8 The diagram shows a sketch of the curve with equation \(y = 6 ^ { x }\).
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- Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
- Explain, with the aid of a diagram, whether your approximate value will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\).
- Describe a single geometrical transformation that maps the graph of \(y = 6 ^ { x }\) onto the graph of \(y = 6 ^ { 3 x }\).
- The line \(y = 84\) intersects the curve \(y = 6 ^ { 3 x }\) at the point \(A\). By using logarithms, find the \(x\)-coordinate of \(A\), giving your answer to three decimal places.
(4 marks)
- The graph of \(y = 6 ^ { x }\) is translated by \(\left[ \begin{array} { c } 1
- 2 \end{array} \right]\) to give the graph of the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).