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Errors in approximations Suppose is to be approximated
Chapter 4, Problem 45E(choose chapter or problem)
Suppose \(f(x)=\sqrt[3]{x}\) is to be approximated near x = 8. Find the linear approximation to f at x = 8. 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 8.
Questions & Answers
QUESTION:
Suppose \(f(x)=\sqrt[3]{x}\) is to be approximated near x = 8. Find the linear approximation to f at x = 8. 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 8.
ANSWER:
SOLUTION STEP 1 .The linear approximation formula is given by y = L(x) = f(a)+f (a)(xa) Given f(x) = and a = 8. 3 f(a) = 8 = 2 Then f (x) = 12 (a) = 12 = 1 3x3 3(8) 12 Therefore 1 L(x) = 2+ (1