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This is a closed book final examination for math 200 course at the university of british columbia held in april, 2010. The examination consists of 8 questions covering various topics in calculus, including partial derivatives, linear approximation, tangent planes, critical points, level sets, rate of change, maxima and minima, and triple integrals.
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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
Student Number Section
Special Instructions:
No books, notes or calculators are allowed. Use backs of sheets if extra space needed.
Rules governing examinations
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
(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
G f^ (x, y)dA^ as
(b) a sum of iterated integrals
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 =
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.