C4 June 2008 Q1
1.
Figure 1 shows part of the curve with equation \(y = \mathrm{e}^{0.5x^2}\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).
| \(x\) | 0 | 0.4 | 0.8 | 1.2 | 1.6 | 2 |
|---|---|---|---|---|---|---|
| \(y\) | \(\mathrm{e}^0\) | \(\mathrm{e}^{0.08}\) | \(\mathrm{e}^{0.72}\) | \(\mathrm{e}^2\) |
| x | 0 | 0.4 | 0.8 | 1.2 | 1.6 | 2 |
|---|---|---|---|---|---|---|
| y | \(\mathrm{e}^0\) | \(\mathrm{e}^{0.08}\) | \(\mathrm{e}^{0.32}\) | \(\mathrm{e}^{0.72}\) | \(\mathrm{e}^{1.28}\) | \(\mathrm{e}^2\) |
| or y | 1 | 1.08329… | 1.37713… | 2.05443… | 3.59664… | 7.38906… |
| Scheme | Marks |
|---|---|
| Either \(\mathrm{e}^{0.32}\) and \(\mathrm{e}^{1.28}\) or awrt 1.38 and 3.60 (or a mixture of e’s and decimals) | B1 |
| (1) |
Way 1
| Scheme | Marks |
|---|---|
| Area \(\approx\) \(\dfrac{1}{2} \times 0.4;\ \times \underline{\left[\mathrm{e}^0 + 2\left(\mathrm{e}^{0.08} + \mathrm{e}^{0.32} + \mathrm{e}^{0.72} + \mathrm{e}^{1.28}\right) + \mathrm{e}^2\right]}\) | B1; M1ft |
| \(= 0.2 \times 24.61203164\ldots = 4.922406\ldots = \underline{4.922}\) (4sf) | A1 cao |
| (3) | |
| (4 marks) |
Notes
B1: Outside brackets \(\tfrac{1}{2} \times 0.4\) or 0.2
M1ft: For structure of trapezium rule \(\underline{[\ldots\ldots\ldots]}\);
A1 cao: 4.922
Aliter (b) Way 2
| Scheme | Marks |
|---|---|
| Area \(\approx 0.4 \times \left[\tfrac{\mathrm{e}^0 + \mathrm{e}^{0.08}}{2} + \tfrac{\mathrm{e}^{0.08} + \mathrm{e}^{0.32}}{2} + \tfrac{\mathrm{e}^{0.32} + \mathrm{e}^{0.72}}{2} + \tfrac{\mathrm{e}^{0.72} + \mathrm{e}^{1.28}}{2} + \tfrac{\mathrm{e}^{1.28} + \mathrm{e}^2}{2}\right]\) | B1 |
| which is equivalent to: Area \(\approx\) \(\dfrac{1}{2} \times 0.4;\ \times \underline{\left[\mathrm{e}^0 + 2\left(\mathrm{e}^{0.08} + \mathrm{e}^{0.32} + \mathrm{e}^{0.72} + \mathrm{e}^{1.28}\right) + \mathrm{e}^2\right]}\) | M1ft |
| \(= 0.2 \times 24.61203164\ldots = 4.922406\ldots = \underline{4.922}\) (4sf) | A1 cao |
| (3) |
B1: 0.4 and a divisor of 2 on all terms inside brackets. M1ft: One of first and last ordinates, two of the middle ordinates inside brackets ignoring the 2. A1 cao: 4.922
Note an expression like Area \(\approx \dfrac{1}{2} \times 0.4 + \mathrm{e}^0 + 2\left(\mathrm{e}^{0.08} + \mathrm{e}^{0.32} + \mathrm{e}^{0.72} + \mathrm{e}^{1.28}\right) + \mathrm{e}^2\) would score B1M1A0
Allow one term missing (slip!) in the ( ) brackets for … (the printed note ends at “for”)
The M1 mark for structure is for the material found in the curly brackets ie \(\left[\text{first } y \text{ ordinate} + 2(\text{intermediate ft } y \text{ ordinate}) + \text{final } y \text{ ordinate}\right]\)