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2220hw1

# 2220hw1 - z = 4 cos xy at the point π 3 1 2 32 Find...

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Math 2220 Problem Set 1 Spring 2010 Section 2.1 : In the following two problems you are asked to do two things: (a) determine several level curves f ( x, y ) = c , being sure to label each curve with the appropriate value of c , and (b) sketch the graph, using the information from (a). 12. f ( x, y ) = x 2 + y 2 - 9. 18. f ( x, y ) = 3 - 2 x - y . Section 2.2 : In the next four problems, evaluate the limit if it exists, or explain why the limit does not exist. 8. lim ( x,y ) (0 , 0) | y | radicalbig x 2 + y 2 14 . lim ( x,y ) (0 , 0) xy x 2 + y 2 15. lim ( x,y ) (0 , 0) x 4 - y 4 x 2 + y 2 16 . lim ( x,y ) (0 , 0) x 2 x 2 + y 2 Section 2.3 : 2. Calculate ∂f/∂x and ∂f/∂y for f ( x, y ) = e x 2 + y 2 . 4. Calculate ∂f/∂x and ∂f/∂y for f ( x, y ) = x 3 - y 2 1 + x 2 + 3 y 4 . 8. Calculate ∂F/∂x , ∂F/∂y , and ∂F/∂z for F ( x, y, z ) = xyz . 12. Do the same for F ( x, y, z ) = sin x 2 y 3 z 4 . 24. Find the matrix D f ( a ) of partial derivatives for f ( x, y ) = ( x 2 y, x + y 2 , cos πxy ) at a = (2 , - 1). 30. Find an equation for the plane tangent to the graph of
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Unformatted text preview: z = 4 cos xy at the point ( π/ 3 , 1 , 2). 32. Find equations for the planes tangent to z = x 2-6 x + y 3 that are parallel to the plane 4 x-12 y + z = 7. 34. Suppose you have the following information about a di±erentiable function f ( x, y ): f (2 , 3) = 12, f (1 . 98 , 3) = 12 . 1, and f (2 , 3 . 01) = 12 . 2. (a) Give an approximate equation for the plane tangent to the graph of f at (2 , 3 , 12). (b) Use the result of part (a) to estimate f (1 . 98 , 2 . 98). 36. (a) Use a linear function h ( x, y ) to approximate the value of f ( x, y ) = 3 + cos πxy at ( x, y ) = (0 . 98 , . 51). (b) How accurate is the approximation?...
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