| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Differentiate exponential functions |
| Difficulty | Standard +0.3 This is a structured multi-part question testing standard C3 techniques: proving even functions, quotient rule differentiation, algebraic manipulation, integration by substitution, and solving equations. While part (iii) requires algebraic insight to manipulate the expression and part (v) involves some problem-solving, each part follows predictable methods with clear guidance. The substitution is explicitly given, and the question scaffolds students through each step, making it slightly easier than a typical C3 question. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.07j Differentiate exponentials: e^(kx) and a^(kx)1.08h Integration by substitution |
\includegraphics{figure_8}
Fig. 8 shows the curve $y = f(x)$, where $f(x) = \frac{1}{e^x + e^{-x} + 2}$.
\begin{enumerate}[label=(\roman*)]
\item Show algebraically that $f(x)$ is an even function, and state how this property relates to the curve $y = f(x)$. [3]
\item Find $f'(x)$. [3]
\item Show that $f(x) = \frac{e^x}{(e^x + 1)^2}$. [2]
\item Hence, using the substitution $u = e^x + 1$, or otherwise, find the exact area enclosed by the curve $y = f(x)$, the $x$-axis, and the lines $x = 0$ and $x = 1$. [5]
\item Show that there is only one point of intersection of the curves $y = f(x)$ and $y = \frac{1}{4}e^x$, and find its coordinates. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 2011 Q8 [18]}}