OCR MEI C2 2008 January — Question 12 12 marks

Exam BoardOCR MEI
ModuleC2 (Core Mathematics 2)
Year2008
SessionJanuary
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicApplied differentiation
TypeTangent, normal and triangle area
DifficultyModerate -0.8 This is a straightforward C2 question with routine tasks: finding intersection points, calculating triangle area, integration for area between curves, and completing a standard first principles differentiation exercise. All parts require only direct application of standard techniques with no problem-solving insight or novel approaches needed.
Spec1.02q Use intersection points: of graphs to solve equations1.07g Differentiation from first principles: for small positive integer powers of x1.08e Area between curve and x-axis: using definite integrals
12
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-5_652_764_269_733} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
    (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
    (B) Find the area of the region bounded by the line and the curve.
  2. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
    (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$f ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (D) State what this limit represents.
iA
\(x^4 = 8x\)
\((2, 16)\) c.a.o.
AnswerMarks Guidance
\(PQ = 16\) and completion to show \(\frac{1}{2} \times 2 \times 16 = 16\)M1, A1, A1 NB answer 16 given
iB
\(\frac{x^5}{5}\)
evaluating their integral at their co-ord of P and zero [or \(32/5\) o.e.]
AnswerMarks Guidance
\(9.6\) o.e.M1, M1, A1 ft only if integral attempted, not for \(x^4\) or differentiation c.a.o.
iiA
AnswerMarks Guidance
\(6x^2h^2 + 4xh^3 + h^3\)2 B1 for two terms correct.
iiB
AnswerMarks Guidance
\(4x^3 + 6x^2h + 4xh^2 + h^3\)2 B1 for three terms correct
iiC
AnswerMarks Guidance
\(4x^3\)1 —
iiD
AnswerMarks Guidance
gradient of [tangent to] curve1 — 
**iA**

$x^4 = 8x$
$(2, 16)$ c.a.o.
$PQ = 16$ and completion to show $\frac{1}{2} \times 2 \times 16 = 16$ | M1, A1, A1 | NB answer 16 given | 3

**iB**

$\frac{x^5}{5}$
evaluating their integral at their co-ord of P and zero [or $32/5$ o.e.]
$9.6$ o.e. | M1, M1, A1 | ft only if integral attempted, not for $x^4$ or differentiation c.a.o. | 3

**iiA**

$6x^2h^2 + 4xh^3 + h^3$ | 2 | B1 for two terms correct. | 2

**iiB**

$4x^3 + 6x^2h + 4xh^2 + h^3$ | 2 | B1 for three terms correct | 2

**iiC**

$4x^3$ | 1 | — | 1

**iiD**

gradient of [tangent to] curve | 1 | — | 1
12
\begin{enumerate}[label=(\roman*)]
\item \begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-5_652_764_269_733}
\captionsetup{labelformat=empty}
\caption{Fig. 12}
\end{center}
\end{figure}

Fig. 12 shows part of the curve $y = x ^ { 4 }$ and the line $y = 8 x$, which intersect at the origin and the point P .\\
(A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.\\
(B) Find the area of the region bounded by the line and the curve.
\item You are given that $\mathrm { f } ( x ) = x ^ { 4 }$.\\
(A) Complete this identity for $\mathrm { f } ( x + h )$.

$$f ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$

(B) Simplify $\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }$.\\
(C) Find $\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }$.\\
(D) State what this limit represents.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C2 2008 Q12 [12]}}
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