linearization

linearization - LINEAR APPROXIMATION HOMEWORK: 13.7: 12,...

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LINEAR APPROXIMATION Math21a, O. Knill HOMEWORK: 13.7: 12, 16, 30, 34, 38 LINEAR APPROXIMATION. 1D: The linear approximation of a function f ( x ) at a point x 0 is the linear function L ( x ) = f ( x 0 ) + f ( x 0 )( x x 0 ) . The graph of L is tangent to the graph of f at x 0 . 2D: The linear approximation of a function f ( x, y ) at ( x 0 , y 0 ) is L ( x, y ) = f ( x 0 , y 0 ) + f x ( x 0 , y 0 )( x x 0 ) + f y ( x 0 , y 0 )( y y 0 ) The level curve of g is tangent to the level curve of f at ( x 0 , y 0 ). The graph of L is tangent to the graph of f . 3D: The linear approximation of a function f ( x, y, z ) at ( x 0 , y 0 , z 0 ) by L ( x, y, z ) = f ( x 0 , y 0 , z 0 ) + f x ( x 0 , y 0 , z 0 )( x x 0 ) + f y ( x 0 , y 0 , z 0 )( y y 0 ) + f z ( x 0 , y 0 , z 0 )( z z 0 ) . The level surface of L is tangent to the level surface of f at ( x 0 , y 0 , z 0 ). Using f = a f x , f y A , the linearization can be written as L ( vx ) = f ( vx 0 ) + f ( vx 0 ) · ( vx vx 0 ) HOW CAN IT BE USED? Linearization is important because linear functions are easier to deal with. Using linearization, one can estimate function values near known points. JUSTIFYING THE LINEAR APPROXIMATION. If the second variable y = y 0 is ±xed, then we have a one-dimensional situation where the only variable is x . Now f ( x, y 0 ) = f ( x 0 , y 0 ) + f x ( x 0 , y 0 )( x x 0 ) is the linear approximation. Similarly, if x = x 0 is ±xed y is the single variable, then f ( x 0 , y ) = f ( x 0 , y 0 ) + f y ( x 0 , y
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