MAT 331 Homework Exercises. Week 5 (Oct 19, 99).

NOTE: [No Maple] means that the problem does not involve Maple, except as a word processor to write your solution. In this case you can alternatively turn in a short paper, if you are more comfortable with that.

#17 (exp. 10/26)

[No Maple] Recall that for a differentiable function of two variables, f, the level curve corresponding to the constant c is defined as the set of all points (x, y) for which f (x, y) = c. Prove that the gradient of f is always orthogonal to the level curve. That is, in each point (x, y) of the level curve corresponding to c, $\nabla$f (x, y) is orthogonal to the tangent vector to the curve in (x, y).

#18 (exp. 10/26)

Find all the critical points of g(x, y) = (2x² - x)(y² - 1) and determine which ones are local maximum, minimum or saddle points. What are the absolute maximum and minimum of g? [Hint: You may find array, matrix and other commands form the linalg library useful.]

#19 (exp. 10/26)

Define a Maple function h(x, y) that is 0 in the first quadrant, y sin x in the third quadrant and xy in the rest of the plane.

#20 (exp. 11/2)

Write a Maple procedure that takes as input two integers a, b, and returns the following objects: l, s, p. Here l is the list of all primes between a + 1 and b(extremes included), s their sum, and p their product. Also, have Maple print on the screen ``Above average'' if the number of these primes is greater than ${\frac{b}{\log b}}$ - ${\frac{a}{\log a}}$; ``Below average'' otherwise. [Hint: See isprime, print. Check op on how to add an element to a list]