(a) Given that \(y=4 \sin 2 x \cos 2 x\), find the value of \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) when \(x=\frac{\pi}{6}\).

(b) A curve has equation \(y=4 \sin 2 x \cos 2 x\).

The normal to the curve at the point where \(x=\frac{\pi}{6}\) meets the \(x\)-axis at the point \(P\). Find the exact coordinates of \(P\).

(a) (i) Given that \(y=3 \sin ^{2} x+\cos x\), show that \(y+\cot x \frac{\mathrm{~d} y}{\mathrm{~d} x}=k\left(1+\cos ^{2} x\right), \quad\) where \(k\) is an integer.

(ii) Using your value of \(k\), solve the equation \(k\left(1+\cos ^{2} x\right)=4\) for \(-\pi \leqslant x \leqslant \pi\). (b) (i) Differentiate \(y=\tan (x-\sqrt{x})\) with respect to \(x\). (ii) Hence find \(\int \frac{2 \sqrt{x}-1}{\sqrt{x} \cos ^{2}(x-\sqrt{x})} \mathrm{d} x\).

Given that \(y=\tan \frac{x}{2}\), find the exact value of \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) when \(x=\frac{\pi}{3}\).

A curve has equation \(y=x\sin2x\).

(a) Find \(\frac{\mathrm dy}{\mathrm dx}\).

(b) Find the equation of the tangent to the curve at \(x=\frac\pi4\).

(c) Use your answer to part (a) to find the exact value of

\(\displaystyle \int_0^{\pi/6}2x\cos2x\,\mathrm dx.\)

(a) The function f is defined by \(\mathrm{f}(x)=\frac{1+2 \sin ^{2} x}{\cos ^{2} x}\) for \(-\frac{\pi}{2}\lt x\lt \frac{\pi}{2}\). (i) Show that \(\mathrm{f}(x)\) can be written as \(a \tan ^{2} x+b\), where \(a\) and \(b\) are integers. (ii) Hence solve the equation \(\mathrm{f}(x)=4\).

(iii) Hence also find the gradient of the curve \(y=\mathrm{f}(x)\) at each of the points where \(y=4\). (b) Solve the equation \(50 \cos ^{2} \theta=5 \sin \theta+47\) for \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\).

Variables \(x\) and \(y\) are related by the equation

\(y=2+\tan(1-x),\)

where \(0\leq x\leq\frac{\pi}{2}\). Given that \(x\) is increasing at a constant rate of \(0.04\) radians per second, find the corresponding rate of change of \(y\) when \(y=3\).

A curve has equation

\(y=\cos\frac{x}{4},\)

where \(x\) is in radians. The normal to the curve at the point where \(x=\frac{4\pi}{3}\) cuts the \(x\)-axis at the point \(P\). Find the exact coordinates of \(P\).

The function \(f\) is defined by \(f(x)=3\sin^2x-2\cos x\) for \(2\le x\le4\), where \(x\) is in radians.

(a) Find the \(x\)-coordinate of the stationary point on the curve \(y=f(x)\).

(b) Solve the equation \(f(x)=1-3\cos x\).

You are given that

\(y=\frac1{\cos2x}.\)

(a) Show that

\(\frac{dy}{dx}=\frac{k\sin2x}{\cos^22x},\)

where \(k\) is a constant to be found.

(b) Find the values of \(x\) such that

\(\frac{dy}{dx}=\frac5{\sin2x}\)

for \(0\lt x\lt \frac{\pi}{2}\).

The curve \(y=3\tan^2 x\) is defined for \(0^\circ\lt x\lt 360^\circ\).

(a) Show that

\(\frac{dy}{dx}=m\tan x\operatorname{sec}^2x,\)

where \(m\) is a constant to be found.

(b) Find all the values of \(x\) for which

\(\frac{dy}{dx}=3\operatorname{sec}x\operatorname{cosec}x.\)

It is given that \(y=\dfrac{\tan 3x}{\sin x}\).

(a) Find the exact value of \(\dfrac{dy}{dx}\) when \(x=\dfrac{\pi}{3}\).

(b) Hence find the approximate change in \(y\) as \(x\) increases from \(\dfrac{\pi}{3}\) to \(\dfrac{\pi}{3}+h\), where \(h\) is small.

(c) Given that \(x\) is increasing at the rate of \(3\) units per second, find the corresponding rate of change in \(y\) when \(x=\dfrac{\pi}{3}\), giving your answer in its simplest surd form.

Differentiate \(\tan 3x\cos\dfrac{x}{2}\) with respect to \(x\).

Given that \(y=2\sin 3x+\cos 3x\), show that

\(\displaystyle \frac{d^2y}{dx^2}+\frac{dy}{dx}+3y=k\sin 3x,\)

where \(k\) is a constant to be determined.

The curve \(y=4+5\sin 3x\) passes through the point \(P\), where \(x=\dfrac13\pi\).

(i) Find \(\dfrac{dy}{dx}\).

(ii) Find the equation of the tangent to the curve at \(P\), giving your answer in the form \(y=mx+c\).

(i) Differentiate \(1+\tan\left(\dfrac{x}{3}\right)\) with respect to \(x\).

(ii) Hence find \(\displaystyle\int \operatorname{sec}^2\left(\dfrac{x}{3}\right)\,dx\).

In this question, the units of \(x\) are radians and the units of \(y\) are centimetres.

It is given that \(y=(1+\cos3x)^{10}\).

(i) Find the value of \(\dfrac{dy}{dx}\) when \(x=\dfrac{\pi}{2}\).

Given also that \(y\) is increasing at a rate of \(6\text{ cm s}^{-1}\) when \(x=\dfrac{\pi}{2}\),

(ii) find the corresponding rate of change of \(x\).

A curve has equation

\(y=\frac{x^3}{\sin 2x}.\)

Find

(i) \(\dfrac{dy}{dx}\),

(ii) the equation of the tangent to the curve at the point where \(x=\dfrac{\pi}{4}\).

(i) Differentiate \((\cos x)^{-1}\) with respect to \(x\).

(ii) Hence find \(\dfrac{dy}{dx}\), given that \(y=\tan x+4(\cos x)^{-1}\).

(iii) Using your answer to part (ii), find the values of \(x\) in the range \(0\leq x\leq2\pi\) such that \(\dfrac{dy}{dx}=4\).

(i) Without using a calculator, solve the equation \(6c^3-7c^2+1=0\).

It is given that \(y=\tan x+6\sin x\).

(ii) Find \(\dfrac{dy}{dx}\).

(iii) If \(\dfrac{dy}{dx}=7\), show that \(6\cos^3x-7\cos^2x+1=0\).

(iv) Hence solve the equation \(\dfrac{dy}{dx}=7\), for \(0\leq x\leq\pi\) radians.