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Unformatted text preview: 4.5: Linear Approximations, Differentials and Newtons Method 2 x cos x 2 2   + 1 ( 29 2 x 0.567 x 0.322 y = + 1 For any function f ( x ), the tangent is a close approximation of the function for some small distance from the tangent point. y x x a = ( 29 ( 29 f x f a = We call the equation of the tangent the linearization of the function. The linearization is the equation of the tangent line, and you can use the old formulas if you like. Start with the point/slope equation: ( 29 1 1 y y m x x = 1 x a = ( 29 1 y f a = ( 29 m f a = ( 29 ( 29 ( 29 y f a f a x a  = ( 29 ( 29 ( 29 y f a f a x a = + ( 29 ( 29 ( 29 ( 29 L x f a f a x a = + linearization of f at a ( 29 ( 29 f x L x is the standard linear approximation of f at a. Linearization Example Find the linear approximation of f(x) = x 2 at x = 1. Use the approximation to 1.1 2 . ( 29 ( 29 ( 29 ( 29 L x f a f a x a = + Linearization Example Find the linear approximation of f(x) = x 2 at x = 1. Use the approximation to 1.1 2 . f x ( 29 = 2 (129 = 1 ( 29 = 2 1 ( 29 = 2 ( 29 = 1 + 2  1 ( 29 1.1 ( 29 = 1 + 2 1.1 1 ( 29 = 1.2 1.1 ( 29 = 1.1 2 = 1.21 ! 2 1.5 1 0.50.5 1 1 2 ( 29 ( 29 ( 29 ( 29 L x f a f a x a = + Linearization Example Use linearization to approximate 123 Linearization Example Use linearization to approximate 123 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 f x = x f 121 = 11 1 1 f x = x f 121 = 2 22 1 L x = 11 + x 121 22 1 1 L 123 = 11 + 123 121 = 11 + = 22 11 and 123 = 11.09 11.09 very cl se! 5 o  Approximating Binomial Powers...
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This note was uploaded on 02/16/2012 for the course CALCULUS 0064 taught by Professor Waldron during the Fall '10 term at Broward College.
 Fall '10
 waldron
 Approximation, Linear Approximation

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