A2 June 2023 Paper 1 Q4
4
(a)
(i) Given that \(\mathrm{f}(x) = \sqrt{1 + 2x}\), find \(\mathrm{f}^{\prime}(x)\) and \(\mathrm{f}^{\prime\prime}(x)\). [2]
(ii) Hence, find the first three terms of the Maclaurin series for \(\sqrt{1 + 2x}\). [2]
(b) Hence, using a suitable value for \(x\), show that \(\sqrt{5} \approx \dfrac{143}{64}\). [2]
| Scheme | Marks | AO |
|---|---|---|
| (i) \(\mathrm{f}^{\prime}(x) = (1 + 2x)^{-\frac{1}{2}}\) | B1 | 1.1 |
| \(\mathrm{f}^{\prime\prime}(x) = -(1 + 2x)^{-\frac{3}{2}}\) | B1 | 1.1 |
| [2] | ||
| (ii) \(\mathrm{f}(0) = 1, \mathrm{f}^{\prime}(0) = 1, \mathrm{f}^{\prime\prime}(0) = -1\) \(1 + x - \dfrac{1}{2}x^2\) | M1 A1 | 1.1 1.1 |
| [2] |
Notes
(a)(ii)
M1: Their \(\mathrm{f}(0)\), \(\mathrm{f}^{\prime}(0)\) and \(\mathrm{f}^{\prime\prime}(0)\) evaluated and substituted into Maclaurin
A1: Must come from correct expressions for \(\mathrm{f}^{\prime}(x)\), \(\mathrm{f}^{\prime\prime}(x)\); cannot come from binomial expansion. Ignore subsequent terms.
| Scheme | Marks | AO |
|---|---|---|
| \(\sqrt{1 + 2 \times \frac{1}{8}}\) or \(1 + \frac{1}{8} - \frac{1}{2}\left(\frac{1}{8}\right)^2\) | M1 | 3.1a |
| \(\Rightarrow \sqrt{5} \approx \frac{143}{64}\) | A1 | 2.2a |
| [2] |
Notes
M1: using \(x = \frac{1}{8}\) in their expansion
A1: AG