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Unformatted text preview: 4.5: Linear Approximations, Differentials and Newton’s Method 2 x cos x 2 2 ξσιν ξ  σιν ξ  χοσξ ξ + 1 ( 29 2 x ≈ 0.567 x ≈0.322 y = ξ+ 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 = ξ 2 φ(129 = 1 ′ φ ξ ( 29 = 2ξ ′ φ 1 ( 29 = 2 Λ ξ ( 29 = 1 + 2 ξ  1 ( 29 Λ 1.1 ( 29 = 1 + 2 1.1 1 ( 29 = 1.2 φ 1.1 ( 29 = 1.1 2 = 1.21 σο ουραππροξιμ ατιον ισωερψχλοσε! 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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 Fall '10
 waldron
 Calculus, Numerical Analysis, Approximation, Linear Approximation, Isaac Newton, Rootfinding algorithm

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