| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiation from First Principles |
| Type | First principles: x⁴ and higher power terms |
| Difficulty | Moderate -0.8 This is a structured, multi-part question that guides students through finding a tangent (routine differentiation), calculating a chord gradient (basic algebra), and understanding differentiation from first principles via binomial expansion. While it covers multiple techniques, each step is straightforward and heavily scaffolded, making it easier than average for A-level. |
| Spec | 1.07g Differentiation from first principles: for small positive integer powers of x1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = 4x^3\) | M1 | |
| When \(x = 2, \frac{dy}{dx} = 32\) s.o.i. | A1 | i.s.w. |
| When \(x = 2, y = 16\) s.o.i. | B1 | |
| \(y = 32x - 48\) c.a.o. | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(34.481\) | 2 marks | M1 for \(\frac{2(1^2-2^2)}{0.1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(16 + 32h + 24h^2 + 8h^3 + h^4\) c.a.o. | 3 marks | B2 for 4 terms correct; B1 for 3 terms correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(32 + 24h + 8h^2 + h^3\) or ft | 2 marks | B1 if one error |
| Answer | Marks |
|---|---|
| As \(h \to 0\), result \(\to\) their 32 from (iii)(B) | 1 mark |
| Gradient of tangent is limit of gradient of chord | 1 mark |
## (i)
$\frac{dy}{dx} = 4x^3$ | M1 |
When $x = 2, \frac{dy}{dx} = 32$ s.o.i. | A1 | i.s.w.
When $x = 2, y = 16$ s.o.i. | B1 |
$y = 32x - 48$ c.a.o. | A1 |
## (ii)
$34.481$ | 2 marks | M1 for $\frac{2(1^2-2^2)}{0.1}$
## (iii) (A)
$16 + 32h + 24h^2 + 8h^3 + h^4$ c.a.o. | 3 marks | B2 for 4 terms correct; B1 for 3 terms correct
## (iii) (B)
$32 + 24h + 8h^2 + h^3$ or ft | 2 marks | B1 if one error
## (iii) (C)
As $h \to 0$, result $\to$ their 32 from (iii)(B) | 1 mark |
Gradient of tangent is limit of gradient of chord | 1 mark |\begin{enumerate}[label=(\roman*)]
\item Find the equation of the tangent to the curve $y = x^4$ at the point where $x = 2$. Give your answer in the form $y = mx + c$. [4]
\item Calculate the gradient of the chord joining the points on the curve $y = x^4$ where $x = 2$ and $x = 2.1$. [2]
\item \begin{enumerate}[label=(\Alph*)]
\item Expand $(2 + h)^4$. [3]
\item Simplify $\frac{(2 + h)^4 - 2^4}{h}$. [2]
\item Show how your result in part (iii) $(B)$ can be used to find the gradient of $y = x^4$ at the point where $x = 2$. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 2010 Q10 [13]}}