Standard +0.3 This requires product rule differentiation of exponential and trigonometric functions, then solving a transcendental equation for stationary points. While it involves multiple techniques (product rule, setting derivative to zero, solving tan x = -2), these are standard C4 procedures with no novel insight required. The numerical solving aspect and restricted domain make it slightly above average difficulty but still routine for this level.
3 The equation of a curve is \(y = \mathrm { e } ^ { 2 x } \cos x\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of any stationary points for which \(- \pi \leqslant x \leqslant \pi\). Give your answers correct to 3 significant figures.
\(\frac{dy}{dx}=\pm ke^{2x}\cos x \pm e^{2x}\sin x\)
M1*
\(k\) is any constant; Product Rule
\(\frac{dy}{dx}=2e^{2x}\cos x - e^{2x}\sin x\) oe
A1
Their \(\frac{dy}{dx}=0\)
M1dep*
\(\tan x=2\) or \(\cos x=(\pm)\frac{1}{\sqrt{5}}\) or \(\sin x=(\pm)\frac{2}{\sqrt{5}}\)
A1
Ignore omission of "\(e^{2x}=0\) has no solution"; or \(\sqrt{5}\cos(x+\tan^{-1}\frac{1}{2})=0\)
\(x=1.11\) and \(-2.03\) cao
A1
\((1.11, 4.09)\) and/or \((-2.03,-0.00765)\); if A0A0, SC1 for all 4 values to greater precision
\(y=4.09\) and \(-0.00765\) cao
A1
Or A1 for each correct pair of coordinates; extra values within range incur penalty of one mark
# Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx}=\pm ke^{2x}\cos x \pm e^{2x}\sin x$ | M1* | $k$ is any constant; Product Rule |
| $\frac{dy}{dx}=2e^{2x}\cos x - e^{2x}\sin x$ oe | A1 | |
| Their $\frac{dy}{dx}=0$ | M1dep* | |
| $\tan x=2$ or $\cos x=(\pm)\frac{1}{\sqrt{5}}$ or $\sin x=(\pm)\frac{2}{\sqrt{5}}$ | A1 | Ignore omission of "$e^{2x}=0$ has no solution"; or $\sqrt{5}\cos(x+\tan^{-1}\frac{1}{2})=0$ |
| $x=1.11$ and $-2.03$ cao | A1 | $(1.11, 4.09)$ and/or $(-2.03,-0.00765)$; if **A0A0**, **SC1** for all 4 values to greater precision |
| $y=4.09$ and $-0.00765$ cao | A1 | Or **A1** for each correct pair of coordinates; extra values within range incur penalty of one mark |
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3 The equation of a curve is $y = \mathrm { e } ^ { 2 x } \cos x$. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence find the coordinates of any stationary points for which $- \pi \leqslant x \leqslant \pi$. Give your answers correct to 3 significant figures.
\hfill \mbox{\textit{OCR C4 2015 Q3 [6]}}