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Unformatted text preview: 14. Linearization and Differentials P. K. Lamm 10/13/11 (22:05) p. 1 / 6 Lecture Notes: 14. Linearization and Differentials These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Linear Approximation to a Curve Suppose we have a curve y = f ( x ) where f is differentiable at x . The tangent line to the curve at the point ( x ,f ( x )) has slope f ( x ) , with equation given by y- f ( x ) = f ( x )( x- x ) . or y = f ( x ) + f ( x )( x- x ) . For x ≈ x , we expect the y-value of the curve to be close to the y-value of the tangent line, or f ( x ) ≈ f ( x ) + f ( x )( x- x ) , for x “near” x . (1) We call the function L ( x ) = f ( x ) + f ( x )( x- x ) the linear approximation to y = f ( x ) for x at x . It is nothing more than the y-value of the tangent line to the curve at ( x ,f ( x )). 1 14. Linearization and Differentials P. K. Lamm 10/13/11 (22:05) p. 2 / 6 Question: What do we mean when we say that x must be “near” x ? The answer depends on the function f ( x ) and the point x , as the following graphs show. Obviously in all cases, the closer x is to x the better the linear approximation can be expected to be. Example 1.1: Find the linear approximation of y = sin x at x = 0. Here f ( x ) = sin x, x = 0 . Since f ( x ) = cos x , the linear approximation is L ( x ) = f (0) + f (0)( x- 0) = sin0 + cos0 · ( x- 0) = 0 + 1 · x = x Thus, f ( x ) = sin x ≈ x, for x near 0 ....
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.
- Fall '10