121616949-math.145 - The linear approximation to y = x 2...

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6.4 Linear Approximations 131 Newton’s method is one example of the usefulness of the tangent line as an approximation to a curve. Here we explore another such application. Recall that the tangent line to f ( x ) at a point x = a is given by L ( x ) = f ( a )( x - a ) + f ( a ). The tangent line in this context is also called the linear approximation to f at a . If f is differentiable at a then L is a good approximation of f so long as x is “not too far” from a . Put another way, if f is differentiable at a then under a microscope f will look very much like a straight line. Figure 6.19 shows a tangent line to y = x 2 at three different magnifications. If we want to approximate f ( b ), because computing it exactly is difficult, we can approximate the value using a linear approximation, provided that we can compute the tangent line at some a close to b . Figure 6.19 The linear approximation to
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Unformatted text preview: The linear approximation to y = x 2 . EXAMPLE 6.22 Let f ( x ) = √ x + 4. Then f ± ( x ) = 1 / (2 √ x + 4). The linear ap-proximation to f at x = 5 is L ( x ) = 1 / (2 √ 5 + 4)( x-5) + √ 5 + 4 = ( x-5) / 6 + 3. As an immediate application we can approximate square roots of numbers near 9 by hand. To estimate √ 10, we substitute 6 into the linear approximation instead of into f ( x ), so √ 6 + 4 ≈ (6-5) / 6 + 3 = 19 / 6 ≈ 3 . 1 6. This rounds to 3 . 17 while the square root of 10 is actually 3 . 16 to two decimal places, so this estimate is only accurate to one decimal place. This is not too surprising, as 10 is really not very close to 9; on the other hand, for many calculations, 3 . 2 would be accurate enough....
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