Iterative method for special point

7 A curve has equation \(\mathrm { e } ^ { 2 x } y - \mathrm { e } ^ { y } = 100\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \mathrm { e } ^ { 2 x } y } { \mathrm { e } ^ { y } - \mathrm { e } ^ { 2 x } }\).
2. 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.
3. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \ln 10 - \frac { 1 } { 2 } \ln ( 2 x - 1 )$$
4. 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.

7 The curve with equation \(\mathrm { e } ^ { 2 x } - 18 x + y ^ { 3 } + y = 11\) has a stationary point at \(( p , q )\).

1. Find the exact value of \(p\).
2. Show that \(q = \sqrt [ 3 ] { 2 + 18 \ln 3 - q }\).
3. Show by calculation that the value of \(q\) lies between 2.5 and 3.0.
4. Use an iterative formula, based on the equation in (b), to find the value of \(q\) correct to 4 significant figures. Give the result of each iteration to 6 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.

6 \includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-4_476_709_251_683} The diagram shows the curve with equation \(x = \ln \left( y ^ { 3 } + 2 y \right)\). At the point \(P\) on the curve, the gradient is 4 and it is given that \(P\) is close to the point with coordinates (7.5,12).

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
2. Show that the \(y\)-coordinate of \(P\) satisfies the equation $$y = \frac { 12 y ^ { 2 } + 8 } { y ^ { 2 } + 2 }$$
3. By first using an iterative process based on the equation in part (ii), find the coordinates of \(P\), giving each coordinate correct to 3 decimal places.