MAT291F
Calculus III
Quiz #2
Wednesday, October 30, 2002

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Duration: 45 Minutes

Last Name: ____________________________________

First Name: ____________________________________

Student No: ____________________________________

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1. (5 Marks)

Let , . Estimate a point at which and .

Solution:

2. (5 Marks)
Find the absolute maximum and minimum of the function:

on ,

Solution:

i.

ii.

iii. Conclusion:
Abs min of -1/4
Abs max of 2

3. (5 Marks)
Decide whether or not the following vector function has a potential function, and find a potential function if there is one.

Solution:

4. (5 Marks)
Evaluate the double integral

Solution:

This question was omitted in the marking due to the fact that it is impossible to evaluate with the tools that we have learned so far. For you keeners out there though, the integral evaluates to: -2.16517

I did this calculation using mathematica, and apparently, the integral can only be evaluated numerically. Interestingly, the numerical value that the integral evaluates to is precisely -2.16517. In other words, no rounding was performed. I know that this is contradictory to the very nature of numerical integration, but regardless of the precision that I chose for the integral in mathematica, -2.16517 was always returned. This leads me to believe that it is indeed what the numerical integral converges to. By the way, if you do figure out a way to evaluate the integral, don't be surprised if you end up having to use complex integration. I did see complex numbers appear while mathematica was performing the integral.