OCR MEI C2 2007 June — Question 2 5 marks

Exam BoardOCR MEI
ModuleC2 (Core Mathematics 2)
Year2007
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTangents, normals and gradients
TypeFind second derivative
DifficultyEasy -1.3 This is a straightforward differentiation exercise requiring only routine application of the power rule twice, followed by basic arithmetic substitution. The calculation at x=36 involves simple fractional arithmetic with no conceptual challenge—well below average difficulty for A-level.
Spec1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums
2 Given that \(y = 6 x ^ { \frac { 3 } { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show, without using a calculator, that when \(x = 36\) the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) is \(\frac { 3 } { 4 }\).
Question 2:
AnswerMarks Guidance
\(\frac{\mathrm{d}y}{\mathrm{d}x} = 9x^{\frac{1}{2}}\)M1 A1 M1 for attempt to differentiate
\(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = \frac{9}{2}x^{-\frac{1}{2}}\)M1 A1 M1 for differentiating their \(\frac{\mathrm{d}y}{\mathrm{d}x}\)
When \(x = 36\): \(\frac{9}{2} \times \frac{1}{6} = \frac{9}{12} = \frac{3}{4}\)A1 Must show working without calculator 
## Question 2:

$\frac{\mathrm{d}y}{\mathrm{d}x} = 9x^{\frac{1}{2}}$ | M1 A1 | M1 for attempt to differentiate

$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = \frac{9}{2}x^{-\frac{1}{2}}$ | M1 A1 | M1 for differentiating their $\frac{\mathrm{d}y}{\mathrm{d}x}$

When $x = 36$: $\frac{9}{2} \times \frac{1}{6} = \frac{9}{12} = \frac{3}{4}$ | A1 | Must show working without calculator

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2 Given that $y = 6 x ^ { \frac { 3 } { 2 } }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.\\
Show, without using a calculator, that when $x = 36$ the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ is $\frac { 3 } { 4 }$.

\hfill \mbox{\textit{OCR MEI C2 2007 Q2 [5]}}
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Q1 3 Q2 5 Q3 4 Q4 4 Q5 5 Q6 5 Q7 5 Q8 5 Q9 12 Q10 12 Q11 12