Chapter 5, Problem 51RE

a.

To determine

To find: an upper bound for $\left|f^{(4)}(x)\right|$ using the graph if $f(x)=\sin (\sin x)$

Answer to Problem 51RE

  $\max \left|f^{(4)}(x)\right|\approx 3.7678$ .

Explanation of Solution

Given information: Given curve is $f(x)=\sin (\sin x)$ .

Calculation:

  $\begin{array}{l}f(x)=\sin (\sin x).\\ f^{(4)}(x)=\sin (x)(3\cos (2x)+4)\cos (\sin (x))+\frac{1}{8}\sin (\sin (x))(32\cos (2x)+\cos (4x)+7)\end{array}$

Graph the absolute value of it: $y=abs(f^{(4)}(x))$ .

  

From the graph:

  $\max \left|f^{(4)}(x)\right|\approx \mathrm{3.7678.}$

b.

To determine

To approximate: ${\displaystyle \underset{0}{\overset{\pi }{\int }}f(x)dx}$ using Simpson’s Rule with n = 10 and use part (a) to estimate the error.

Answer to Problem 51RE

  $S_6\approx \mathrm{1.7867.}$

  $\left|E_s\right|\le 0.000641$ .

Explanation of Solution

Given information:

Calculation:

  $\begin{array}{c}{\displaystyle \underset{0}{\overset{\pi }{\int }}\sin (\sin x)}dx,n=10.\\ \text{Interval}\text{width}:\\ △x=\frac{b-a}{n}=\frac{\pi -0}{10}\approx \mathrm{0.314159.}\end{array}$

Calculate values of the function from 0 to $\pi$ in increments of $△x$ .

x	f(x)
0.000000	0.000000
0.314159	0.304122
0.628319	0.554519
0.942478	0.723609
1.256637	0.814030
1.570796	0.841471
1.884956	0.814030
2.199115	0.723609
2.513274	0.554519
2.827433	0.304122
3.141593	0.000000

Write out the Simpson’s rule formula with n +1 =11 terms, then fill in the numbers.

  $\begin{array}{c}{S}_6={\displaystyle \underset{1}{\overset{4}{\int }}\frac{e^x}{x}}dx\approx \frac{△x}{3}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+4f(x_5)+2f(x_6)\\ +4f(x_7)+2f(x_8)+4f(x_9)+f(x_{10}))\\ \approx \frac{0.314159}{3}[0+4(0.304122)+2(0.554519)+4(0.723609)+2(0.81403)\\ +4(0.841471)+2(0.81403)+4(0.723609)+2(0.554519)+4(0.304122)+0)\\ S_6\approx \mathrm{1.7867.}\end{array}$

For the Simpson’s rule error bounds.

  $\begin{array}{l}K=\max \left|f^{(4)}(x)\right|\approx \mathrm{3.7678.}\\ \left|E_s\right|\le \frac{K{(b-a)}^5}{180n^5}\le \frac{3.7678{(\pi -0)}^5}{180n^4}\le \mathrm{0.000641.}\\ \left|E_s\right|\le \mathrm{0.000641.}\end{array}$

c.

To determine

To find: how large should n be to guarantee that the size of the error in using $S_n$ is less than 0.00001.

Answer to Problem 51RE

  $n\ge 30$ .

Explanation of Solution

Given information:

Calculation:

For the error less than 0.00001, solve the Simpson’s error formula for n :

  $\begin{array}{c}\left|E_s\right|\le \frac{K{(b-a)}^5}{180n^5}\le \frac{3.7678{(\pi -0)}^5}{180n^4}\le \mathrm{0.00001.}\\ \frac{3.7678{(\pi -0)}^5}{180(0.00001)}\le n^4\\ $ n\ge {(\frac{3.7678{\pi }^5}{180(0.00001)})}^{\frac{1}{4}}$n\ge \mathrm{28.291.}\end{array}$

For the Simpson’s Rule, round up to the next whole even number, so :

  $n\ge 30.$