prac2C

prac2C - u and v . c) (3) Find xw x + yw y in case w = v 5...

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18.02 Practice Exam 2 B Problem 1. Let f ( x, y ) = x 2 y 2 - x . a) (5) Find f at (2 , 1) b) (5) Write the equation for the tangent plane to the graph of f at (2 , 1 , 2). c) (5) Use a linear approximation to ±nd the approximate value of f (1 . 9 , 1 . 1). d) (5) Find the directional derivative of f at (2 , 1) in the direction of - ˆ ı . Problem 2. (10) On the contour plot below, mark the portion of the level curve f = 2000 on which ∂f ∂y 0. Problem 3. a) (10) Find the critical points of w = - 3 x 2 - 4 xy - y 2 - 12 y + 16 x and say what type each critical point is. b) (10) Find the point of the ±rst quadrant x 0, y 0 at which w is largest. Justify your answer. Problem 4. Let u = y/x , v = x 2 + y 2 , w = w ( u, v ). a) (10) Express the partial derivatives w x and w y in terms of w u and w v (and x and y ). b) (7) Express xw x + yw y in terms of w u and w v . Write the coe²cients as functions of
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Unformatted text preview: u and v . c) (3) Find xw x + yw y in case w = v 5 . Problem 5. a) (10) Find the Lagrange multiplier equations for the point of the surface x 4 + y 4 + z 4 + xy + yz + zx = 6 at which x is largest. (Do not solve.) b) (5) Given that x is largest at the point ( x , y , z ), ±nd the equation for the tangent plane to the surface at that point. Problem 6. Suppose that x 2 + y 3-z 4 = 1 and z 3 + zx + xy = 3. a) (8) Take the total di³erential of each of these equations. b) (7) The two surfaces in part (a) intersect in a curve along which y is a function of x . Find dy/dx at ( x, y, z ) = (1 , 1 , 1)....
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This note was uploaded on 04/18/2008 for the course 18 18.02 taught by Professor Auroux during the Fall '08 term at MIT.

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