Chapter 4 | Applications of Derivatives
355
Figure 4.7 (a) The tangent line to f ( x ) =1/ x at x =2 provides a good approximation to f for x near 2. (b) At x =2.1, the value of y on the tangent line to f ( x ) =1/ x is 0.475. The actual value of f (2.1) is 1/2.1, which is approximately 0.47619.
In general, for a differentiable function f , the equation of the tangent line to f at x = a can be used to approximate f ( x ) for x near a . Therefore, we can write f ( x ) ≈ f ( a )+ f ′( a )( x − a ) for x near a . We call the linear function (4.1) L ( x ) = f ( a )+ f ′( a )( x − a ) the linear approximation , or tangent line approximation , of f at x = a . This function L is also known as the linearization of f at x = a . To show how useful the linear approximation can be, we look at how to find the linear approximation for f ( x ) = x at x =9. Example 4.5 Linear Approximation of x
Find the linear approximation of f ( x ) = x at x =9 and use the approximation to estimate 9.1.
Solution Since we are looking for the linear approximation at x =9, using Equation 4.1 we know the linear approximation is given by L ( x ) = f (9)+ f ′(9)( x −9). We need to find f (9) and f ′(9).
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