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# sn7_1 - Math 2433 Week 7 Popper007 1 Have you signed up for...

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Math 2433 – Week 7 Popper007 1. Have you signed up for a time for your midterm yet? a. YES b. NO (then go do it NOW and come back and choose A) 2. Calculate the partial derivatives. a) b) c) d) e) 15.1 - DIFFERENTIABILITY AND GRADIENT We say that f is differentiable at x if there exists a vector y such that f ( x + h ) - f ( x ) = y i h + o ( h ). We will say that g ( h ) is o ( h ) if 0 (h) lim 0 h h g = Example: For 2 ( , ) 3 f x y x y = + :

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Let f be differentiable at x . The gradient of f at x is the unique vector f ( x ) such that f ( x + h ) - f ( x ) = f ( x ) i h + o ( h ). Continuing the previous example: More examples:

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15.2 Gradients and Directional Derivatives Properties of gradients: Directional Derivatives: Note that u f gives the rate of change of f in the direction of u . And
Example: 1. Find the directional derivative at the point P in the direction indicated. 2 2 ( , ) 3 at P(1,1) in the direction of f x y x y = + i j

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Note that the directional derivative in a direction u is the component of the gradient vector in that direction.
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