# Calc04_5 - 4.5 Linear Approximations Differentials and...

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4.5: Linear Approximations, Differentials and Newton’s Method Greg Kelly, Hanford High School, Richland, Washington

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For any function f ( x ), the tangent is a close approximation of the function for some small distance from the tangent point. y x 0 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.

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Important linearizations for x near zero: ( 29 1 k x + 1 kx + sin x cos x tan x x 1 x ( 29 1 2 1 1 1 1 2 x x x + = + + ( 29 ( 29 1 3 4 4 3 4 4 1 5 1 5 1 5 1 5 1 3 3 x x x x + = + + = + ( 29 f x ( 29 L x This formula also leads to non-linear approximations:
Differentials: When we first started to talk about derivatives, we said that becomes when the change in x and change in y become very small.

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