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Unformatted text preview: Math 20C — Second Midterm Solutions (Version #1) Problem 1. Let f ( x,y ) = x 2 + y 3 . (a) Compute the gradient vector ∇ f as a function of x and y . We have f x = 2 x and f y = 3 y 2 , so ∇ f = h 2 x, 3 y 2 i . (b) Find an equation of the tangent plane to the graph of f at the point (3 , 1 , 10) . The equation of a tangent plane at ( a,b,c ) is z = c + f x ( a,b ) · ( x a )+ f y ( a,b ) · ( y b ), so here we have z = 10 + 6( x 3) + 3( y 1) = 6 x + 3 y 11 . (c) Compute an approximate value for f (3 . 01 , . 99) . Using the equation of the tangent plane, which is an approximation to f , we have f (3 . 01 , . 99) ≈ 10 + 6(0 . 01) + 3( . 01) = 10 + 0 . 06 . 03 = 10 . 03 . Problem 2. Consider the following contour map of a function f , with its contour lines labeled by the corresponding function values. A B C E D 16 16 16 20 20 20 24 28 12 12 16 16 20 20 24 24 12 1 (a) The function f has critical points at A , B , and C . For each of these points, say whether it is a local maximum, a local minimum, or a saddle point. A and B are local maxima, since they are in the center of concentric contour lines and any direction you go from them is down. C...
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 Fall '07
 BoonYeap
 Derivative, fxy, fxx, fyy

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