OCR MEI FP2 2009 January — Question 1 19 marks

Exam BoardOCR MEI
ModuleFP2 (Further Pure Mathematics 2)
Year2009
SessionJanuary
Marks19
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTaylor series
TypeSeries for reciprocal functions
DifficultyStandard +0.3 This is a straightforward Further Maths question testing standard techniques: deriving Maclaurin series by differentiation (routine), finding coefficients by multiplying series and equating to 1 (algebraic manipulation), and applying the arctan derivative formula to standard integrals. All parts follow textbook methods with no novel insight required, making it slightly easier than average even for FM.
Spec4.08a Maclaurin series: find series for function4.08h Integration: inverse trig/hyperbolic substitutions
1
    1. By considering the derivatives of \(\cos x\), show that the Maclaurin expansion of \(\cos x\) begins $$1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 4 }$$
    2. The Maclaurin expansion of \(\sec x\) begins $$1 + a x ^ { 2 } + b x ^ { 4 }$$ where \(a\) and \(b\) are constants. Explain why, for sufficiently small \(x\), $$\left( 1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 4 } \right) \left( 1 + a x ^ { 2 } + b x ^ { 4 } \right) \approx 1$$ Hence find the values of \(a\) and \(b\).
    1. Given that \(y = \arctan \left( \frac { x } { a } \right)\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a } { a ^ { 2 } + x ^ { 2 } }\).
    2. Find the exact values of the following integrals. $$\begin{aligned} & \text { (A) } \int _ { - 2 } ^ { 2 } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x \\ & \text { (B) } \int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 4 } { 1 + 4 x ^ { 2 } } \mathrm {~d} x \end{aligned}$$
1
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item By considering the derivatives of $\cos x$, show that the Maclaurin expansion of $\cos x$ begins

$$1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 4 }$$
\item The Maclaurin expansion of $\sec x$ begins

$$1 + a x ^ { 2 } + b x ^ { 4 }$$

where $a$ and $b$ are constants. Explain why, for sufficiently small $x$,

$$\left( 1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 4 } \right) \left( 1 + a x ^ { 2 } + b x ^ { 4 } \right) \approx 1$$

Hence find the values of $a$ and $b$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Given that $y = \arctan \left( \frac { x } { a } \right)$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a } { a ^ { 2 } + x ^ { 2 } }$.
\item Find the exact values of the following integrals.

$$\begin{aligned}
& \text { (A) } \int _ { - 2 } ^ { 2 } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x \\
& \text { (B) } \int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 4 } { 1 + 4 x ^ { 2 } } \mathrm {~d} x
\end{aligned}$$
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR MEI FP2 2009 Q1 [19]}}
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Q1 19 Q2 18 Q3 17 Q4 18 Q5 18