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MVC SHOW ALL WORK !! Calculator Allowed. Quiz #3 NAME: 1. Let f ( x, y) x3 cos( xy) a. Find f x ( x, y ), f y ( x, y ), and f xy ( x, y ) b. Evaluate f 1, . 2 2. Evaluate the following limit or show that it does not exist. x2 y a. Lim 2 x , y 0,0 x y 2 b. MVC x2 y x , y 0,0 x 4 y 2 Lim x 3 y 4 if ( x, y ) (0,0) 3. Let f ( x, y ) x 2 y 2 . Find f x (0,0) and f y (0,0) or show that they do if ( x, y) (0,0) 0 not exist.. 4. Find the directional derivative of F ( x, y, z) v 5. MVC 1 3 Evaluate i 1 3 j 1 3 x y z in the direction of the vector k , at the point 1,2,5 . w at the point (r , s) (1, 1) if w x 2 y y 2 z and x 2r s, y r 2s, and z 2rs . s 6. Find a tangent plane to the surface z xy 2 yx 2 at the point (2,1,6) . 7. Find a point on the paraboloid 2 x z 2 3 y 2 0 at which the tangent plane is parallel to the plane 5 x y z 1 8. Prove that if z f ( x, y), with x(s, t ) s 2 t 2 and y(s, t ) t 2 s 2 , then f f t s 0 s t MVC 9. For the contour map for z f ( x, y ) shown below, estimate each of the following quantities. Explain briefly how you are getting your answer. (a) f x (2, 1) and f y (2, 1) (b) f (2, 1) (c) Du f (2, 1) , where u is a unit vector in the direction of f (2, 1) (d) Sketch the vector f (2, 1) on the contour map for f using (2, 1) as the initial point. (e) Sketch a unit vector v with initial point (2, 1) such that Dv f (2, 1) 0 . 7 5 3 1 -1 -3 -5 3 MVC Concepts: 10. Let f : 2 3 be given by f ( x1 , x2 ) ( x1 x2 , x1 x2 , x12 x2 ) and let x : by x (t1 , t2 , t3 ) (t1t2 , t ) . 2 1 a) Write an explicit formula for x f b) Find the derivative, D x f . MVC 3 2 be given