Question 9 - A-Level Maths

9. A curve $C$ has equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = x + 2 \ln ( \mathrm { e } - x )$$

1. Show that the equation of the normal to $C$ at the point where $C$ crosses the $y$-axis is given by $$y = \left( \frac { \mathrm { e } } { 2 - \mathrm { e } } \right) x + 2$$
2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer.
The equation $\mathrm { f } ( x ) = 0$ has one positive root, $\alpha$.
1. Show that $\alpha$ lies between 2 and 3 Fully justify your answer.
2. Show that the roots of $\mathrm { f } ( x ) = 0$ satisfy the equation $$x = \mathrm { e } - \mathrm { e } ^ { - \frac { x } { 2 } }$$ [2 marks]
3. Use the recurrence relation $$x _ { n + 1 } = \mathrm { e } - \mathrm { e } ^ { - \frac { x _ { n } } { 2 } }$$ with $x _ { 1 } = 2$ to find the values of $x _ { 2 }$ and $x _ { 3 }$ giving your answers to three decimal places.
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4. Figure 1 below shows a sketch of the graphs of $y = e - e ^ { - \frac { x } { 2 } }$ and $y = x$, and the position of $x _ { 1 }$ On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of $x _ { 2 }$ and $x _ { 3 }$ on the $x$-axis.
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  \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{1d67c98c-e81c-4967-8a0b-a78afd95a0aa-22_1236_1566_1519_360}
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