# a5 - b If the climber walks northeast will she ascend or...

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Math 237 Assignment 5 Due: Friday, Oct 23th 1. Let f ( x, y ) = xy 3 - x 2 y . a) Find the directional derivative of f at (1 , - 2) in the direction of the vector ~v = (3 , 4). b) Is there a direction at (1 , - 2) in which the rate of change of f is equal to 13? Justify your answer. c) Find the direction at (1 , - 2) in which the rate of change of f is the greatest. 2. Let f ( x, y ) = 2 x 2 + xy 3 . a) Find the rate of change of f at the point (1 , 2) in the direction of the vector (1 , - 3). b) In what direction from ( - 2 , 1) does f change most rapidly and what is the maximum rate of change. 3. On a mountain with equation z = 2000 - 0 . 02 x 2 - 0 . 04 y 2 where x , y , and z are in meters, a climber is at the point ( - 20 , 5 , 1991). The positive x -axis points east, and the positive y -axis north. a) If the climber walks due west, will she begin to ascend or descend?
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Unformatted text preview: b) If the climber walks northeast, will she ascend or descend? At what rate? c) In what directions could she begin walking to travel a level path? 4. f ( x, y ) = ± y 3 x 2 + y 2 , if ( x, y ) 6 = (0 , 0) , if ( x, y ) = (0 , 0) . Find the directional derivative of f at (0 , 0) in the direction of (1 , 1). 5. Consider the surface x 2 + y 2-z 2 = 0. Find the equation of the tangent plane to the surface at the point (-3 , 4 , 5). 6. Find the equation of the tangent plane to the surface xz-yz 3 + yz 2 = 2 at (2 ,-1 , 1). 7. Prove that the level curves of the functions f and g deﬁned by f ( x, y ) = y x 2 , x 6 = 0 g ( x, y ) = x 2 + 2 y 2 intersect orthogonally....
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