| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This is a straightforward C4 question testing standard techniques: locating a stationary point by testing given bounds (routine substitution), integrating to find f(x) using a constant of integration, and computing a definite integral. All steps are mechanical applications of core calculus methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08d Evaluate definite integrals: between limits |
\includegraphics{figure_1}
The curve $C$ has equation $y = f(x)$, $x \in \mathbb{R}$. Figure 1 shows the part of $C$ for which $0 \leq x \leq 2$.
Given that
$$\frac{dy}{dx} = e^x - 2x^2,$$
and that $C$ has a single maximum, at $x = k$,
\begin{enumerate}[label=(\alph*)]
\item show that $1.48 < k < 1.49$. [3]
\end{enumerate}
Given also that the point $(0, 5)$ lies on $C$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find $f(x)$. [4]
\end{enumerate}
The finite region $R$ is bounded by $C$, the coordinate axes and the line $x = 2$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Use integration to find the exact area of $R$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q5 [11]}}