| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Taylor series about π/3 or π/6 |
| Difficulty | Standard +0.8 This is a Further Maths F2 Taylor series question requiring multiple derivatives of a composite trigonometric function, evaluation at a non-standard point (π/3), and application to estimate sin(1/2). While the technique is standard for FM students, the calculation is moderately involved with potential for arithmetic errors, and the connection between parts (a) and (b) requires insight (setting x = π/3 + h where 3h/2 = 1/2). This places it somewhat above average difficulty even for Further Maths. |
| Spec | 4.08a Maclaurin series: find series for function |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f'(x) = \frac{3}{2}\cos\left(\frac{3}{2}x\right)\), \(f''(x) = -\frac{9}{4}\sin\left(\frac{3}{2}x\right)\), \(f'''(x) = -\frac{27}{8}\cos\left(\frac{3}{2}x\right)\), \(f''''(x) = \frac{81}{16}\sin\left(\frac{3}{2}x\right)\) | M1A2 | M1: Attempt first 4 derivatives; pattern \(\sin\to\cos\to\sin\to\cos\to\sin\), ignore signs/coefficients. A1: \(f' = \frac{3}{2}\cos\left(\frac{3}{2}x\right)\) and \(f'' = -\frac{9}{4}\sin\left(\frac{3}{2}x\right)\). A1: \(f''' = -\frac{27}{8}\cos\left(\frac{3}{2}x\right)\) and \(f'''' = \frac{81}{16}\sin\left(\frac{3}{2}x\right)\) |
| \(y\!\left(\frac{\pi}{3}\right)=1,\ y'\!\left(\frac{\pi}{3}\right)=0,\ y''\!\left(\frac{\pi}{3}\right)=-\frac{9}{4},\ y'''\!\left(\frac{\pi}{3}\right)=0,\ y''''\!\left(\frac{\pi}{3}\right)=\frac{81}{16}\) | M1 | Attempts at least 1 derivative evaluated at \(x=\frac{\pi}{3}\) |
| \(f(x) = 1 - \frac{9}{8}\left(x-\frac{\pi}{3}\right)^2 + \frac{27}{128}\left(x-\frac{\pi}{3}\right)^4\) | dM1A1 | dM1: Correct use of Taylor series \(f(x)=f\!\left(\frac{\pi}{3}\right)+\left(x-\frac{\pi}{3}\right)f'\!\left(\frac{\pi}{3}\right)+\left(x-\frac{\pi}{3}\right)^2\frac{f''\!\left(\frac{\pi}{3}\right)}{2!}+\cdots\); evidence of at least one term of correct structure \(\left(x-\frac{\pi}{3}\right)^n\frac{f^n\!\left(\frac{\pi}{3}\right)}{n!}\); dependent on previous M. A1: Correct expansion; allow equivalent single fractions for \(\frac{9}{8}\) and \(\frac{27}{128}\); allow decimals 1.125 and 0.2109375 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f\!\left(\frac{1}{3}\right) = 0.4815\) | M1A1 | M1: Attempts \(f\!\left(\frac{1}{3}\right)\) or states \(x=\frac{1}{3}\). A1: 0.4815 cao |
# Question 4:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(x) = \frac{3}{2}\cos\left(\frac{3}{2}x\right)$, $f''(x) = -\frac{9}{4}\sin\left(\frac{3}{2}x\right)$, $f'''(x) = -\frac{27}{8}\cos\left(\frac{3}{2}x\right)$, $f''''(x) = \frac{81}{16}\sin\left(\frac{3}{2}x\right)$ | M1A2 | M1: Attempt first 4 derivatives; pattern $\sin\to\cos\to\sin\to\cos\to\sin$, ignore signs/coefficients. A1: $f' = \frac{3}{2}\cos\left(\frac{3}{2}x\right)$ and $f'' = -\frac{9}{4}\sin\left(\frac{3}{2}x\right)$. A1: $f''' = -\frac{27}{8}\cos\left(\frac{3}{2}x\right)$ and $f'''' = \frac{81}{16}\sin\left(\frac{3}{2}x\right)$ |
| $y\!\left(\frac{\pi}{3}\right)=1,\ y'\!\left(\frac{\pi}{3}\right)=0,\ y''\!\left(\frac{\pi}{3}\right)=-\frac{9}{4},\ y'''\!\left(\frac{\pi}{3}\right)=0,\ y''''\!\left(\frac{\pi}{3}\right)=\frac{81}{16}$ | M1 | Attempts at least 1 derivative evaluated at $x=\frac{\pi}{3}$ |
| $f(x) = 1 - \frac{9}{8}\left(x-\frac{\pi}{3}\right)^2 + \frac{27}{128}\left(x-\frac{\pi}{3}\right)^4$ | dM1A1 | dM1: Correct use of Taylor series $f(x)=f\!\left(\frac{\pi}{3}\right)+\left(x-\frac{\pi}{3}\right)f'\!\left(\frac{\pi}{3}\right)+\left(x-\frac{\pi}{3}\right)^2\frac{f''\!\left(\frac{\pi}{3}\right)}{2!}+\cdots$; evidence of at least one term of correct structure $\left(x-\frac{\pi}{3}\right)^n\frac{f^n\!\left(\frac{\pi}{3}\right)}{n!}$; dependent on previous M. A1: Correct expansion; allow equivalent single fractions for $\frac{9}{8}$ and $\frac{27}{128}$; allow decimals 1.125 and 0.2109375 |
**Total: (6)**
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f\!\left(\frac{1}{3}\right) = 0.4815$ | M1A1 | M1: Attempts $f\!\left(\frac{1}{3}\right)$ or states $x=\frac{1}{3}$. A1: 0.4815 cao |
**Total: (2)**
---4.
$$f ( x ) = \sin \left( \frac { 3 } { 2 } x \right)$$
\begin{enumerate}[label=(\alph*)]
\item Find the Taylor series expansion for $\mathrm { f } ( x )$ about $\frac { \pi } { 3 }$ in ascending powers of $\left( x - \frac { \pi } { 3 } \right)$ up to and including the term in $\left( x - \frac { \pi } { 3 } \right) ^ { 4 }$
\item Hence obtain an estimate of $\sin \frac { 1 } { 2 }$, giving your answer to 4 decimal places.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel F2 2016 Q4 [8]}}