| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - mixed transcendental products |
| Difficulty | Standard +0.3 This is a straightforward application of the product rule to find stationary points of e^(2x)cos(x). Students must differentiate using product and chain rules, set the derivative to zero, and solve a simple trigonometric equation (tan x = 2). While it involves transcendental functions, it's a standard textbook exercise with no novel insight required, making it slightly easier than average. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = e^{2x}\cos x\) | M1 | product rule used |
| \(\Rightarrow \frac{dy}{dx} = 2e^{2x}\cos x - e^{2x}\sin x\) | A1 | cao – mark final ans |
| \(\frac{dy}{dx} = 0 \Rightarrow e^{2x}(2\cos x - \sin x) = 0\) | M1 | their derivative = 0 |
| \(\Rightarrow 2\cos x = \sin x\) | M1 | sin x / cos x = tan x used |
| \(\Rightarrow 2 = \sin x/\cos x = \tan x\) | A1 | 1.1 or 0.35π or better, or arctan 2, not 63.4°; but condone ans given in both degrees and radians here |
| \(\Rightarrow y = 4.09\) | A1 cao | art 4.1 |
| Total: [6] | consistent with their derivs e.g. 2e^{2x} - e^{2x}tan x is A0; no choice; 1.1071487 …, 0.352416…π, penalise incorrect rounding |
$y = e^{2x}\cos x$ | M1 | product rule used
$\Rightarrow \frac{dy}{dx} = 2e^{2x}\cos x - e^{2x}\sin x$ | A1 | cao – mark final ans
$\frac{dy}{dx} = 0 \Rightarrow e^{2x}(2\cos x - \sin x) = 0$ | M1 | their derivative = 0
$\Rightarrow 2\cos x = \sin x$ | M1 | sin x / cos x = tan x used
$\Rightarrow 2 = \sin x/\cos x = \tan x$ | A1 | 1.1 or 0.35π or better, or arctan 2, not 63.4°; but condone ans given in both degrees and radians here
$\Rightarrow y = 4.09$ | A1 cao | art 4.1
**Total: [6]** | | consistent with their derivs e.g. 2e^{2x} - e^{2x}tan x is A0; no choice; 1.1071487 …, 0.352416…π, penalise incorrect rounding
---1 Fig. 1 shows part of the curve $y = \mathrm { e } ^ { 2 x } \cos x$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{955bebfb-04a3-4cd9-a33e-a8ba4b73e2ba-2_670_1029_404_504}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
Find the coordinates of the turning point P .
\hfill \mbox{\textit{OCR MEI C3 2015 Q1 [6]}}