8. A curve has the equation \(y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , \quad x \neq 0\).
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
[0pt] - Show that the curve has a stationary point in the interval [1.3,1.4].
The point \(A\) on the curve has \(x\)-coordinate 2 .
- Show that the tangent to the curve at \(A\) passes through the origin.
The tangent to the curve at \(A\) intersects the curve again at the point \(B\).
The \(x\)-coordinate of \(B\) is to be estimated using the iterative formula
$$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$
with \(x _ { 0 } = - 1\). - Find \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 7 significant figures and hence state the \(x\)-coordinate of \(B\) to 5 significant figures.