Example use the gradient to approximate the point on the curve x x 2y 1 9 closest to the point 1 1 This is not a lagrange multiplier problem we want to bring f down while moving as little as possible meaning we want to move in the direction f degreases fastest Lecture 17 Topics Anything up to but not including double integrals Gradient f x y z Chain rule Tangent plane we need a point and a normal The gradient of f gives a normal Example Directional derivative Example Linear approximation Minimizing and maximizing Minimize f 2x 3y z such that g x y z 1 0 x y z 0 Solution one solve Critical points are where Solution two use legrange multiplier We know this is a minimum because 6 given that in nity is the value of the boundary Taking the derivative while keeping variables constant Find Where do f x f y 0
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