| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.3 This is a straightforward application of the trapezium rule with a given formula. Part (a) requires calculator work to evaluate a function at given points, part (b) is simple verification by substitution, and part (c) is a standard trapezium rule calculation with provided ordinates. The formula itself is complex but students only need to evaluate it, not manipulate it. This is slightly easier than average because it's a routine textbook exercise with no problem-solving or conceptual challenges. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(t\) | 0 | 5 | 10 | 15 | 20 | 25 |
| \(v\) | 1.56 | 7.23 | 17.36 |
The speed, $v$ m s$^{-1}$, of a lorry at time $t$ seconds is modelled by
$$v = 5(e^{0.1t} - 1) \sin (0.1t), \quad 0 \leq t \leq 30.$$
\begin{enumerate}[label=(\alph*)]
\item Copy and complete the following table, showing the speed of the lorry at 5 second intervals. Use radian measure for $0.1t$ and give your values of $v$ to 2 decimal places where appropriate.
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$t$ & 0 & 5 & 10 & 15 & 20 & 25 \\
\hline
$v$ & & 1.56 & 7.23 & 17.36 & & \\
\hline
\end{tabular}
[3]
\item Verify that, according to this model, the lorry is moving more slowly at $t = 25$ than at $t = 24.5$. [1]
\end{enumerate}
The distance, $s$ metres, travelled by the lorry during the first 25 seconds is given by
$$s = \int_0^{25} v \, dt.$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Estimate $s$ by using the trapezium rule with all the values from your table. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q16 [8]}}