| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Derivative of inverse trig function |
| Difficulty | Challenging +1.2 This is a Further Maths question involving inverse trig differentiation and tangent equations. Part (a) requires chain rule with arcsec derivative (a standard FP3 result), part (b) is routine sketching, and part (c) involves finding a tangent line equation with exact values. While it requires multiple techniques and careful algebraic manipulation, each step follows standard procedures without requiring novel insight or particularly complex reasoning. |
| Spec | 1.07m Tangents and normals: gradient and equations4.08g Derivatives: inverse trig and hyperbolic functions |
The curve $C$ has equation
$$y = \operatorname{arcsec} e^x, \quad x > 0, \quad 0 < y < \frac{1}{2}\pi.$$
\begin{enumerate}[label=(\alph*)]
\item Prove that $\frac{dy}{dx} = \frac{1}{\sqrt{e^{2x} - 1}}$. [5]
\item Sketch the graph of $C$. [2]
\end{enumerate}
The point $A$ on $C$ has $x$-coordinate $\ln 2$. The tangent to $C$ at $A$ intersects the $y$-axis at the point $B$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the exact value of the $y$-coordinate of $B$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q14 [11]}}