# B compute the value of the directional derivative d u

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(b) Compute the value of the directional derivative D u f ( 1,2) when u is the unit vector in the plane that is in the direction of the gradient of f at (-1,2). When u is the unit vector in the plane that is in the direction of the gradient of f at (-1,2), we have D u f ( 1,2) f( 1,2) 5 since . f( x , y ) < 2 x ,1> when f( x , y ) y x 2 Note: To compute the directional derivative in most other directions, you might want to have the unit vector in hand. Here, we don’t need it thanks to an earlier analysis.

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TEST2/MAC2313 Page 4 of 5 ______________________________________________________________________ 7. (10 pts.) (a) Use an appropriate form of chain rule to find z / v when z = sin( x )sin( y ) when x = u + v and y = u 2 - v 2 . z v z x x v z y y v cos( x )sin( y )(1) sin( x )cos( y )( 2 v ) cos( u v )sin( u 2 v 2 ) sin( u v )cos( u 2 v 2 )(2 v ). (b) Assume that F ( x , y , z ) = 0 defines z implicitly as a function of x and y . Show that if F / z 0, then z x F / x F / z . Using classical curly d notation, since x and y are independent variables and z is a function of x and y , we have F ( x , y , z ) x 0 F x x x F y y x F z z x 0 F x F z z x 0 z x F / x F / z . ______________________________________________________________________ 8. (10 pts.). (a) Use limit laws and continuity properties to evaluate the following limit. lim ( x , y ) ( 1/4, π ) ( xy 2 sec 2 ( xy )) ( 1 4 )( π ) 2 sec 2 ( π 4 ) π 2 2 (b) Evaluate the limit, if it exists, by converting to polar coordinates.
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