4 [Figure 1, printed on the insert, is provided for use in this question.]
- Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to \(\int _ { 1 } ^ { 2 } 3 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
- The curve \(y = 3 ^ { x }\) intersects the line \(y = x + 3\) at the point where \(x = \alpha\).
- Show that \(\alpha\) lies between 0.5 and 1.5.
- Show that the equation \(3 ^ { x } = x + 3\) can be rearranged into the form
$$x = \frac { \ln ( x + 3 ) } { \ln 3 }$$
- Use the iteration \(x _ { n + 1 } = \frac { \ln \left( x _ { n } + 3 \right) } { \ln 3 }\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\) to two significant figures.
- The sketch on Figure 1 shows part of the graphs of \(y = \frac { \ln ( x + 3 ) } { \ln 3 }\) 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.