Question 2 - A-Level Maths

Edexcel C1 2014 January — Question 2 5 marks

Exam Board	Edexcel
Module	C1 (Core Mathematics 1)
Year	2014
Session	January
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Tangents, normals and gradients
Type	Find second derivative
Difficulty	Easy -1.3 This is a straightforward C1 differentiation question requiring only direct application of the power rule twice. Students must rewrite 4/√x as 4x^(-1/2), then differentiate mechanically using standard index rules with no problem-solving or conceptual insight needed.
Spec	1.07d Second derivatives: d^2y/dx^2 notation 1.07i Differentiate x^n: for rational n and sums

2. $$y = 2 x ^ { 2 } - \frac { 4 } { \sqrt { } x } + 1 , \quad x > 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving each term in its simplest form.
Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$, giving each term in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{6081d81b-51d2-4140-9834-71ef7fd700b0-05_104_97_2613_1784}

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Answer	Marks	Guidance
(a)	$\frac{dy}{dx} = 2 \times 2x - 4x^{-\frac{1}{2}}(+0)$	M1
$\frac{dy}{dx} = 4x + 2x^{-\frac{3}{2}}$ or $4x + \frac{2}{x^{\frac{3}{2}}}$ 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 ie $\frac{dy}{dx} = ...$. However if the lhs is incorrect withhold the last A1.
(b)	$x^n \to x^{n+1}$	M1
$\frac{d^2y}{dx^2} = 4 - 3x^{-\frac{5}{2}}$ 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 $4 - \frac{3}{x^{\frac{5}{2}}}$ or $-3 \times x^{-2.5} + 4$. There is no requirement to give the lhs ie $\frac{d^2y}{dx^2} = ...$. However if the lhs is incorrect withhold the last A1.

(a) | $\frac{dy}{dx} = 2 \times 2x - 4x^{-\frac{1}{2}}(+0)$ | 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.

| $\frac{dy}{dx} = 4x + 2x^{-\frac{3}{2}}$ or $4x + \frac{2}{x^{\frac{3}{2}}}$ 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 ie $\frac{dy}{dx} = ...$. However if the lhs is incorrect withhold the last A1.

(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 $4 - \frac{3}{x^{\frac{5}{2}}}$ or $-3 \times x^{-2.5} + 4$. There is no requirement to give the lhs ie $\frac{d^2y}{dx^2} = ...$. However if the lhs is incorrect withhold the last A1.

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2.

$$y = 2 x ^ { 2 } - \frac { 4 } { \sqrt { } x } + 1 , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving each term in its simplest form.
\item Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$, giving each term in its simplest form.\\

\includegraphics[max width=\textwidth, alt={}, center]{6081d81b-51d2-4140-9834-71ef7fd700b0-05_104_97_2613_1784}
\end{enumerate}

\hfill \mbox{\textit{Edexcel C1 2014 Q2 [5]}}

This paper (10 questions)

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Q1 4 Q2 5 Q3 7 Q4 4 Q5 5 Q6 11 Q7 10 Q8 7 Q9 12 Q10 10

Answer	Marks	Guidance
(a)	\(\frac{dy}{dx} = 2 \times 2x - 4x^{-\frac{1}{2}}(+0)\)	M1
\(\frac{dy}{dx} = 4x + 2x^{-\frac{3}{2}}\) or \(4x + \frac{2}{x^{\frac{3}{2}}}\) 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 ie \(\frac{dy}{dx} = ...\). However if the lhs is incorrect withhold the last A1.
(b)	\(x^n \to x^{n+1}\)	M1
\(\frac{d^2y}{dx^2} = 4 - 3x^{-\frac{5}{2}}\) 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 \(4 - \frac{3}{x^{\frac{5}{2}}}\) or \(-3 \times x^{-2.5} + 4\). There is no requirement to give the lhs ie \(\frac{d^2y}{dx^2} = ...\). However if the lhs is incorrect withhold the last A1.