C3 January 2008 Q2

2. A curve \(C\) has equation\[y = \mathrm{e}^{2x}\tan x, \qquad x \neq (2n + 1)\frac{\pi}{2}.\]

(a) Show that the turning points on \(C\) occur where \(\tan x = -1\). (6)

(b) Find an equation of the tangent to \(C\) at the point where \(x = 0\). (2)

Mark scheme (a)

Scheme	Marks
\(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2\mathrm{e}^{2x}\tan x + \mathrm{e}^{2x}\sec^2 x\)	M1 A1+A1
\(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0 \Rightarrow 2\mathrm{e}^{2x}\tan x + \mathrm{e}^{2x}\sec^2 x = 0\)	M1
\(2\tan x + 1 + \tan^2 x = 0\)	A1
\((\tan x + 1)^2 = 0\) \(\tan x = -1\ \ \ast\) cso	A1
	(6)

Scheme	Marks
\(\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)_0 = 1\)	M1
Equation of tangent at \((0, 0)\) is \(y = x\)	A1
	(2)
	(8 marks)