5 Fig. 5 shows part of the curve \(y = \operatorname { cosec } x\) together with the \(x\) - and \(y\)-axes.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-06_732_625_317_244}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{figure}
- For the section of the curve which is shown in Fig. 5, write down
- the equations of the two vertical asymptotes,
- the coordinates of the minimum point.
- Show that the equation \(x = \operatorname { cosec } x\) has a root which lies between \(x = 1\) and \(x = 2\).
- Use the iteration \(\mathrm { x } _ { \mathrm { n } + 1 } = \operatorname { cosec } \left( \mathrm { x } _ { \mathrm { n } } \right)\), with \(x _ { 0 } = 1\), to find
- the values of \(x _ { 1 }\) and \(x _ { 2 }\), correct to 5 decimal places,
- this root of the equation, correct to 3 decimal places.
- There is another root of \(x = \operatorname { cosec } x\) which lies between \(x = 2\) and \(x = 3\).
Determine whether the iteration \(\mathrm { x } _ { \mathrm { n } + 1 } = \operatorname { cosec } \left( \mathrm { x } _ { \mathrm { n } } \right)\) with \(x _ { 0 } = 2.5\) converges to this root.
- Sketch the staircase or cobweb diagram for the iteration, starting with \(x _ { 0 } = 2.5\), on the diagram in the Printed Answer Booklet.