| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 8 |
| Topic | Areas by integration |
| Type | Trapezium rule estimation |
| Difficulty | Standard +0.3 This is a straightforward calculus question requiring standard differentiation (quotient or product rule after rewriting), solving f'(x)=0 for the minimum, and applying the trapezium rule. All techniques are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.09f Trapezium rule: numerical integration |
\includegraphics{figure_12}
The figure above shows the curve $C$ with equation
$$f(x) = \frac{x+4}{\sqrt{x}}, \quad x > 0.$$
\begin{enumerate}[label=(\alph*)]
\item Determine the coordinates of the minimum point of $C$, labelled as $M$. [5]
\end{enumerate}
The point $N$ lies on the $x$ axis so that $MN$ is parallel to the $y$ axis. The finite region $R$ is bounded by $C$, the $x$ axis, the straight line segment $MN$ and the straight line with equation $x = 1$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Use the trapezium rule with 4 strips of equal width to estimate the area of $R$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q12 [8]}}