AS June 2019 Q2
2. The following algorithm produces a numerical approximation for the integral
\[I = \int_{\text{A}}^{\text{B}} x^4 \,\mathrm{d}x\]| Step 1 | Start |
| Step 2 | Input the values of A, B and N |
| Step 3 | Let H = (B – A) / N |
| Step 4 | Let C = H / 2 |
| Step 5 | Let D = 0 |
| Step 6 | Let D = D + A4 + B4 |
| Step 7 | Let E = A |
| Step 8 | Let E = E + H |
| Step 9 | If E = B go to Step 12 |
| Step 10 | Let D = D + 2 × E4 |
| Step 11 | Go to Step 8 |
| Step 12 | Let F = C × D |
| Step 13 | Output F |
| Step 14 | Stop |
For the case when A = 1, B = 3 and N = 4,
You may not need to use all the rows in this table.
It may not be necessary to complete all boxes in each row.
| A | B | N | H | C | D | E | F |
|---|---|---|---|---|---|---|---|
| Scheme | Marks | AO | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
(i)
| M1 A1 A1 | 1.1b 1.1b 1.1b | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (ii) Final output = 50.5625 | A1 | 1.1b | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (4) |
Notes
(a)(i) M1: At least three rows of cells completed (so at least two values of D and E given) with either a correct first row or 82 found for D – condone repeated values in all columns or a single value in each row
A1: CAO – the values in the second, third and fourth rows correct (so up to the 92.125 in column D and the 2 in column E) – accept exact equivalent fractions
A1: CAO – all values correct in columns A to E – accept exact equivalent fractions
(ii) A1: CAO (output = 50.5625) (or equivalent e.g. \(50\frac{9}{16}\)) – allow if stated only in column F
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle\int_1^3 x^4\,\mathrm{d}x = 48.4\) | B1 | 1.1b |
| \(\left(\dfrac{50.5625 - 48.4}{48.4}\right) \times 100\) | M1 | 1.1a |
| 4.47% | A1ft | 3.2b |
| (3) | ||
| (7 marks) |
Notes
B1: CAO (48.4)
M1: Correct method (including multiplying by 100) using candidate’s final output from (a)(ii) and their value for \(I\)
A1ft: Follow through their final output from (a)(ii) (for reference: 4.4679752…) must be using 48.4 - dependent on M mark in (a) and percentage error being < 10% (answer must be given to 3 significant figures)