8 Kareem wants to solve the equation \(\sin 4 x + \mathrm { e } ^ { - x } + 0.75 = 0\). He uses his calculator to create the following table of values for \(\mathrm { f } ( x ) = \sin 4 x + \mathrm { e } ^ { - x } + 0.75\).
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| \(\mathrm { f } ( x )\) | 1.750 | 0.361 | 1.875 | 0.263 | 0.480 | 1.670 | - 0.153 |
He argues that because \(\mathrm { f } ( 6 )\) is the first negative value in the table, there is a root of the equation between 5 and 6 .
- Comment on the validity of his argument.
The diagram shows the graph of \(y = \sin 4 x + e ^ { - x } + 0.75\).
\includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-07_538_1260_920_244} - Explain why Kareem failed to find other roots between 0 and 6 .
Kareem decides to use the Newton-Raphson method to find the root close to 3 .
- Determine the iterative formula he should use for this equation.
- Use the Newton-Raphson method with \(x _ { 0 } = 3\) to find a root of the equation \(\mathrm { f } ( x ) = 0\). Show three iterations and give your answer to a suitable degree of accuracy.
Kareem uses the Newton-Raphson method with \(x _ { 0 } = 5\) and also with \(x _ { 0 } = 6\) to try to find the root which lies between 5 and 6 . He produces the following tables.
| \(x _ { 0 }\) | 5 |
| \(x _ { 1 }\) | 3.97288 |
| \(x _ { 2 }\) | 4.12125 |
| \(x _ { 0 }\) | 6 |
| \(x _ { 1 }\) | 6.09036 |
| \(x _ { 2 }\) | 6.07110 |
- For the iteration beginning with \(x _ { 0 } = 5\), represent the process on the graph in the Printed Answer Booklet.
- Explain why the method has failed to find the root which lies between 5 and 6 .
- Explain how Kareem can adapt his method to find the root between 5 and 6 .