Exam 2

# Exam 2 - =-2 x 2 3 xy y 2-4 x 3 y-1(a(7 pts Find all...

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MATH 241 (Sec. 02): CALCULUS III Department of Mathematics, UMCP Fall 2007 Exam 2 Handed out : Friday, 10/26/07 READ CAREFULLY AND WORK ON ALL PROBLEMS. Justify your answers. If you get stuck to a problem, move on to another one. Cross out what is not part of your ±nal answer. Total time: 50min . 1. (10 pts) Consider the function z = x - 1 e - y/x for x n = 0. Show that z satis±es the equation ∂z ∂x = y 2 z ∂y 2 + ∂z ∂y . 2. (a)(5 pts) Consider the function f ( x, y ) = sin( xy ). Find the direction in which f increases most rapidly at the point ( x 0 , y 0 ) = (1 , π ). What is the maximal directional derivative of f at this point? (b)(5 pts) Now consider the function f ( x, y, z ) = sin( xy ) + cos( xz ). Find an equation for the plane tangent to the level surface f ( x, y, z ) = 0 at the point ( x 0 , y 0 , z 0 ) = ( π/ 4 , 4 , 2). 3. (10 pts) Consider the function f ( x, y
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Unformatted text preview: ) =-2 x 2 + 3 xy + y 2-4 x + 3 y-1. (a)(7 pts) Find all critical points of f . Specify which of these points correspond to relative maximum values, relative minimum values and saddle points of f . Hint: Apply the second partials test. (b)(3 pts) Suppose that f takes values in the region R de±ned by x 2 + y 2 ≤ 4. Explain brie²y why f has a maximum and a minimum value in R . Are such values attained in the interior or at the boundary of R ? Hint: In part (b), you may use your result from (a). Then, the answer to (b) is really short! 4. (10 pts) Consider the function w = f ( y-x, y-z, z-x ). Assume that partial derivatives of f exist. Compute the sum A = ∂w ∂x + ∂w ∂y + ∂w ∂z ....
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## This note was uploaded on 04/08/2008 for the course MATH 241 taught by Professor Wolfe during the Fall '08 term at Maryland.

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