| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find derivative of simple polynomial (integer powers) |
| Difficulty | Moderate -0.8 This is a straightforward numerical differentiation question requiring only basic skills: drawing a tangent by eye, calculating gradient from a line, and computing a chord gradient using two points. All calculations involve simple exponential evaluation (2^x) and gradient formula. No conceptual depth or problem-solving required—purely mechanical application of standard techniques taught early in C2. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.07a Derivative as gradient: of tangent to curve |
| Answer | Marks | Guidance |
|---|---|---|
| ruled line touching curve at \(x = 2\); their \(\frac{y_2 - y_1}{x_2 - x_1}\) from their tangent; answer in range 2.5 to 3.0 inclusive | M1, M1, A1 | intent to touch, but must not clearly cut curve; may be on graph or in working; must use correct points from their line; their tangent may be at another point; both M1s must be awarded; M0 for reciprocal; (value is approx 2.773) |
| Answer | Marks | Guidance |
|---|---|---|
| 3.482202253... and 4.59479342... rot to 3 or more sf; 2.78 to 2.7815 or 2.8 | B1, B1 | mark the final answer; 2.781477917... |
### Part (i)
ruled line touching curve at $x = 2$; their $\frac{y_2 - y_1}{x_2 - x_1}$ from their tangent; answer in range 2.5 to 3.0 inclusive | M1, M1, A1 | intent to touch, but must not clearly cut curve; may be on graph or in working; must use correct points from their line; their tangent may be at another point; both M1s must be awarded; M0 for reciprocal; (value is approx 2.773)
### Part (ii)
3.482202253... and 4.59479342... rot to 3 or more sf; 2.78 to 2.7815 or 2.8 | B1, B1 | mark the final answer; 2.781477917...\includegraphics{figure_5}
Fig. 5 shows the graph of $y = 2^x$.
\begin{enumerate}[label=(\roman*)]
\item On the copy of Fig. 5, draw by eye a tangent to the curve at the point where $x = 2$. Hence find an estimate of the gradient of $y = 2^x$ when $x = 2$. [3]
\item Calculate the $y$-values on the curve when $x = 1.8$ and $x = 2.2$. Hence calculate another approximation to the gradient of $y = 2^x$ when $x = 2$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 2013 Q5 [5]}}