(iii) Evaluate I by any method.
