   Chapter 11.11, Problem 28E

Chapter
Section
Textbook Problem

# Use the Alternating Series Estimation Theorem or Taylor’s inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically.28. cos x ≈ 1 − x 2 2 + x 4 24   ( | error |   < 0.005 )

To determine

To estimate: The range of values of x for which cosx1x22+x424 is accurate within the error 0.005 and check the answer graphically.

Explanation

Formula used:

Taylor’s inequality:

If |f(n+1)(x)|M, then |Rn(x)|M(n+1)!|xa|n+1.

Calculation:

Generally, cosx=1x22+x44!+.

Since the maximum value of cos function is 1, the value of M=1.

The Maclaurin series is a Taylor series at a=0.

The approximation is given as, cosx1x22+x424.

The Maclaurin series of cosine function has zero on the odd terms.

Up to 4th term is specified in the given approximation.

Since fifth term is zero, consider the sixth term.

That is, use n=5 in Taylor’s inequality as follows.

|Rn(x)|M(n+1)!|xa|n+1|

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