| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find tangent line equation |
| Difficulty | Moderate -0.3 This is a straightforward C3 differentiation question requiring standard techniques: differentiating ln(3x) and e^x, finding a tangent equation, and locating a y-intercept. Part (a) is routine substitution to verify an inequality. The algebraic manipulation is minimal and all steps follow textbook procedures with no novel insight required. Slightly easier than average due to its predictable structure. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.09a Sign change methods: locate roots |
\includegraphics{figure_1}
Figure 1 shows a sketch of the curve with equation $y = f(x)$, where
$$f(x) = 10 + \ln(3x) - \frac{1}{2}e^x, \quad 0.1 \leq x \leq 3.3.$$
Given that $f(k) = 0$,
\begin{enumerate}[label=(\alph*)]
\item show, by calculation, that $3.1 < k < 3.2$. [2]
\item Find $f'(x)$. [3]
\end{enumerate}
The tangent to the graph at $x = 1$ intersects the $y$-axis at the point $P$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item
\begin{enumerate}[label=(\roman*)]
\item Find an equation of this tangent.
\item Find the exact $y$-coordinate of $P$, giving your answer in the form $a + \ln b$. [5]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q13 [10]}}