| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find second derivative |
| Difficulty | Moderate -0.3 This is a straightforward application of the chain rule and product rule with no conceptual challenges. Part (i) is routine differentiation to find stationary points, and part (ii) requires a second derivative calculation. While the algebra involves some care due to the power of 8, the techniques are standard C3 material with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Obtain derivative of form \(kx(x^2+1)^7\) | M1 | any constant k |
| Obtain \(16x(x^2+1)^7\) | A1 | or equiv |
| Equate first derivative to 0 and confirm \(x=0\) or substitute \(x=0\) into first derivative to verify first derivative zero | M1 | AG; allow for deriv of form \(kx(x^2+1)^7\) |
| Refer, in some way, to \(x^2+1=0\) having no root | A1 | 4 or equiv |
| (ii) Attempt use of product rule | *M1 | obtaining \(\ldots + \ldots\) form |
| Obtain \(16(x^2+1)^7 + \ldots\) | A1∇ | follow their \(kx(x^2+1)^7\) |
| Obtain \(\ldots + 224x^3(x^2+1)^6\) | A1∇ | follow their \(kx(x^2+1)^7\); or unsimplified equiv |
| Substitute 0 in attempt at second derivative | M1 | dep *M |
| Obtain 16 | A1 | 5 from second derivative which is correct at some point |
(i) Obtain derivative of form $kx(x^2+1)^7$ | M1 | any constant k
Obtain $16x(x^2+1)^7$ | A1 | or equiv
Equate first derivative to 0 and confirm $x=0$ or substitute $x=0$ into first derivative to verify first derivative zero | M1 | AG; allow for deriv of form $kx(x^2+1)^7$
Refer, in some way, to $x^2+1=0$ having no root | A1 | 4 or equiv
(ii) Attempt use of product rule | *M1 | obtaining $\ldots + \ldots$ form
Obtain $16(x^2+1)^7 + \ldots$ | A1∇ | follow their $kx(x^2+1)^7$
Obtain $\ldots + 224x^3(x^2+1)^6$ | A1∇ | follow their $kx(x^2+1)^7$; or unsimplified equiv
Substitute 0 in attempt at second derivative | M1 | dep *M
Obtain 16 | A1 | 5 from second derivative which is correct at some point
**Total: 9**
---The equation of a curve is $y = (x^2 + 1)^8$.
\begin{enumerate}[label=(\roman*)]
\item Find an expression for $\frac{dy}{dx}$ and hence show that the only stationary point on the curve is the point for which $x = 0$. [4]
\item Find an expression for $\frac{d^2y}{dx^2}$ and hence find the value of $\frac{d^2y}{dx^2}$ at the stationary point. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2010 Q5 [9]}}