math192prelim2jenn - The derivative of f at Po(xo,yo in the...

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The derivative of f at P o (x o ,y o ) in the direction of the unit vector u=u 1 i+u 2 j is the number df ds u,Po = 0 lim sf 1 2 ( , ) ( , ) o o o o f x su y su f x y s + + - provided the limit exits. The gradient vector (gradient) of f(x,y) at a point P o (x o ,y o ) is the vector df df f dx dy = + i j obtained by evaluating the partial derivatives of f at P o . If the partial derivatives of f(x,y) are defined at P o (x o ,y o ), then , ( ) , o p u Po df f ds = ѷ u the scalar product of the gradient f a P o and u. Properties of the Directional Derivative D u f= cos f f θ = ѷ u 1. The function f increases most rapidly when cos θ =1, or when u is the direction of h f. That is, at each point P in its domain, f increases most rapidly in the direction of the gradient vector f f at P. The derivatives in this direction is D u f = ( 29 cos 0 . f f = 2. Similarly, f decreases most rapidly in the direction of - h f. The derivative in this direction is D u f = ( cos( )) . f f π = - 3. Any direction u orthogonal to the gradient is a direction of a zero change in f because θ then equals / 2 π and D u f= ( 29 cos / 2 0 0 f f π = = ѷ At every point (x o ,y o ) in the domain f(x,y), the gradient of f is normal to the level curve through (x o ,y o ). The tangent plane at the point P o (x o ,y o ,z o ) on the level surface f(x,y,z)=c is the plane through P o normal to o P f f . The normal line of the surface at P o is the line through P o is the line through P o parallel to o p f f .
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