| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Second derivative and nature determination |
| Difficulty | Moderate -0.8 This is a straightforward differentiation question requiring basic power rule application (rewriting 4/x as 4x^{-1}) and evaluating the second derivative at a point. It involves routine calculus techniques with no problem-solving or conceptual challenges, making it easier than the average A-level question. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f'(x) = -4x^{-2}-3\) | M1 | Attempt to differentiate |
| A1 | \(-4x^{-2}\) | |
| A1 | Fully correct derivative (no "\(+c\)") |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f''(x) = 8x^{-3}\) | M1* | Attempts to differentiate their (i). Must involve reducing power of an \(x\) term by 1 |
| A1 | Correct derivative. \(f''(x)\) must involve \(x\) | |
| \(f''\!\left(\frac{1}{2}\right) = \frac{8}{\left(\frac{1}{2}\right)^3}\) | M1dep | Substitutes \(x=\frac{1}{2}\) correctly into their \(f''(x)\); allow "invisible brackets" |
| \(= 64\) | A1 | www |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f'(x) = -4x^{-2}-3$ | M1 | Attempt to differentiate |
| | A1 | $-4x^{-2}$ |
| | A1 | Fully correct derivative (no "$+c$") |
**Guidance:** $kx^{-2}$ or $-3$ correctly obtained
---
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f''(x) = 8x^{-3}$ | M1* | Attempts to differentiate their (i). Must involve reducing power of an $x$ term by 1 |
| | A1 | Correct derivative. $f''(x)$ must involve $x$ |
| $f''\!\left(\frac{1}{2}\right) = \frac{8}{\left(\frac{1}{2}\right)^3}$ | M1dep | Substitutes $x=\frac{1}{2}$ correctly into their $f''(x)$; allow "invisible brackets" |
| $= 64$ | A1 | www |
---6 Given that $\mathrm { f } ( x ) = \frac { 4 } { x } - 3 x + 2$,\\
(i) find $\mathrm { f } ^ { \prime } ( x )$,\\
(ii) find $\mathrm { f } ^ { \prime \prime } \left( \frac { 1 } { 2 } \right)$.
\hfill \mbox{\textit{OCR C1 2012 Q6 [7]}}