[Maximum number: 5]

Find the exact coordinates of the stationary point on the curve with equation $y=5 x \mathrm{e}^{\frac{1}{2} x}$ .

The equation of a curve is $y=\frac{\mathrm{e}^{2 x}}{1+\mathrm{e}^{2 x}}$ . Show that the gradient of the curve at the point for which $x=\ln 3$ is $\frac{9}{50}$ .

[Maximum number: 4]

The parametric equations of a curve are

x=3\left(1+\sin ^{2} t\right), \quad y=2 \cos ^{3} t

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of t, simplifying your answer as far as possible.

The curve $y=\frac{\ln x}{x^{3}}$ has one stationary point. Find the x-coordinate of this point.

[Maximum number: 4]

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in each of the following cases:

(a)

$y=\ln (1+\sin 2 x)$ ,

[ 2 ]

(b)

$y=\frac{\tan x}{x}$ .

[ 2 ]

The parametric equations of a curve are

x=\frac{t}{2 t+3}, \quad y=\mathrm{e}^{-2 t}

Find the gradient of the curve at the point for which t=0.

[Maximum number: 5]

The equation of a curve is $\cos 3 x+5 \sin y=3$ .
Find the gradient of the curve at the point $\left(\frac{1}{9} \pi, \frac{1}{6} \pi\right)$ .

[Maximum number: 5]

The diagram shows the curve with equation

y=5 \sin 2 x-3 \tan 2 x

for values of x such that $0 \leqslant x<\frac{1}{4} \pi$ . Find the x-coordinate of the stationary point M, giving your answer correct to 3 significant figures.

[Maximum number: 7]

The parametric equations of a curve are

x=t+\ln (t+2), \quad y=(t-1) \mathrm{e}^{-2 t}

where t>-2.

(a)

Express $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of t, simplifying your answer.

[ 5 ]

(b)

Find the exact y-coordinate of the stationary point of the curve.

[ 2 ]

[Maximum number: 5]

The parametric equations of a curve are

x=3-\cos 2 \theta, \quad y=2 \theta+\sin 2 \theta,

for $0<\theta<\frac{1}{2} \pi$ .
Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\cot \theta$ .