2224-Sec12_5-HWT

2224-Sec12_5-HWT - Mat h 22 24 Multiva ri a ble C alc Se c....

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I. Directional Derivatives A. What if you want to calculate the slope at any point moving in any direction, not just in the direction of x or y . 1. Find the slope moving in NE direction from (4,-6) -8 -6 -4 2 80 140 209 4 71 124 186 6 58 102 152 2. B. Def n : The derivative of f at P 0 ( x 0 , y 0 ) in the direction of the unit vector ! u = u 1 ! i + u 2 ! j is the number ( D ! u f ) P 0 = df ds ! " # $ % ! u , P 0 = lim s 0 f ( x 0 + su 1 , y 0 + su 2 ) ( f ( x 0 , y 0 ) s ! " # $ % , provided the limit exists. C. Theorem: If f is a differentiable function of x and y , then f has a directional derivative in the direction of any unit vector ! u = u 1 ! i + u 2 ! j and D ! u f ( x , y ) = f x ( x , y ) u 1 + f y ( x , y ) u 2 D. Example: Find the directional derivative D ! u f ( x , y ) if f ( x , y ) = x 3 ! 3 xy + 4 y 2 in the direction of ! v
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2224-Sec12_5-HWT - Mat h 22 24 Multiva ri a ble C alc Se c....

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