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lecture9 - Lecture 9: Directional Derivatives and the...

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Lecture 9: Directional Derivatives and the Gradient May 19, 2009 Lecture 9: Directional Derivatives and the Gradient
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Objectives 1 Compute the directional derivative of a function. 2 Interpret geometrically the directional derivative. 3 Construct the gradient vector of a multivariable function 4 Use the gradient to maximize the directional derivative. 5 Use the gradient in three dimensions to find tangent planes and normal lines to level surfaces. Lecture 9: Directional Derivatives and the Gradient
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Partial derivatives f x ( x ) = lim h 0 f ( x + h , y ) - f ( x , y ) h f y ( x ) = lim h 0 f ( x , y + h ) - f ( x , y ) h Derivatives in the directions of ˆ i and ˆ j Lecture 9: Directional Derivatives and the Gradient
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Directional derivative Suppose that u = h a , b i is a unit vector. We define the directional derivative D u f ( x 0 , y o ) = lim h 0 f ( x 0 + ah , y 0 + bh ) - f ( x 0 , y 0 ) h Lecture 9: Directional Derivatives and the Gradient
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Directional derivative Suppose that u = h a , b i is a unit vector. We define the directional derivative D u f ( x 0 , y o ) = lim h 0 f ( x 0 + ah , y 0 + bh ) - f ( x 0 , y 0 ) h We compute the directional derivative as follows: D u f ( x 0 , y o ) = f x ( x 0 , y 0 ) a + f y ( x 0 , y 0 ) b Lecture 9: Directional Derivatives and the Gradient
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Directional derivative Suppose that u = h a , b i is a unit vector. We define the directional derivative D u f ( x 0 , y o ) = lim h 0 f ( x 0 + ah , y 0 + bh ) - f ( x 0 , y 0 ) h We compute the directional derivative as follows: D u f ( x 0 , y o ) = f x ( x 0 , y 0 ) a + f y ( x 0 , y 0 ) b IMPORTANT: u must be a unit vector Lecture 9: Directional Derivatives and the Gradient
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Gradient vector If f is a function of two variables, the gradient of f is defined as f ( x , y ) = h f x ( x , y ) , f y ( x , y ) i = f x ˆ i + f y ˆ j Lecture 9: Directional Derivatives and the Gradient
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Gradient vector If f is a function of two variables, the gradient of f is defined as f ( x , y ) = h f x ( x , y ) , f y ( x , y ) i
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lecture9 - Lecture 9: Directional Derivatives and the...

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