Find intersection point coordinates

6 The curve with equation \(y = \frac { 6 } { x ^ { 2 } }\) intersects the line \(y = x + 1\) at the point \(P\).

1. Verify by calculation that the \(x\)-coordinate of \(P\) lies between 1.4 and 1.6.
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \sqrt { } \left( \frac { 6 } { x + 1 } \right)$$
3. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 6 } { x _ { n } + 1 } \right)$$ with initial value \(x _ { 1 } = 1.5\), to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6. \begin{figure}[h] \end{figure} Figure 1 is a sketch showing part of the curve with equation \(y = 2 ^ { x + 1 } - 3\) and part of the line with equation \(y = 17 - x\). The curve and the line intersect at the point \(A\).

1. Show that the \(x\) coordinate of \(A\) satisfies the equation $$x = \frac { \ln ( 20 - x ) } { \ln 2 } - 1$$
2. Use the iterative formula $$x _ { n + 1 } = \frac { \ln \left( 20 - x _ { n } \right) } { \ln 2 } - 1 , \quad x _ { 0 } = 3$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
3. Use your answer to part (b) to deduce the coordinates of the point \(A\), giving your answers to one decimal place.

4

1. Sketch the graphs of \(y = 1 + \mathrm { e } ^ { 2 x }\) and \(y = | x - 4 |\) on the same diagram.
2. The two graphs meet at the point \(P\) .
   Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 2 } \ln ( 3 - x )\) . \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-06_2716_38_109_2012}
3. Use an iterative formula, based on the equation in part (b), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Use an initial value of 0.45 and give the result of each iteration to 5 significant figures.