## (solution) Hello, Can you help me solve those questions in detail. Thank you.

Hello,

Can you help me solve those questions in detail. Thank you.

1. Let W R ! R2, .t/ D 2.cos t; sin t/. - Sketch and describe the graph of . 2. Find an approximate value for 

 a ´ 3:99 .4:98/2 sin.4:03?/

by setting up a suitable function f .x; y; z/ and using an appropriate linear approximation of f .

3. Consider the function

f W R 2 ! R; f .x; y/ D 4x 2 C y 2 (a) Plot and describe a contour map of f .  (b) Plot and describe the graph of f .  (c) Plot and describe the intersection of the graph of f with the xz-plane.  (d) Give a function R 2 ! R whose graph is the tangent plane to the graph of f at the point .1; 2; 8/.  (e) Find the direction of maximal increase of f at .1; 2/.  (f) Find the directional derivative of f at .1; 2/ in the direction toward the point .5; 5/.  4. Determine if any of the following functions is harmonic.

(a) f W R 3 ! R, f .x; y; z/ D x 2 C y 2 C z 2 . (b) uW R ! R, u.x; y/ D x

2 3  3xy .  2 5. Find the local maximum and minimum values of

f W D ! R;  f .x; y/ D x 2 C y 2 C 1

;

x2y 2 where D consists of all points .x; y/ in R 2 such that x; y ¤ 0.

6. Find the absolute extrema of 

f .x; y/ D 5 C 4x 2x 2 3y 2 on the region D consisting of all points .x; y/ in R 2 with x 2 C y 2 16.

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