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12.3 Partial Derivatives

12.3 Partial Derivatives - 1 2.3 P a r t ia l D e r iv a t...

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12.3 Partial Derivatives Below is a chart of elevation above sea level at any point (x, y) in feet. (+x = East, +y = North) x y ‐4 ‐2 0 2 4 ‐6 152 193 209 193 152 ‐4 186 236 256 236 186 ‐2 209 266 288 266 209 0 218 277 300 277 218 2 209 266 288 266 209 4 186 236 256 236 186 6 152 193 209 193 152 What is the slope going north at the point (0, ‐2)? Slope = change in elevation change in y What is the slope going east?

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For better approximations, find values closer to (0, ‐2). x y ‐2.2 ‐2 ‐1.8 ‐.2 271.8 278 280.2 0 270.8 277 279.2 .2 269.8 276 278.2 Slope north: Slope east: Def: For z = f(x, y) the rate of change of f at (a, b) in the +x direction (with y constant) is the partial derivative of f with respect to x. f x = f x ( a , b ) = lim h 0 f ( a + h , b ) f ( a , b ) h if the limit exists.
Similarly the partial derivative of f with respect to y is: f y = f y ( a , b ) = lim h 0 f ( a , b + h ) f ( a , b ) h if the limit exists. So from our chart f x 0, 2 ( ) = f 2, 2 ( ) f (0, 2) 2 f x 0, 2 ( ) = f 0.2, 2 ( ) f (0, 2) 0.2 GeometricInterpretation

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We can use contours to estimate f x and f y . Ex. F(x,y) = 2x + y Ex. F(x, y) = x 2 + y 2 - 3 - 2 - 1 0 1 2 3 - 3 - 2 - 1 0 1 2 3 - 2 - 1 0 1 2 - 2 - 1

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