Exercises

1.

For each of the following functions,

i

Print out a plot of the level curves.

ii

Calculate the partial derivatives f_x and f_y at the given points and for each point, build the gradient vector:

$$\nabla f(x,y) = f_x(x,y){\bf i} + f_y(x,y){\bf j}$$

(You may use the grad command in Maple, but you will have to look up its syntax in Maple Help.)

iii

Sketch arrows (vectors) by hand at each of the chosen points in part b showing the direction of the gradient and explain why the gradient vectors are perpendicular to the level curves (you may have to scale the gradient vector).

(a)

$$f(x,y) = \frac{x^2-y^2}{x^2+y^2}$$, at (10,2), (-10,2), (-10,-2), and (10,-2)

(b)

$$g(x,y) = 4x^2 + 9y^2$$, at (2,4), (-2,4), (-2,-4), and (2,-4)

(c)

$$h(x,y) = (1+x^2+y^2)e^{1-x^2-y^2}$$ at (1,1), (-1,1), (-1,-1), and (1,-1)

2.

Find the absolute extrema of each of the following functions on the closed bounded set S in the xy-plane.

(a)

f(x,y) = x² - 4xy +y³ + 4y where S is the square region $0 \leq x \leq 2$, $0 \leq y \leq 2$.

(b)

$$f(x,y) = e^{\frac{x^2}{4}-\frac{y^2}{9}}$$ where S is the region $$\frac{x^2}{4} + \frac{y^2}{9} \leq 1$$.

3.

Find and classify the critical points of the following functions as a relative maximum, relative minimum, or saddle point.

(a)

f(x,y) = x³ + y³ - 3x² - 18y² + 81y + 7

(b)

$$f(x,y) = (2xy)e^{-\frac{1}{8}(4x^2+y^2)}$$