practice_midterm2_solutions.pdf

practice_midterm2_solutions.pdf - Sample Midterm Exam 2...

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Sample Midterm Exam 2 — Solutions Math 32A/3, Fall 2014 This is a collection of problems that would not be unreasonable for a real midterm exam. WARNING: the inclusion (or exclusion) of a certain topic or type of problem on this sample exam does not guarantee its inclusion (or exclusion) on the actual exam! 1. Let f ( x, y ) = 1 p x - y 2 . (a) Describe the domain of f (as a set). In the box below, indicate the domain and sketch at least three level curves. [Your sketch only needs to be qualitatively correct.] (b) Let ( a, b ) = (10 , 3). Show that f is differentiable at ( a, b ) and find an equation for the tangent plane above that point. Solution: (a) (Compare with § 15.1 # 5-12, 29-36) The domain is { ( x, y ) | x - y 2 > 0 } , with strict inequality to avoid division by zero. The function is always positive, and we can make it as big as we want by taking y = 0 and x arbitrarily small. Then level curves are of the form c = 1 x - y 2 , where c > 0. This can be rewritten as x = y 2 + 1 c 2 . These curves are therefore parabolas, opening towards the x -axis, shifted to the right by 1 /c 2 . (b) (Compare with § 15.4 # 3-10) The partial derivatives of f are f x ( x, y ) = - 1 2 ( x - y 2 ) - 3 / 2 , f y ( x, y ) = y ( x - y 2 ) - 3 / 2 .

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