| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | October |
| Marks | 11 |
| Topic | Chain Rule |
| Type | Find stationary points and nature |
| Difficulty | Standard +0.3 Part (i) requires quotient rule differentiation and solving a quadratic to find stationary points - standard A-level technique with straightforward algebra. Part (ii) involves chain rule with exponential functions and substitution - routine calculus application. Both parts are textbook-style exercises requiring competent technique but no novel insight, making this slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
\begin{enumerate}[label=(\roman*)]
\item The curve $C$ has equation
$$y = \frac{x}{9 + x^2}.$$
Use calculus to find the coordinates of the turning points of $C$. [6]
\item Given that
$$y = (1 + e^{2x})^{\frac{3}{2}},$$
find the value of $\frac{dy}{dx}$ at $x = \frac{1}{2} \ln 3$. [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q2 [11]}}