| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find second derivative |
| Difficulty | Easy -1.2 This is a straightforward differentiation exercise requiring only direct application of the power rule twice. The square root needs to be rewritten as x^(1/2), but this is standard C1 technique with no problem-solving or conceptual challenge beyond basic recall. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{dy}{dx} = 15x^4 - \frac{1}{2}x^{-1}\) | B1, B1, B1, 3 | \(15x^4\); \(kx^{-1}\); \(cx^4 - \frac{1}{2}x^{-1}\) only |
| (ii) \(\frac{d^2y}{dx^2} = 60x^3 + \frac{1}{4}x^{-3}\) | M1, A1, 2 | Attempt to differentiate their 2 term \(\frac{dy}{dx}\) and get one correctly differentiated term; \(60x^3 + \frac{1}{4}x^{-3}\) |
**(i)** $\frac{dy}{dx} = 15x^4 - \frac{1}{2}x^{-1}$ | B1, B1, B1, 3 | $15x^4$; $kx^{-1}$; $cx^4 - \frac{1}{2}x^{-1}$ only
**(ii)** $\frac{d^2y}{dx^2} = 60x^3 + \frac{1}{4}x^{-3}$ | M1, A1, 2 | Attempt to differentiate their 2 term $\frac{dy}{dx}$ and get one correctly differentiated term; $60x^3 + \frac{1}{4}x^{-3}$3 Given that $y = 3 x ^ { 5 } - \sqrt { x } + 15$, find\\
(i) $\frac { \mathrm { d } y } { \mathrm {~d} x }$,\\
(ii) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
\hfill \mbox{\textit{OCR C1 2006 Q3 [5]}}