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Simon Fraser University
MATH 251
Summer 2004
Second Midterm Examination
Instructor: A. Belshaw Date: July 7, 2004
Name:
Student number:
Signature:
Instructions
1. DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO
SO.
- Fill in the information above.
- This booklet contains 5 printed pages in addition to this cover page.
- Do all your work in this test booklet. Show all your work. Use the backs of the pages if necessary.
- No books, no notes, no calculators and no devices.
- Students observed writing anything after the call to stop writing will be subject to summary penalties.
1 2 3 4 5 Total
[4] 1. (a) Give the equation of the plane tangent to the surface z = 4−x^2 −y^2 at the point (1, 1 , 2).
[2] (b) What is the direction of most rapid increase of the function f (x, y) = 4 − x^2 − y^2 at the point (3, 2)?
[4] (c) In what direction(s) is the directional derivative of f (x, y) = 4 − x^2 − y^2 at the point (3, 2) equal to zero?
[8] 3. (a) Find and classify the critical points of
f (x, y) = 3x − 3 y + x^2 − xy + 2y^2.
[6] (b) The area of an ellipse is given by A = πab, where a and b are the lengths of the semiaxes and the equation of the ellipse is
x^2 a^2
y^2 b^2
Consider an cylinder of height h = 3, above an ellipse in the x- y plane with a = 2 and b = 4. How fast is the volume V of the cylinder increasing or decreasing if
dh dt
da dt
= 2, and db dt
[8] 4. (a) Use double integration to find the volume under the surface z = x^2 + y over the region bounded by x = y^2 − 2 y and y = x.
[4] (b) Find ∫ π 0
∫ (^) π
0
(ex^ + cos y) dx dy
