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practice_midterm2.pdf

practice_midterm2.pdf - Sample Midterm Exam 2 Math 32A/3...

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Sample Midterm Exam 2 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.
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2. (a) Show that the function f ( x, y ) = e y cos x is harmonic: Δ f = f xx + f yy = 0. (b) Find a function z = g ( x, y ) that satisfies g x ( x, y ) = x 2 y = g y ( x, y ), or else show that such a function cannot exist. Justify your answer! [Hint: consider Clairaut’s Theorem.]
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3. Parametrize the osculating circle for the graph of f ( x ) = sin(2 x ) at x = π/ 4. [Hint: you can find N indirectly by using that
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