| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2004 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Taylor series about π/4 |
| Difficulty | Standard +0.8 This is a multi-step Further Maths question requiring successive differentiation of tan x (using quotient/chain rule repeatedly), evaluation at π/4, Taylor series construction, and numerical approximation. While the framework is standard FP2 material, the algebraic manipulation of derivatives of tan x and careful substitution into Taylor's theorem requires sustained accuracy across multiple steps, placing it moderately above average difficulty. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)4.08a Maclaurin series: find series for function |
(a) $f'(x) = \sec^2 x$ M1 A1
$f''(x) = 2\sec x(\sec x \tan x)$ (or equiv.) A1
(Alternative forms: $2\sec^2 x(\sec^2 x) + 2\tan x(2\sec^2 x \tan x)$ or $2\sec^4 x + 4\sec^2 x \tan^2 x$ or $6\sec^4 x - 4\sec^2 x$ or $2 + 8\tan^2 x + 6\tan^4 x$)
(b) $\tan\frac{\pi}{4} = 1$ or $\sec\frac{\pi}{4} = \sqrt{2}$ B1
$\tan x = f\left(\frac{\pi}{4}\right) + f'\left(\frac{\pi}{4}\right)\left(x-\frac{\pi}{4}\right) + \frac{1}{2}f''\left(\frac{\pi}{4}\right)\left(x-\frac{\pi}{4}\right)^2 + \frac{1}{6}f'''\left(\frac{\pi}{4}\right)\left(x-\frac{\pi}{4}\right)^3$ M1
$= 1 + 2\left(x-\frac{\pi}{4}\right) + 2\left(x-\frac{\pi}{4}\right)^2 + \frac{8}{3}\left(x-\frac{\pi}{4}\right)^3$ A1 (cso)
(Allow equiv. fractions)
(c) $x = \frac{3\pi}{10}$, so use $\frac{3\pi}{10} - \frac{\pi}{4} = \frac{\pi}{20}$ M1
$\tan\frac{3\pi}{10} \approx 1 + \frac{\pi}{10} + 2 \times \frac{\pi^2}{400} + \frac{8}{3} \times \frac{\pi^3}{8000} = 1 + \frac{\pi}{10} + \frac{\pi^2}{200} + \frac{\pi^3}{3000}$ (*) A1 (cso)
---10. Given that $y = \tan x$,
\begin{enumerate}[label=(\alph*)]
\item find $\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$.
\item Find the Taylor series expansion of $\tan x$ in ascending powers of $\left( x - \frac { \pi } { 4 } \right)$ up to and including the term in $\left( x - \frac { \pi } { 4 } \right) ^ { 3 }$.
\item Hence show that $\tan \frac { 3 \pi } { 10 } \approx 1 + \frac { \pi } { 10 } + \frac { \pi ^ { 2 } } { 200 } + \frac { \pi ^ { 3 } } { 3000 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2004 Q10 [8]}}