2 Fig. 2.1 shows the graph of \(y = x ^ { 2 } \mathrm { e } ^ { 2 x } - 5 x ^ { 2 } + 0.5\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4838f71e-a1d0-4695-89d2-c7ebb47edd77-4_757_545_315_248}
\captionsetup{labelformat=empty}
\caption{Fig. 2.1}
\end{figure}
There are three roots of the equation \(x ^ { 2 } \mathrm { e } ^ { 2 x } - 5 x ^ { 2 } + 0.5 = 0\). The roots are \(\alpha , \beta\) and \(\gamma\), where \(\alpha < \beta < \gamma\).
- Explain why it is not possible to use the method of false position with \(x _ { 0 } = 0\) and \(x _ { 1 } = 1\) to find \(\beta\) and \(\gamma\).
The graph of the function indicates that the root \(\gamma\) lies in the interval [0.6, 0.8]. Fig. 2.2 shows some spreadsheet output using the method of false position using these values as starting points.
\begin{table}[h]
| A | B | C | D | E | F |
| 1 | a | f(a) | b | f(b) | approx | |
| 2 | 0.6 | -0.10476 | 0.8 | 0.469941 | 0.636457 | -0.07876 |
| 3 | 0.636457 | -0.07876 | 0.8 | 0.469941 | 0.659931 | -0.04748 |
| 4 | 0.659931 | -0.04748 | 0.8 | 0.469941 | 0.672783 | -0.0249 |
| 5 | 0.672783 | -0.0249 | 0.8 | 0.469941 | 0.679184 | -0.01211 |
| 6 | 0.679184 | -0.01211 | 0.8 | 0.469941 | 0.682218 | -0.00567 |
| 7 | 0.682218 | -0.00567 | 0.8 | 0.469941 | 0.683623 | -0.00261 |
| 8 | 0.683623 | -0.00261 | 0.8 | 0.469941 | 0.684266 | -0.00119 |
| 9 | 0.684266 | -0.00119 | 0.8 | 0.469941 | 0.684559 | -0.00054 |
| 10 | 0.684559 | -0.00054 | 0.8 | 0.469941 | 0.684692 | -0.00025 |
| 11 | 0.684692 | -0.00025 | 0.8 | 0.469941 | 0.684753 | -0.00011 |
| 12 | 0.684753 | -0.00011 | 0.8 | 0.469941 | 0.68478 | \(- 5.1 \mathrm { E } - 05\) |
\captionsetup{labelformat=empty}
\caption{Fig. 2.2}
- Without doing any further calculation, write down the smallest possible interval which is certain to contain \(\gamma\).
- State what is being calculated in column F.
The formula in cell A3 is \(\quad = \operatorname { IF } ( \mathrm { F } 2 < 0 , \mathrm { E } 2 , \mathrm {~A} 2 )\).
- Explain the purpose of this formula in the application of the method of false position.
The method of false position uses the same formula for obtaining new approximations as the secant method.
- Explain how the method of false position differs from the secant method.
- Give one advantage and one disadvantage of using the method of false position instead of the secant method.