| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiation from First Principles |
| Type | Chord gradient estimation |
| Difficulty | Moderate -0.8 This is a straightforward application of chord gradient calculation (y₂-y₁)/(x₂-x₁) with simple arithmetic involving square roots. Part (ii) requires minimal insight—just choosing a closer x-value like 4.01. No calculus derivation needed, purely numerical computation with clear instructions. Easier than average A-level questions which typically require multiple techniques or conceptual understanding. |
| Spec | 1.07a Derivative as gradient: of tangent to curve |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(m = \frac{\sqrt{1+2\times 4.1}-\sqrt{1+2\times 4}}{4.1-4}\) s.o.i | M1 | no marks for use of Chain Rule or any other attempt to differentiate |
| \(\text{grad} = \frac{\sqrt{9.2}-\sqrt{9}}{4.1-4}\) s.o.i | M1 | SC2 for 0.33… appearing only embedded in equation of chord |
| \(0.3315\) cao | A1 | |
| (ii) selection of value in \((4, 4.1)\) and 4 or of two values in \([3.9, 4.1]\) centred on 4 | M1 | allow selection of 4 and value in \((3.9, 4)\) |
| answer closer to \(\frac{1}{3}\) than \(0.3315(...)\) | A1 |
(i) $m = \frac{\sqrt{1+2\times 4.1}-\sqrt{1+2\times 4}}{4.1-4}$ s.o.i | M1 | no marks for use of Chain Rule or any other attempt to differentiate
$\text{grad} = \frac{\sqrt{9.2}-\sqrt{9}}{4.1-4}$ s.o.i | M1 | SC2 for 0.33… appearing only embedded in equation of chord
$0.3315$ cao | A1 |
(ii) selection of value in $(4, 4.1)$ and 4 or of two values in $[3.9, 4.1]$ centred on 4 | M1 | allow selection of 4 and value in $(3.9, 4)$
answer closer to $\frac{1}{3}$ than $0.3315(...)$ | A1 |3 The equation of a curve is $y = \sqrt { 1 + 2 x }$.\\
(i) Calculate the gradient of the chord joining the points on the curve where $x = 4$ and $x = 4.1$. Give your answer correct to 4 decimal places.\\
(ii) Showing the points you use, calculate the gradient of another chord of the curve which is a closer approximation to the gradient of the curve when $x = 4$.
\hfill \mbox{\textit{OCR MEI C2 2011 Q3 [5]}}