| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find second derivative |
| Difficulty | Moderate -0.8 This is a straightforward differentiation exercise requiring only standard power rule application. Part (i) involves rewriting the term with a square root as a power (x^{-1/2}) and differentiating three terms—routine for C1. Part (ii) simply requires differentiating the answer from (i) again. No problem-solving, geometric interpretation, or novel insight needed; purely mechanical application of basic differentiation rules. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| i) \(y = 6x^3 + 4x^{-\frac{1}{2}} + 5x\); \(\frac{dy}{dx} = 18x^2 - 2x^{-\frac{3}{2}} + 5\) | B1, M1, A1, A1 [4] | \(\frac{4}{\sqrt{x}} = 4x^{-\frac{1}{2}}\) soi. Attempt to differentiate, any term correct. Two correct terms. Fully correct, no "+c" |
| ii) \(\frac{d^2y}{dx^2} = 36x + 3x^{-\frac{5}{2}}\) | M1, A1 [2] | Attempt to differentiate their \(\frac{dy}{dx}\). cao www in either part. Any term still involving \(x\) correct – follow through from their expression for the M mark only |
**i)** $y = 6x^3 + 4x^{-\frac{1}{2}} + 5x$; $\frac{dy}{dx} = 18x^2 - 2x^{-\frac{3}{2}} + 5$ | B1, M1, A1, A1 [4] | $\frac{4}{\sqrt{x}} = 4x^{-\frac{1}{2}}$ soi. Attempt to differentiate, any term correct. Two correct terms. Fully correct, no "+c"
**ii)** $\frac{d^2y}{dx^2} = 36x + 3x^{-\frac{5}{2}}$ | M1, A1 [2] | Attempt to differentiate their $\frac{dy}{dx}$. cao www in either part. Any term still involving $x$ correct – follow through from their expression for the M mark onlyGiven that $y = 6x^3 + \frac{4}{\sqrt{x}} + 5x$, find
\begin{enumerate}[label=(\roman*)]
\item $\frac{\text{d}y}{\text{d}x}$, [4]
\item $\frac{\text{d}^2y}{\text{d}x^2}$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 2014 Q6 [6]}}