- (a) Given that \(a\) is a constant, \(a > 1\), sketch the graph of
$$y = a ^ { x } , \quad x \in \mathbb { R }$$
On your diagram show the coordinates of the point where the graph crosses the \(y\)-axis.
(2)
The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 ^ { x }\)
| \(x\) | - 4 | - 2 | 0 | 2 | 4 |
| \(y\) | 0.0625 | 0.25 | 1 | 4 | 16 |
(b) Use the trapezium rule, with all of the values of \(y\) from the table, to find an approximate value, to 2 decimal places, for
$$\int _ { - 4 } ^ { 4 } 2 ^ { x } \mathrm {~d} x$$
(c) Use the answer to part (b) to find an approximate value for
- \(\int _ { - 4 } ^ { 4 } 2 ^ { x + 2 } \mathrm {~d} x\)
- \(\int _ { - 4 } ^ { 4 } \left( 3 + 2 ^ { x } \right) \mathrm { d } x\)
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