10 In this question you must show detailed reasoning.
The equation of a curve is
$$y = \frac { \sin 2 x - x } { x \sin x }$$
- Use the small angle approximation given in the list of formulae on pages 2-3 of this question paper to show that
$$\int _ { 0.01 } ^ { 0.05 } \mathrm { ydx } \approx \ln 5$$
- Use the same small angle approximation to show that
$$\frac { d y } { d x } \approx - 10000 \text { at the point where } x = 0.01 \text {. }$$
The equation \(y = 0\) has a root near \(x = 1\). Joan uses the Newton-Raphson method to find this root. The output from the spreadsheet she uses is shown in Fig. 10.1.
\begin{table}[h]
| \(n\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| \(\mathrm { x } _ { \mathrm { n } }\) | 1 | 0.958509 | 0.950084 | 0.948261 | 0.94786 | 0.947772 | 0.947753 | 0.947748 |
\captionsetup{labelformat=empty}
\caption{Fig. 10.1}
| \(x\) | \(y\) |
| 0.9477475 | \(- 7.79967 \mathrm { E } - 07\) |
| 0.9477485 | \(- 2.90821 \mathrm { E } - 06\) |
| \(x\) | \(y\) |
| 0.947745 | \(4.54066 \mathrm { E } - 06\) |
| 0.947755 | \(- 1.67417 \mathrm { E } - 05\) |
\captionsetup{labelformat=empty}
\caption{Fig. 10.2}
- Consider the information in Fig. 10.1 and Fig. 10.2.
- Write 4.54066E-06 in standard mathematical notation.
- State the value of the root as accurately as you can, justifying your answer.