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MVC SHOW ALL WORK !! Calculator Allowed. Chapter 2 Quiz NAME: #1. (6 pts) Find the equation of the tangent plane to the surface with equation z 3 zy x 2e xy 5 at the point (2, 0,1) . #2. (6 pts) Find the directional derivative of F ( x, y) sin( x 2 y 2 ) in the direction of the vector v i j , at the point , . #3. (5 pts) Write a formula for z if z f ( x, y ) and x x(r , s, t ), y y (r , s, t ). s #4. (8 pts) a. b. Evaluate each of the following limits, or show that it does not exist. lim x , y 0,0 lim e y sin 2 x x ( x , y ) (0,0) x3 y x 2 y 2 x2 y 2 #5(6 pt’s) Prove that if z f ( x, y ) , where x r cos and y r sin , then 2 1 z z z z 2 r x y r 2 2 y x2 if ( x, y ) (0, 0) #6. (5 pts) Find f x (0, 0) if f ( x, y ) x 0 if ( x, y ) (0, 0) 2 xe y and P = (1,0,1). Find u such that Du [ f (1,0,1)] is a maximum. 2z2 3 Evaluate Du [ f (1,0,1)] for this vector u . #7(6 pt’s) Let f ( x, y, z ) #8 (6 pts) a. Find a function z f ( x, y ) and a point (a, b) such that f (a, b) (0, 0) . Be careful to clearly state the point (a, b) . Please give a function f such that f ( x, y ) (0, 0) for all ( x, y ) . b. What is true about your function (or the graph of your function) at the point (a, b) ? Explain. c. Is your answer in b. a general result? That is, if f : your answer in b. always be true? 2 with f (a, b) (0, 0) , will 9. (3 pts) Find a function f : 3 such that 3 5 0 f ( p) 5 1 2 p, 0 2 6 for all p 3