Question

1. Consider the function.

(a) Draw the level curves of this function for levels c = 0, 1, 2. Please clearly label each

level curve with the appropriate value of c.

(b) Use the previous answer to sketch the graph

(c) Find all first and second order derivatives of this function. (Please label all your derivatives clearly.)

(d) Find the equation of the tangent plane to

2.. Let $f(x,y)=y^2/(x^2+y^2)$

(a) Show that does not exist.

(b) Show that does exist and find the limit.

f(x,y) = f(x, y)at(Z, y) = (2. 1) ( ) lim fx, y lim yf(x, y)

Answer 1

The sketch of the level curves are shown below: (b) The graph of f(x, y)-(y-2x) is shown below (c) Differentiate f(x, y) partially to get 左=-4(y-2x),左=8.fi =2(yー2x).ふ=2,寿=-4 (d) At (2,1) f 2,1(1-4)-12 (2,1) 2(1-4)-6 So the equation tangent at (2,1) is --f(2,1)=시2,1)(x-2)+ (2,1)(y-1) -9 12(x 2)-6(v-1) z-9 12x 24-6y +6 12x-61 9