| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Show that derivative equals expression |
| Difficulty | Standard +0.3 This is a structured multi-part question covering standard C3 techniques (odd functions, quotient rule differentiation, integration with substitution, and implicit differentiation). While it requires multiple skills, each part follows predictable methods with clear guidance ('show that' questions provide the answer). The implicit differentiation in part (iv) adds mild interest, but overall this is slightly easier than average due to its scaffolded nature and routine application of techniques. |
| Spec | 1.07q Product and quotient rules: differentiation1.07s Parametric and implicit differentiation1.08h Integration by substitution |
Fig. 8 shows the curve $y = f(x)$, where $f(x) = \frac{x}{\sqrt{2 + x^2}}$.
\includegraphics{figure_8}
\begin{enumerate}[label=(\roman*)]
\item Show algebraically that $f(x)$ is an odd function. Interpret this result geometrically. [3]
\item Show that $f'(x) = \frac{2}{(2 + x^2)^{\frac{3}{2}}}$. Hence find the exact gradient of the curve at the origin. [5]
\item Find the exact area of the region bounded by the curve, the $x$-axis and the line $x = 1$. [4]
\item \begin{enumerate}[label=(\alph*)]
\item Show that if $y = \frac{x}{\sqrt{2 + x^2}}$, then $\frac{1}{y^2} = \frac{2}{x^2} + 1$. [2]
\item Differentiate $\frac{1}{y^2} = \frac{2}{x^2} + 1$ implicitly to show that $\frac{dy}{dx} = \frac{2y^3}{x^3}$. Explain why this expression cannot be used to find the gradient of the curve at the origin. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 2014 Q8 [18]}}