A2 October 2021 Paper 1 Q5
5
(a) Use a Maclaurin series to find a quadratic approximation for \(\ln(1 + 2x)\). [1]
(b) Find the percentage error in using the approximation in part (a) to calculate \(\ln(1.2)\). [3]
(c) Jane uses the Maclaurin series in part (a) to try to calculate an approximation for \(\ln 3\).
Explain whether her method is valid. [2]
Explain whether her method is valid. [2]
| Scheme | Marks | AO |
|---|---|---|
| \(\ln(1 + 2x) \approx 2x - \dfrac{1}{2}(2x)^2 = 2x - 2x^2\) | B1 | 1.1 |
| [1] |
| Scheme | Marks | AO |
|---|---|---|
| \(\ln(1.2) \approx 0.18\) | B1ft | 1.1 |
| % error \(= 100 \times \dfrac{0.18 - \ln(1.2)}{\ln(1.2)}\) | M1 | 1.1 |
| \(= (-)1.27\%\) | A1 | 1.1 |
| [3] |
| Scheme | Marks | AO |
|---|---|---|
| \(-1 \lt 2x \leqslant 1 \Rightarrow -\frac{1}{2} \lt x \leqslant \frac{1}{2}\) | M1 | 1.1 |
| the Maclaurin series is not convergent for this approximation when \(x = 1\) | A1 | 2.3 |
| [2] |
Notes
M1: substitute \(x = 1\) in quadratic