| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Integration with differentiation context |
| Difficulty | Standard +0.3 This is a structured multi-part question covering standard C3 techniques: verifying coordinates (trivial), product rule differentiation (routine), finding turning points (standard), and substitution integration (straightforward with clear guidance). Each part builds systematically with explicit instructions, requiring competent execution of learned methods rather than problem-solving insight. Slightly easier than average due to the scaffolded structure and routine nature of all components. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals1.08h Integration by substitution |
7 Fig. 7 shows part of the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = x \sqrt { 1 + x }$. The curve meets the $x$-axis at the origin and at the point P .
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{666dc19e-f293-4738-8530-fce90df23d17-4_491_881_476_588}
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\caption{Fig. 7}
\end{center}
\end{figure}
(i) Verify that the point P has coordinates $( - 1,0 )$. Hence state the domain of the function $\mathrm { f } ( x )$.\\
(ii) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 + 3 x } { 2 \sqrt { 1 + x } }$.\\
(iii) Find the exact coordinates of the turning point of the curve. Hence write down the range of the function.\\
(iv) Use the substitution $u = 1 + x$ to show that
$$\int _ { - 1 } ^ { 0 } x \sqrt { 1 + x } \mathrm {~d} x = \int _ { 0 } ^ { 1 } \left( u ^ { \frac { 3 } { 2 } } - u ^ { \frac { 1 } { 2 } } \right) \mathrm { d } u$$
Hence find the area of the region enclosed by the curve and the $x$-axis.
\hfill \mbox{\textit{OCR MEI C3 2007 Q7 [18]}}