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Chapter 4 | Applications of Derivatives
4.2 | Linear Approximations and Differentials Learning Objectives 4.2.1 Describe the linear approximation to a function at a point. 4.2.2 Write the linearization of a given function. 4.2.3 Draw a graph that illustrates the use of differentials to approximate the change in a quantity. 4.2.4 Calculate the relative error and percentage error in using a differential approximation. We have just seen how derivatives allow us to compare related quantities that are changing over time. In this section, we examine another application of derivatives: the ability to approximate functions locally by linear functions. Linear functions are the easiest functions with which to work, so they provide a useful tool for approximating function values. In addition, the ideas presented in this section are generalized later in the text when we study how to approximate functions by higher- degree polynomials Introduction to Power Series and Functions (http://cnx.org/content/m53760/latest/) . Linear Approximation of a Function at a Point Consider a function f that is differentiable at a point x = a . Recall that the tangent line to the graph of f at a is given by the equation y = f ( a )+ f ′( a )( x − a ). For example, consider the function f ( x ) = 1 x at a =2. Since f is differentiable at x =2 and f ′( x ) = − 1 x 2 , we see that f ′(2) = − 1 4 . Therefore, the tangent line to the graph of f at a =2 is given by the equation y = 1 2 − 1 4 ( x −2). Figure4.7 (a) shows a graph of f ( x ) = 1 x along with the tangent line to f at x =2. Note that for x near 2, the graph of the tangent line is close to the graph of f . As a result, we can use the equation of the tangent line to approximate f ( x ) for x near 2. For example, if x =2.1, the y value of the corresponding point on the tangent line is y = 1 2 − 1 4(2.1 − 2) = 0.475. The actual value of f (2.1) is given by f (2.1) = 1 2.1 ≈ 0.47619. Therefore, the tangent line gives us a fairly good approximation of f (2.1) ( Figure 4.7 (b)). However, note that for values of x far from 2, the equation of the tangent line does not give us a good approximation. For example, if x =10, the y -value of the corresponding point on the tangent line is y = 1 2 − 1 4 (10−2) = 1 2 −2=−1.5, whereas the value of the function at x =10 is f (10) =0.1.
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