| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | January |
| 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 introduction to differentiation from first principles with clear scaffolding: part (i) is basic coordinate geometry (finding chord gradient), part (ii) tests conceptual understanding that closer points give better approximations, and part (iii) is routine differentiation using the power rule. All steps are standard techniques with no problem-solving insight required, making it easier than average. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| \(y_B = \frac{4}{2.1^2} = 0.9070...\) | M1 | Correct method for gradient of chord |
| Gradient \(= \frac{0.9070 - 1}{2.1 - 2} = -0.93\) | A1 | cao to 2 d.p. |
| Answer | Marks | Guidance |
|---|---|---|
| Any \(x\) value between 2 and 2.1 e.g. \(x = 2.05\) | B1 | Closer to \(x = 2\) than B |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = 4x^{-2}\), \(\frac{dy}{dx} = -8x^{-3}\) | M1 | Attempt differentiation |
| At \(x = 2\): gradient \(= -8/8 = -1\) | A1 |
## Question 5(i):
$y_B = \frac{4}{2.1^2} = 0.9070...$ | M1 | Correct method for gradient of chord
Gradient $= \frac{0.9070 - 1}{2.1 - 2} = -0.93$ | A1 | cao to 2 d.p.
## Question 5(ii):
Any $x$ value between 2 and 2.1 e.g. $x = 2.05$ | B1 | Closer to $x = 2$ than B
## Question 5(iii):
$y = 4x^{-2}$, $\frac{dy}{dx} = -8x^{-3}$ | M1 | Attempt differentiation
At $x = 2$: gradient $= -8/8 = -1$ | A1 |
---5 A is the point $( 2,1 )$ on the curve $y = \frac { 4 } { x ^ { 2 } }$.\\
B is the point on the same curve with $x$-coordinate 2.1.\\
(i) Calculate the gradient of the chord AB of the curve. Give your answer correct to 2 decimal places.\\
(ii) Give the $x$-coordinate of a point C on the curve for which the gradient of chord AC is a better approximation to the gradient of the curve at A .\\
(iii) Use calculus to find the gradient of the curve at A .
\hfill \mbox{\textit{OCR MEI C2 2007 Q5 [5]}}