# 53_mt2_smpl - by the equation x 4 ax 2 + b = 0. 6. Find and...

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Practice Exam 1. Derive Kepler’s 3rd law: The squares of periods of revolution of planets around the Sun are proportional to the cubes of the major semiaxes of their elliptic orbits. 2. Give an example of a function f with f xy n = f yx . 3. Find all di±erentiable functions f ( x, y ) such that f x = y exp xy . 4. Compute the gradient of the function 1 / | v r | . 5. Find all values and partial derivatives at the point ( a, b ) = (5 , 4) of the function x ( a, b ) de²ned implicitly
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Unformatted text preview: by the equation x 4 ax 2 + b = 0. 6. Find and classify all critical points of the function f = xy x 3 / 3 y 3 / 3. 7. Find the global maximum and minimum values of the function x 2 + y 2 in the region 1 x 3 , 2 y 4 . 8. Find all critical points and critical values of the function x + y + z under the constraint 2 xy 2 z 3 = 1....
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