University of Toronto
Department of Electrical and Computer Engineering
MAT291F - Calculus III

Midterm - November 11, 2002

Duration: Two Hours          6:00 - 8:00 pm

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Last Name: ____________________________________

Given Names: __________________________________

Student No: ____________________________________

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1a. (10 Marks)
Find the equation of the tangent plane to the graph of at the point .

Solution:

1b. (5 Marks)
Find all the points on the graph in part (a) of which the tangent plane is horizontal.

Solution:

2a. (5 Marks)
If the equation is used to define z (implicitly) as a function of x and y, find:

Solution:

2b. (10 Marks)

Solution:

3. (17 Marks)
Lef . Find the points in where the absolute maximum and the absolute minimum of f over D occur.

Solution:

4. (18 Marks)
Find the absolute extreme values of the function on the ellipsoid .

Solution:

I didn't get the correct answer for this question on the quiz. I redid the question to what I now think is the correct answer, but I have not seen a correct solution for the problem, so I can only hope that the solution that I am providing here is correct.

It might be instructive to note the following level surface of f = 1 (note that it is not one of the extreme values of f). The graph of g is also in the plot. It is the ellipsoid that can be seen at the origin. The extreme value where x = y = z = 0 is of course the graph of nothing (or perhaps just a single point at the origin) and is therefore a minimum on f.

5. (17 Marks)
Evaluate the double integral

Solution:

6. (18 Marks)
Find the volume of the solid above the xy-plane, inside the cone , and inside the cylinder .

Solution:

Note: This graph assumes a=1 and is for visualization purposes only.