| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2009 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find tangent line equation |
| Difficulty | Hard +2.3 This AEA question requires implicit differentiation of a transcendental function (taking ln of both sides to differentiate x^(sin x)), finding a tangent equation, then proving it touches the curve infinitely many times—requiring analysis of when the tangent line equals the curve and showing infinitely many solutions exist. The proof component demands significant mathematical insight beyond routine calculus. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07l Derivative of ln(x): and related functions1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
2. The curve $C$ has equation $y = x ^ { \sin x } , \quad x > 0$.
\begin{enumerate}[label=(\alph*)]
\item Find the equation of the tangent to $C$ at the point where $x = \frac { \pi } { 2 }$.
\item Prove that this tangent touches $C$ at infinitely many points.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2009 Q2 [9]}}