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# l_notes09 - Lecture IX Linear Approximation 1 Onevariable...

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Lecture IX Linear Approximation 1 One-variable Functions Let f be a one-variable function on a domain D . If f is differentiable at x = c , we say f has a linear approximation, which we define in the following way. Definition 1 If f is a function differentiable at c , for any x define Δ f = f ( x ) f ( c ) and Δ x = x c . There exist a scalar A c and a function c ( x ) such that Δ f = A c Δ x + c ( x x and lim c ( x ) = 0 . x c Then Δ f A c = f ( c ) and c ( x ) = Δ x f ( c ) . We say that f has a linear approximation at c and that the expression Δ f app = f ( c x is a linear approximation formula for f at c . The function c is called the relative error function for f at c . 2 Multivariable Functions Definition 2 Let f be a multilinear function and let P be a point in its domain D in E 2 . Define Δ f = f ( a + Δ x, b + Δ y ) f ( a, b ) . We say that f has linear approximation at P if there exist scalars A , B and functions 1 , 2 such that: 1. for all Q D , Δ f = A Δ x + B Δ y + 1 ( Q x + 2 ( Q y . 2. lim Q P 1 ( Q ) = 0 and lim Q P 2 ( Q ) = 0 . Theorem 1 If f has a linear approximation at P , then f has partial derivatives at P and ∂f ∂y | ∂x | P = A , ∂f P = B . 1

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Theorem 2 If f has continuous
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l_notes09 - Lecture IX Linear Approximation 1 Onevariable...

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