Parametric

A function is defined parametrically by the pair of equations:

$x=3t^2+1$ and $y=\cos t.$

Find an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$.

Char goes for a ride on a Ferris wheel. As she rotates around,
her position can be described by the pair of parametric
equations :

$x=5\sqrt{2}\sin \left(\frac{\pi t}{5}\right)$ and $y=10-5\sqrt{2}\cos \left(\frac{\pi t}{5}\right)$

where $t$ is time, in seconds, from the start of the ride.

Find the gradient of the normal to this curve at the point when
$t=6.25$ seconds, after the start of the ride.

*You must use calculus and show any derivatives that you need
to find when solving this problem.*

A curve is defined parametrically by the equations:

$x=2+3t$ and $y=3t-\ln (3t-1)$ where $t>\frac{1}{3}$.

Find the coordinates, $(x,y)$, of any point(s) on the curve where the tangent to the curve has a gradient
of $\frac{1}{2}$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

A curve is defined by the parametric equations $x=\ln (t)$ and $y=6t^3$ where $t>0$.

The point P lies on the curve, and at point P, $\frac{d^{2}y}{d{x}^2}=2$.

Find the exact coordinates of point P.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

A curve is defined parametrically by the equations $x=\frac{1}{(5-t{)}^2}$ and $y=5t-t^2$.

Find the gradient of the tangent to the curve at the point when $t=2$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

A curve is defined by the parametric equations

$x=t^3+1$
$y=t^2+1$

Show that $\frac{\frac{d^{2}y}{d{x}^2}}{{\left(\frac{dy}{dx}\right)}^4}$ is a constant.

A curve is defined parametrically by the parametric equations

$x=5e^{2t}$

$y=2e^{5t}$

Find the gradient of the tangent to this curve at the point where $t=0$.
*You must use calculus and show any derivatives that you need to find when solving this problem.*

A curve is defined parametrically by the equations $x=\sqrt{t+1}$ and $y=\sin 2t$.

Find the gradient of the tangent to the curve at the point when $t=0$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

A curve is defined by the parametric equations

$x=2\cos 2t$ and $y={\tan }^{2}t$.

Find the gradient of the tangent to the curve at the point where $t=\frac{\pi }{4}$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

A curve is defined parametrically by the equations $x=3\cos t$ and $y=\sin 3t$.

Find the gradient of the normal to the curve at the point where $t=\frac{\pi }{4}$.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

If $x=2\sin t$ and $y=\cos 2t$ show that $\frac{dy}{dx}=-2\sin t$.

A curve is defined by the parametric equations:

$x=5\sin t$ and $y=3\tan t$

Find the gradient of the normal to the curve at the point where $t=\frac{\pi }{3}$.

*Show any derivatives that you need to find when solving this problem.*

A curve is defined by the parametric equations:

$x=t^2-t$ and $y=t^3-3t$

Find the coordinates of the point(s) on the curve for which the normal to the curve is parallel
to the $y$-axis.

*You must use calculus and clearly show your working, including any derivatives you need to
find when solving this problem.*