Lecture19 - tangent line approximation Examples 1 Find L x at x = 1 for f x = x 1 x 2 Find L x at x = 2 for f x = x 3-2 x 3 and use this to

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3.10 Linearization Consider the graph of y = x 2 . At the point (1 , 1), the tangent line is y = 2 x - 1. What do you notice about the tangent line and the graph close to (1 , 1)? The tangent line and the graph are almost equal at points close to (1 , 1). Could we use the tangent line to approximate the function? Linearization of y = f ( x ) If y = f ( x ) is differentiable at x = a , then the approximating function is and is called the linearization of f at a . The approximation of f by L is the standard linear approximation of f at a . The point x = a is the center of the approximation. The error of the approximation is found by computing: 1
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Remark: If f is differentiable at a , then the tangent line to f at ( a, f ( a )) is y = f ( a ) + f 0 ( a )( x - a ). Thus the linearization is sometimes called the
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Unformatted text preview: tangent line approximation . Examples 1. Find L ( x ) at x = 1 for f ( x ) = x + 1 x . 2. Find L ( x ) at x = 2 for f ( x ) = x 3-2 x + 3 and use this to estimate f (2 . 01). Find the error of the approximation. 2 3. Find a linearization to f ( x ) = sin-1 x which would be valid for x = π/ 12 4. Find the linearization of f ( x ) = √ 1 + x + sin x and use it to estimate f (0 . 01). Roots and Powers–Special Case at x=0 An important linear approximation for roots and powers at x = 0 is: Proof: 3 Example: Find L ( x ) near x = 0 for f ( x ) = (1-x ) 6 and g ( x ) = (4 + 3 x ) 1 / 3 4...
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This note was uploaded on 03/31/2008 for the course MATH 1205 taught by Professor Fbhinkelmann during the Spring '08 term at Virginia Tech.

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Lecture19 - tangent line approximation Examples 1 Find L x at x = 1 for f x = x 1 x 2 Find L x at x = 2 for f x = x 3-2 x 3 and use this to

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