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Unformatted text preview: LINEAR APPROXIMATIONS AND DIFFERENTIALS Suppose that we have a curve y = f ( x ). The tangent line at ( a,f ( a )) is given by y- f ( a ) = f ( a )( x- a ) . If we zoom the point, we can see that the graph looks like its tangent line. This observation suggests that finding a tangent line may be a method of finding ap- proximate values of functions. The approximation f ( x ) f ( a ) + f ( a )( x- a ) is called the linear approximation of f at a . The linear function L ( x ) = f ( a ) + f ( a )( x- a ) is called the linearization of f at a . Note that L ( a ) = f ( a ) and L ( a ) = f ( a ). Example 1. Find the linearization of the function f ( x ) = x at a = 4 and use it to approximate the numbers 5 and 3 . 99. Solution L ( x ) = f (4) + f (4)( x- 4) = 2 + 1 4 ( x- 4) Thus, when x is near 4, x x 4 + 1 . This suggests 5 2 . 25 and 3 . 99 1 . 9975. Actual values are 5 = 2 . 236 and 3 . 99 = 1 . 997498....
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