By sketching the graphs of \(y = \frac { 1 } { x }\) and \(y = \sec 2 x\) on the axes below, show that the equation
$$\frac { 1 } { x } = \sec 2 x$$
has exactly one solution for \(x > 0\)
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-08_675_771_689_639}
7
By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6
7
Show that the equation can be rearranged to give
$$x = \frac { 1 } { 2 } \cos ^ { - 1 } x$$
7
Use the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 2 } \cos ^ { - 1 } x _ { n }$$
with \(x _ { 1 } = 0.4\), to find \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to four decimal places.
7
(ii) On the graph below, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). [0pt]
[2 marks]
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-09_954_1600_1717_223}