53 Worksheet 9_28 Solutions

53 Worksheet 9_28 Solutions - Math 53: Multivariable...

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Unformatted text preview: Math 53: Multivariable Calculus Solutions for Worksheet 9/28/11: Derivatives EVERYWAY! (Also gradients and things) Exercise 0.1. Find the gradient of f ( x,y ) = sin(2 x + 3 y ), first in general and then at the point (- 6 , 4). Find the rate of change of f at (- 6 , 4) in the direction u = 1 2 ( 3 i + j ) . Solution. The partial derivatives of f are f x ( x,y ) = 2 cos(2 x + 3 y ) and f y ( x,y ) = 3 cos(2 x + 3 y ) , so the gradient is 5 f ( x,y ) = h 2 cos(2 x + 3 y ) , 3 cos(2 x + 3 y ) i . At the point (- 6 , 4), we have 5 f (- 6 , 4) = h 2 cos(- 12 + 12) , 3 cos(- 12 + 12) i = h 2 , 3 i . The rate of change of f at (- 6 , 4) in the direction u = D 3 2 , 1 2 E (which is a unit vector) is D u f (- 6 , 4) = 5 f (- 6 , 4) u = h 2 , 3 i * 3 2 , 1 2 + = 2 3 2 +3 1 2 = 3+ 3 2 . Exercise 0.2. Find the maximum rate of change of f ( x,y,z ) = tan( x + 2 y + 3 z ) at (- 5 , 1 , 1) and the direction in which it occurs....
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53 Worksheet 9_28 Solutions - Math 53: Multivariable...

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