| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2014 |
| 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 standard power rule application. Students must rewrite the term as x^{-1/2}, differentiate twice, and simplify—all routine A-level techniques with no problem-solving or conceptual challenge beyond basic calculus mechanics. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(2x^2 - \frac{4}{\sqrt{x}} + 1 = 2x^2 - 4x^{-\frac{1}{2}} + 1\) | M1 | \(x^n \to x^{n-1}\) for any term. The sight of \(2x^2 \to Ax\) OR \(Cx^{-\frac{1}{2}} \to Dx^{-\frac{3}{2}}\) OR \(1 \to 0\) is sufficient. Do not follow through on an incorrect index of \(-\frac{4}{\sqrt{x}}\) for this mark. |
| \(\frac{dy}{dx} = 2 \times 2x - 4 \times (-\frac{1}{2})x^{-\frac{3}{2}}(+0)\) | M1 | |
| \(\frac{dy}{dx} = 4x + 2x^{-\frac{3}{2}}\) or \(4x + \frac{2}{x^{\frac{3}{2}}}\) or \(oe\) | A1, A1 | One of the first two terms correct and simplified. Either \(4x\) or \(2x^{-\frac{3}{2}}\). Accept equivalents such as \(4 \times x\) and \(2 \times x^{\frac{3}{2}} = \frac{2}{x^{1.5}}\). Ignore \(+c\) for this mark. Do not accept unsimplified terms like \(2 \times 2x\). A completely correct solution with no \(+c\). That is \(4x + 2x^{-\frac{3}{2}}\). Accept simplified equivalent expressions such as \(4 \times x + 2 \times x^{-\frac{3}{2}}\) or \(4x + \frac{2}{x^{\frac{3}{2}}}\). There is no requirement to give the lhs is \(\frac{dy}{dx} = ...\). However if the lhs is incorrect withhold the last A1 |
| (3 marks) | ||
| (b) \(x^n \to x^{n-1}\) | M1 | For either \(4x \to 4\) or \(x^n \to x^{n-1}\) for a fractional term. Follow through on incorrect answers in (a). |
| \(\frac{d^2y}{dx^2} = 4 - 3x^{-\frac{5}{2}}\) or \(4 - \frac{3}{x^{\frac{5}{2}}}\) | A1 | A completely correct solution \(4 - 3x^{-\frac{5}{2}}\). Award for expressions such as \(4 - 3x^{-\frac{5}{2}}\) or \(4 - \frac{3}{x^{\frac{5}{2}}}\). There is no requirement to give the lhs is \(\frac{d^2y}{dx^2} = ...\). However if the lhs is incorrect withhold the last A1 |
| (2 marks) | ||
| (5 marks) |
(a) $2x^2 - \frac{4}{\sqrt{x}} + 1 = 2x^2 - 4x^{-\frac{1}{2}} + 1$ | M1 | $x^n \to x^{n-1}$ for any term. The sight of $2x^2 \to Ax$ OR $Cx^{-\frac{1}{2}} \to Dx^{-\frac{3}{2}}$ OR $1 \to 0$ is sufficient. Do not follow through on an incorrect index of $-\frac{4}{\sqrt{x}}$ for this mark.
$\frac{dy}{dx} = 2 \times 2x - 4 \times (-\frac{1}{2})x^{-\frac{3}{2}}(+0)$ | M1 |
$\frac{dy}{dx} = 4x + 2x^{-\frac{3}{2}}$ or $4x + \frac{2}{x^{\frac{3}{2}}}$ or $oe$ | A1, A1 | One of the first two terms correct and simplified. Either $4x$ or $2x^{-\frac{3}{2}}$. Accept equivalents such as $4 \times x$ and $2 \times x^{\frac{3}{2}} = \frac{2}{x^{1.5}}$. Ignore $+c$ for this mark. Do not accept unsimplified terms like $2 \times 2x$. A completely correct solution with no $+c$. That is $4x + 2x^{-\frac{3}{2}}$. Accept simplified equivalent expressions such as $4 \times x + 2 \times x^{-\frac{3}{2}}$ or $4x + \frac{2}{x^{\frac{3}{2}}}$. There is no requirement to give the lhs is $\frac{dy}{dx} = ...$. However if the lhs is incorrect withhold the last A1
| | **(3 marks)**
(b) $x^n \to x^{n-1}$ | M1 | For either $4x \to 4$ or $x^n \to x^{n-1}$ for a fractional term. Follow through on incorrect answers in (a).
$\frac{d^2y}{dx^2} = 4 - 3x^{-\frac{5}{2}}$ or $4 - \frac{3}{x^{\frac{5}{2}}}$ | A1 | A completely correct solution $4 - 3x^{-\frac{5}{2}}$. Award for expressions such as $4 - 3x^{-\frac{5}{2}}$ or $4 - \frac{3}{x^{\frac{5}{2}}}$. There is no requirement to give the lhs is $\frac{d^2y}{dx^2} = ...$. However if the lhs is incorrect withhold the last A1
| | **(2 marks)**
| | **(5 marks)**
---$y = 2x^2 - \frac{4}{\sqrt{x}} + 1$, $x > 0$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac{dy}{dx}$, giving each term in its simplest form. [3]
\item Find $\frac{d^2y}{dx^2}$, giving each term in its simplest form. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2014 Q2 [5]}}