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  • Describe the linear approximation to a function at a point.
  • Write the linearization of a given function.
  • Draw a graph that illustrates the use of differentials to approximate the change in a quantity.
  • 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 .

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 ) .

[link] (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 ) ( [link] (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 .

This figure has two parts a and b. In figure a, the line f(x) = 1/x is shown with its tangent line at x = 2. In figure b, the area near the tangent point is blown up to show how good of an approximation the tangent is near x = 2.
(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

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 .

Linear approximation of x

Find the linear approximation of f ( x ) = x at x = 9 and use the approximation to estimate 9.1 .

Since we are looking for the linear approximation at x = 9 , using [link] 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 ) .

f ( x ) = x f ( 9 ) = 9 = 3 f ( x ) = 1 2 x f ( 9 ) = 1 2 9 = 1 6

Therefore, the linear approximation is given by [link] .

L ( x ) = 3 + 1 6 ( x 9 )

Using the linear approximation, we can estimate 9.1 by writing

9.1 = f ( 9.1 ) L ( 9.1 ) = 3 + 1 6 ( 9.1 9 ) 3.0167 .
The function f(x) = the square root of x is shown with its tangent at (9, 3). The tangent appears to be a very good approximation from x = 6 to x = 12.
The local linear approximation to f ( x ) = x at x = 9 provides an approximation to f for x near 9.
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Practice Key Terms 7

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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