| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiation from First Principles |
| Type | Chord gradient with h (algebraic) |
| Difficulty | Moderate -0.8 This is a structured, multi-part question that guides students through differentiation from first principles with explicit scaffolding. Parts (i)-(iii) are routine applications of the chord gradient formula and limit definition, (iv) is standard tangent equation finding, and (v) requires basic coordinate geometry. While comprehensive, each step is straightforward with no novel insight required, making it easier than average. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.07g Differentiation from first principles: for small positive integer powers of x1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| i \(6.1\) | 2 | M1 for \(\frac{(3.1^2-7)-(3^2-7)}{3.1-3}\) o.e. |
| ii \(\frac{((3+h)^2-7)-(3^2-7)}{h}\) | M1 | s.o.i. |
| numerator \(= 6h + h^2\) | M1 | |
| \(6 + h\) | A1 | |
| iii as \(h\) tends to 0, gradient tends to 6 o.e. f.t. from "6"+h | M1, A1 | |
| iv \(y - 2 =\) "6"\((x-3)\) o.e. | M1 | 6 may be obtained from \(\frac{dy}{dx}\) |
| \(y = 6x - 16\) | A1 | |
| v At P, \(x = 16/6\) o.e. or f.t. | M1 | |
| At Q, \(x = \sqrt{7}\) | M1 | |
| \(0.021\) cao | A1 |
# Question 12:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **i** $6.1$ | 2 | M1 for $\frac{(3.1^2-7)-(3^2-7)}{3.1-3}$ o.e. | **[2]** |
| **ii** $\frac{((3+h)^2-7)-(3^2-7)}{h}$ | M1 | s.o.i. |
| numerator $= 6h + h^2$ | M1 | |
| $6 + h$ | A1 | | **[3]** |
| **iii** as $h$ tends to 0, gradient tends to 6 o.e. f.t. from "6"+h | M1, A1 | | **[2]** |
| **iv** $y - 2 =$ "6"$(x-3)$ o.e. | M1 | 6 may be obtained from $\frac{dy}{dx}$ |
| $y = 6x - 16$ | A1 | | **[2]** |
| **v** At P, $x = 16/6$ o.e. or f.t. | M1 | |
| At Q, $x = \sqrt{7}$ | M1 | |
| $0.021$ cao | A1 | | **[3]** |12 (i) Calculate the gradient of the chord joining the points on the curve $y = x ^ { 2 } - 7$ for which $x = 3$ and $x = 3.1$.\\
(ii) Given that $\mathrm { f } ( x ) = x ^ { 2 } - 7$, find and simplify $\frac { \mathrm { f } ( 3 + h ) - \mathrm { f } ( 3 ) } { h }$.\\
(iii) Use your result in part (ii) to find the gradient of $y = x ^ { 2 } - 7$ at the point where $x = 3$, showing your reasoning.\\
(iv) Find the equation of the tangent to the curve $y = x ^ { 2 } - 7$ at the point where $x = 3$.\\
(v) This tangent crosses the $x$-axis at the point P . The curve crosses the positive $x$-axis at the point Q . Find the distance PQ , giving your answer correct to 3 decimal places.
\hfill \mbox{\textit{OCR MEI C2 2009 Q12 [12]}}