10.
The function f is defined by \(\mathrm { f } ( x ) = \arccos x\) for \(0 \leq x \leq a\)
The curve with equation \(y = \mathrm { f } ( x )\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-22_769_771_317_648}
- State the value of \(a\)
- On the diagram above, sketch the curve with equation
$$y = \cos x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$
and
sketch the line with equation
$$y = x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ - Explain why the solution to the equation
$$x - \cos x = 0$$
must also be a solution to the equation
$$\cos x = \arccos x$$
- Use the Newton-Raphson method with \(x _ { 0 } = 0\) to find an approximate solution, \(x _ { 3 }\), to the equation
$$x - \cos x = 0$$
Give your answer to four decimal places.
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