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HH_14_4

# HH_14_4 - Math 10C Lecture Examples Section 14.4...

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(11/11/08) Math 10C. Lecture Examples. Section 14.4. Directional derivatives and gradient vectors in the plane Example 1 (a) Find the directional derivative of f ( x , y ) = x 2 + y 2 at ( 1 , 1 ) in the direction of the unit vector u = ( 1 2 2 , 1 2 2 ) (Figure 1). (b) Why is it plausible that the directional derivative is positive? x 1 2 y 1 2 1 u = (big 1 2 2 , 1 2 2 )big 1 s FIGURE 1 Answer: (a) D u f (1 , 1) = 2 2 (b) f ( x,y ) = x 2 + y 2 is increasing in the direction of u at (1 , 1) in Figure 1 because its graph is a circular paraboloid that opens upward. Example 2 What is the derivative of f ( x , y ) = x 2 y 5 at P = ( 2 , 1 ) in the direction toward Q = ( 4 , 0 ) ? Answer: D u f (2 , 1) = 2 5 Example 3 What is the derivative of h ( x , y ) = e xy at (2,3) in the direction at an angle of 2 3 π radians from the positive x -direction? Answer: Figure A3 u = (− 1 2 , 1 2 3 ) D u h (2 , 3) = ( 3 2 + 3) e 6 x y 2 3 π 1 1 2 1 2 3 u Figure A3 Lecture notes to accompany Section 14.4 of Calculus by Hughes-Hallett et al.

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HH_14_4 - Math 10C Lecture Examples Section 14.4...

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