Chapter 10.3, Problem 45E

(a)

To determine

The approximation of P+sin P

Answer to Problem 45E

This is accurate to more than 6 decimals places

Explanation of Solution

Given information:

Let P be an approximation of $\pi$ accurate to n decimal places. Check with a calculator to see that P + sin P gives an approximation correct to at least 3n decimal places!

Formula used:

Sin value is calculated.

Calculation:

Let’s take approximate value of $\pi$ accurate to two decimal places. That is, p = 3. 14. Therefore,

  $P+\sin P=3.141592653$

This is accurate to more than 6 decimals places

Conclusion:

This is accurate to more than 6 decimals places

(b)

To determine

the value of x using theRemainder Estimation Theorem and the Maclaurin series

Answer to Problem 45E

The value is $\left(P+\sin P\right)-\pi <\frac{{\left|x\right|}^3}{6}$

Explanation of Solution

Given information:

Use the Remainder Estimation Theorem and the Maclaurin series for sin x

Formula used:

  $P+\sin P=\left(\pi +x\right)+\sin \left(\pi +x\right)$

Calculation:

Let P = g + x where x the error is in the original estimate

Then,

  $\begin{array}{l}P+\sin P=\left(\pi +x\right)+\sin \left(\pi +x\right)\\ P+\sin P=\pi +x-\sin x\\ \left(P+\sin P\right)-\pi =x-\sin x\mathrm{.............}\left(1\right)\end{array}$

The given function is $f\left(x\right)=\sin x$

Then,

  $P_1\left(x\right)=x$

Using Remainder Estimation Theorem for n = 2

(Use n = 2 instead n = 1 in this way better range of x)

Since,

  $P_2\left(x\right)=x$

Now,

  $f^{\left(3\right)}\left(x\right)=-\cos x$

Therefore,

  $\left|f^{\left(3\right)}\left(x\right)\right|=\left|-\cos x\right|\le 1$ for all real x

Use $M=r=1$ , which gives $\left|R_2\left(x\right)\right|<\frac{{\left|x\right|}^3}{3!}$ where $\left|R_2\left(x\right)\right|=\left|x-\sin x\right|$

Therefore,

  $\left|x-\sin x\right|<\frac{{\left|x\right|}^3}{6}\mathrm{...........}\left(2\right)$

Hence, from equation (1) and (2)

  $\left(P+\sin P\right)-\pi <\frac{{\left|x\right|}^3}{6}$

Conclusion:

The value is $\left(P+\sin P\right)-\pi <\frac{{\left|x\right|}^3}{6}$