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.
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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:
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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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