7 A curve has equation \(\mathrm { e } ^ { 2 x } y - \mathrm { e } ^ { y } = 100\).
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \mathrm { e } ^ { 2 x } y } { \mathrm { e } ^ { y } - \mathrm { e } ^ { 2 x } }\).
- Show that the curve has no stationary points.
It is required to find the \(x\)-coordinate of \(P\), the point on the curve at which the tangent is parallel to the \(y\)-axis. - Show that the \(x\)-coordinate of \(P\) satisfies the equation
$$x = \ln 10 - \frac { 1 } { 2 } \ln ( 2 x - 1 )$$
- Use an iterative formula, based on the equation in part (c), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 significant figures.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.