8 The graph of \(\mathrm { y } = 0.2 \cosh \mathrm { x } - 0.4 \mathrm { x }\) for values of \(x\) from 0 to 3.32 is shown on the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{4023e87c-34b1-4abd-9acc-ede5e4d68c7f-08_988_1561_312_244}
The equation \(0.2 \cosh x - 0.4 x = 0\) has two roots, \(\alpha\) and \(\beta\) where \(\alpha < \beta\), in the interval \(0 < x < 3\). The secant method with \(x _ { 0 } = 1\) and \(x _ { 1 } = 2\) is to be used to find \(\beta\).
- On the copy of the graph in the Printed Answer Booklet, show how the secant method works with these two values of \(x\) to obtain an improved approximation to \(\beta\).
The spreadsheet output in the table below shows the result of applying the secant method with \(x _ { 0 } = 1\) and \(x _ { 1 } = 2\).
| I | J | K | L | M |
| 2 | \(r\) | \(\mathrm { x } _ { \mathrm { r } }\) | f(x) | \(\mathrm { X } _ { \mathrm { r } + 1 }\) | \(\mathrm { f } \left( \mathrm { x } _ { \mathrm { r } + 1 } \right)\) |
| 3 | 0 | 1 | -0.0914 | 2 | -0.0476 |
| 4 | 1 | 2 | -0.0476 | 3.08529 | 0.95784 |
| 5 | 2 | 3.08529 | 0.95784 | 2.05134 | -0.0298 |
| 6 | 3 | 2.05134 | -0.0298 | 2.08259 | -0.0181 |
| 7 | 4 | 2.08259 | -0.0181 | 2.13042 | 0.00155 |
| 8 | 5 | 2.13042 | 0.00155 | 2.12664 | \(- 7 \mathrm { E } - 05\) |
- Write down a suitable cell formula for cell J4.
- Write down a suitable cell formula for cell L4.
- Write down the most accurate approximation to \(\beta\) which is displayed in the table.
- Determine whether your answer to part (d) is correct to 5 decimal places. You should not calculate any more iterates.
- It is decided to use the secant method with starting values \(x _ { 0 } = 1\) and \(\mathrm { x } _ { 1 } = \mathrm { a }\), where \(a > 1\), to find \(\alpha\). State a suitable value for \(a\).