   Chapter 11.11, Problem 24E

Chapter
Section
Textbook Problem

# Use the information from Exercise 16 to estimate sin 38° correct to five decimal places.13–22 (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor’s Inequality to estimate the accuracy of the approximation f(x) ≈ Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 16. f(x) = sin x, a = π/6, n = 4, 0 ≤ x ≤ π/3

To determine

To estimate: The value of sin38° correct to five decimal places.

Explanation

Formula used:

Taylor’s inequality:

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

Calculation:

The given function is f(x)=sinx, centered at a=π6.

Then, f(π6)=12.

From Exercise 16, the Taylor polynomial with degree 4 at a=π6 is f(x)=12+32(xπ6)14(xπ6)2312(xπ6)3+148(xπ6)4.

The fifth derivative is f(5)(x)=cosx.

Use Taylor’s inequality for n=4 and a=π6.

Thus, the Taylor series becomes, if |f(5)(x)|M, then |R4(x)|M5!|xπ6|5.

|f(5)(x)|=|cosx|M=1

Therefore, take the value of M as 1.

x=38°=30°+8°=π6+2π45=19π90

Substitute the value of M in the Taylor’s inequality

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