Errors in approximations Suppose f(x) = 1/(1 + x) is to be

Chapter 4, Problem 46E

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QUESTION:

Suppose f(x) = 1/(1 + x) is to be approximated near x = 0. Find the linear approximation to f at x = 0. Then, complete the following table showing the errors in various approximations. Use a calculator to obtain the exact values. The percent error is \(100 \cdot \mid \text { approximation }-\text { exact }|/| \text { exact } \mid\). Comment on the behavior of the errors as x approaches 0.

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QUESTION:

Suppose f(x) = 1/(1 + x) is to be approximated near x = 0. Find the linear approximation to f at x = 0. Then, complete the following table showing the errors in various approximations. Use a calculator to obtain the exact values. The percent error is \(100 \cdot \mid \text { approximation }-\text { exact }|/| \text { exact } \mid\). Comment on the behavior of the errors as x approaches 0.

ANSWER:

SOLUTION STEP 1 .The linear approximation formula is given by L(x) = f(a) + f(a)(x a) Given f(x) = 1+x and a = 0. f(a) = 1 = 1 1+0 1 1 Then f x) = (1+x) f(a) = 1 = 1 Therefore L(x) = 1 1(x 0) = 1

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