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MVC SHOW ALL WORK !! Quiz #3 NAME: 1. Let f ( x, y) ( x 2 y)5 a. Find f x ( x, y ), f y ( x, y ), and f xy ( x, y ) b. Evaluate f (1, 0) . c. Find the equation of the tangent plane to the graph of f at (1,0). 2. Evaluate each of the following limits, or show that it does not exist. x2 y a. Lim 4 x , y 0,0 x y 3 b. MVC Lim x , y 0,0 x 2 y 2 Ln x2 y 2 3. 4. x2 4 y 2 if ( x, y) (0,0) Let G( x, y) y ( x 2 y ) . 0 if ( x, y ) (0,0) a. Determine with explanation if G is continuous at (0,0) . b. Find Gx (0,0) or show that it does not exist. Find the directional derivative of F ( x, y) x 2 xy in the direction of the vector v 3i 4 j , at the point 1,3. MVC 5. Suppose that you have a function f : ( x, y ) (1, 2) (1.02, 2) (1,1.99) 2 with table of values given below: f ( x, y ) 4 4.1 4.3 a. Estimate the values of f x (1,2) and f y (1, 2) . b. Give an approximation for the tangent plane to the graph of f at (1,2). c. Use the result from b. to approximate f (1.01, 2.01) d. Under what conditions on f is the approximation from b. a “good” approximation? Explain. 6. MVC Write a formula for w if w f ( x, y, z ) and x x(r , s ), y y (r , s ), and z z (r , s ) . s u w and y (u , v, w) , then v v z w z u v x 0 u v w 7. Prove that if z f ( x, y ), with x(u, v, w) 8. Let f ( x, y) x e2 y and P = (0,1,1). Find u such that Du [f(0,1,1)] is a maximum. Evaluate Du [f(0,1,1)] for this vector u . 9. Find a tangent plane to the surface which is defined implicitly by the equation xy 2 yz 2 zx 2 3 at the point (1, 2,1) . MVC 10. The figure below shows the level curves of a function z k x, y . In this plot, the z-values increase as you move up and to the right. a. b. On this contour plot, sketch vector v with initial point (2, 2) in the same direction as k 2, 2 . On this contour plot, sketch a vector w with initial point (2, 2) so that Dw k 2, 2 0 . That is, the directional derivative at (2, 2) in the direction of w is 0. k k c. For this contour plot, which has the greater value at the point (3, 1), or ? x y Briefly explain your choice. MVC