A2 October 2020 Q4
4.
\[\mathrm{f}(x) = x^4\sin(2x)\]Use Leibnitz’s theorem to show that the coefficient of \((x - \pi)^8\) in the Taylor series expansion of \(\mathrm{f}(x)\) about \(\pi\) is
\[\frac{a\pi + b\pi^3}{315}\]where \(a\) and \(b\) are integers to be determined.
(8)
\[\left[\begin{gathered}\textit{The Taylor series expansion of}\;\; \mathrm{f}(x)\;\; \textit{about}\;\; x = k\;\; \textit{is given by}\\ \mathrm{f}(x) = \mathrm{f}(k) + (x - k)\mathrm{f}^{\prime}(k) + \frac{(x - k)^2}{2!}\mathrm{f}^{\prime\prime}(k) + \ldots + \frac{(x - k)^r}{r!}\mathrm{f}^{(r)}(k) + \ldots\end{gathered}\right]\]| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{f}(x) = x^4\sin(2x) \quad u = x^4 \quad v = \sin(2x)\) | ||
| \(u^{\prime} = 4x^3,\ u^{\prime\prime} = 12x^2,\ u^{\prime\prime\prime} = 24x,\ u^{(4)} = 24\) (and \(u^{(n)} = 0\) for \(n \gt 4\)) | M1 | 1.1b |
| \(v^{\prime} = 2\cos(2x),\ v^{\prime\prime} = -4\sin(2x),\ v^{\prime\prime\prime} = -8\cos(2x),\ v^{(4)} = 16\sin(2x),\) \(v^{(5)} = 32\cos(2x),\ v^{(6)} = -64\sin(2x),\ v^{(7)} = -128\cos(2x),\) \(v^{(8)} = 256\sin(2x),\) | M1 A1 A1 | 3.1a 1.1b 1.1b |
| Thus \(\mathrm{f}^{(8)}(x) = x^4 \times 256\sin(2x) + 8 \times 4x^3 \times -128\cos(2x)\) \(+ \dfrac{8 \times 7}{2} \times 12x^2 \times -64\sin(2x) + \dfrac{8 \times 7 \times 6}{6} \times 24x \times 32\cos(2x)\) \(+ \dfrac{8 \times 7 \times 6 \times 5}{24} \times 24 \times 16\sin(2x)\) \(\mathrm{f}^{(8)}(x) = x^4 \times 256\sin(2x) + 8 \times 4x^3 \times -128\cos(2x)\) \(+ 28 \times 12x^2 \times -64\sin(2x) + 56 \times 24x \times 32\cos(2x)\) \(+ 70 \times 24 \times 16\sin(2x)\) | M1 | 2.1 |
| \(\mathrm{f}^{(8)}(\pi) = 0 - 4096\pi^3 - 0 + 1344 \times 2^5\pi + 0\ \left(= -4096\pi^3 + 43008\pi\right)\) | M1 | 1.1b |
| Coefficient is \(\dfrac{\mathrm{f}^{(8)}(\pi)}{8!} = \dfrac{1344 \times 2^5\pi - 4096\pi^3}{8!\text{ or }\{40320\}}\) | M1 | 2.2a |
| \(= \dfrac{336\pi - 32\pi^3}{315}\) (So \(a = 336\) and \(b = -32\)) | A1 | 2.1 |
| (8) | ||
| (8 marks) |
Notes
M1: Establishes the non-disappearing derivatives of \(x^4\). Allow slips in coefficients, but powers must decrease.
M1: Identifies the relevant derivatives for \(\sin(2x)\), up to the 8th derivative or establishes the correct pattern. Look for alternating between sin and cos. Condone use of \(x\).
A1: Correct sizes for the coefficients, allow sign errors for this mark (may be due to incorrect signs when differentiating sin and cos) Must have angle \(2x\).
A1: All derivatives correctly established. (Note the sin terms may be omitted if the student has made clear they will disappear, but if present they must be correct).
M1: Applies Leibnitz’s theorem to get the 8th derivative with their expressions. Binomial coefficients must be present.
M1: Evaluates their 8th derivative at \(\pi\)
M1: Uses Taylor series – divides their value for \(\mathrm{f}^{(8)}(\pi)\) by \(8!\)
A1: Simplifies to the correct answer.
Note: If do not use Leibnitz’s theorem then maximum M0 M0 A0 A0 M0 M1 M1 A0