Standard +0.2 This is a straightforward fixed-point iteration question requiring routine techniques: sign change to locate root, two iterations of a given formula with calculator work, and drawing a standard cobweb diagram. All parts are procedural with no problem-solving or novel insight required, making it easier than average.
The equation \(\mathrm { e } ^ { - x } - 2 + \sqrt { x } = 0\) has a single root, \(\alpha\).
Show that \(\alpha\) lies between 3 and 4 .
Use the recurrence relation \(x _ { n + 1 } = \left( 2 - e ^ { - x _ { n } } \right) ^ { 2 }\), with \(x _ { 1 } = 3.5\), to find \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
The diagram below shows parts of the graphs of \(y = \left( 2 - \mathrm { e } ^ { - x } \right) ^ { 2 }\) and \(y = x\), and a position of \(x _ { 1 }\).
On the diagram, draw a staircase or cobweb diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-03_1100_1402_881_367}
3
\begin{enumerate}[label=(\alph*)]
\item The equation $\mathrm { e } ^ { - x } - 2 + \sqrt { x } = 0$ has a single root, $\alpha$.\\
Show that $\alpha$ lies between 3 and 4 .
\item Use the recurrence relation $x _ { n + 1 } = \left( 2 - e ^ { - x _ { n } } \right) ^ { 2 }$, with $x _ { 1 } = 3.5$, to find $x _ { 2 }$ and $x _ { 3 }$, giving your answers to three decimal places.
\item The diagram below shows parts of the graphs of $y = \left( 2 - \mathrm { e } ^ { - x } \right) ^ { 2 }$ and $y = x$, and a position of $x _ { 1 }$.
On the diagram, draw a staircase or cobweb diagram to show how convergence takes place, indicating the positions of $x _ { 2 }$ and $x _ { 3 }$ on the $x$-axis.\\
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-03_1100_1402_881_367}
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2013 Q3 [4]}}