C2 June 2006 Q5
5.
(a) In the space provided, sketch the graph of \(y = 3^x,\ x \in \mathbb{R}\), showing the coordinates of the point at which the graph meets the \(y\)-axis. (2)
(b) Complete the table, giving the values of \(3^x\) to 3 decimal places.
(2)
| \(x\) | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 |
|---|---|---|---|---|---|---|
| \(3^x\) | 1.246 | 1.552 | 3 |
(c) Use the trapezium rule, with all the values from your table, to find an approximation for the value of\[\int_0^1 3^x\,\mathrm{d}x.\] (4)
| Scheme | Marks |
|---|---|
 | B1 |
| \((0, 1)\), or just 1 on the \(y\)-axis, or seen in table for (b) | B1 |
| (2) |
Notes
Beware the order of marks!
Must be a curve (not a straight line).
Curve must extend to the left of the \(y\)-axis, and must be increasing.
Curve can ‘touch’ the \(x\)-axis, but must not go below it.
Otherwise, be generous in cases of doubt.
The B1 for \((0, 1)\) is independent of the sketch.
| \(x\) | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 |
|---|---|---|---|---|---|---|
| \(3^x\) | 1 | 1.246 | 1.552 | 1.933 | 2.408 | 3 |
| Scheme | Marks |
|---|---|
| Missing values: 1.933, 2.408 (Accept awrt) | B1, B1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{2} \times 0.2, \ \left\{(1 + 3) + 2(1.246 + 1.552 + 1.933 + 2.408)\right\}\) | B1, M1 A1ft |
| \(= 1.8278\) (awrt 1.83) | A1 |
| (4) | |
| (8 marks) |
Notes
Beware the order of marks!
Bracketing mistake: i.e. \(\dfrac{1}{2} \times 0.2(1 + 3) + 2(1.246 + 1.552 + 1.933 + 2.408)\) scores B1 M1 A0 A0 unless the final answer implies that the calculation has been done correctly (then full marks can be given).