| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This is a straightforward C3 calculus question. Part (i) requires recognizing that ln(x)/x = 0 when ln(x) = 0, giving x=1. Part (ii) involves applying the quotient rule to find the derivative, setting it to zero to find x=e, then using the second derivative test. While it requires multiple steps and the quotient rule, these are standard techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(1, 0)\) | B1 | 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = \frac{x \cdot \frac{1}{x} - 1 \cdot \ln x}{x^2} = \frac{1 - \ln x}{x^2}\) | M1 | Quotient rule |
| At Q, gradient is zero, so \(x = e\). Q is \((e, \frac{1}{e})\) | M1, A1 | \(= 0\) |
| \(\frac{d^2y}{dx^2} = \frac{x^2(\frac{-1}{x}) - (1-\ln x) \cdot 2x}{x^4} = \frac{-3 + 2\ln x}{x^3}\) | M1 | Or equivalent methods |
| When \(x = e\), this is \(-ve\), so Q is a maximum. | A1 | For \(\frac{1}{e}\). 6 marks |
### (i)
$P(1, 0)$ | B1 | 1 mark
### (ii)
$\frac{dy}{dx} = \frac{x \cdot \frac{1}{x} - 1 \cdot \ln x}{x^2} = \frac{1 - \ln x}{x^2}$ | M1 | Quotient rule
At Q, gradient is zero, so $x = e$. Q is $(e, \frac{1}{e})$ | M1, A1 | $= 0$
$\frac{d^2y}{dx^2} = \frac{x^2(\frac{-1}{x}) - (1-\ln x) \cdot 2x}{x^4} = \frac{-3 + 2\ln x}{x^3}$ | M1 | Or equivalent methods
When $x = e$, this is $-ve$, so Q is a maximum. | A1 | For $\frac{1}{e}$. 6 marks
---The function f(x) is defined as $f(x) = \frac{\ln x}{x}$. The graph of the function is shown in Fig. 6.
\includegraphics{figure_6}
\begin{enumerate}[label=(\roman*)]
\item Give the coordinates of the point, P, where the curve crosses the $x$-axis. [1]
\item Use calculus to find the coordinates of the stationary point, Q, and show that it is a maximum. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 Q6 [7]}}