Problems

01

Parametric curves: Points with given slope

Where on the image of $(3t^2-2t,t^3-6t)$ does the tangent line have slope $3$?

02

Parametric concavity

Find ${\displaystyle \frac{d^{2}y}{d{x}^2}}$ at $t=1$ for the curve given parametrically by $x=4-t^{-2}$, $y=t^{-1}+t$.

03

Parametric concavity

Find the interval(s) of $t$ on which the parametric curve $c(t)=(t^2,t^3-4t)$ is concave up.

04

Parametric arclength

Find the arclength of the curve given parametrically by $x=2t^2$, $y=3t^2-1$ over the time interval $0\le t\le 4$.

05

Minimum speed of a particle

Suppose a travelling particle has position modelled by the parametric curve:

$$c(t)=(t^3-4t,t^2+1)$$

What is the slowest speed of the particle?

06

Cycloid - Arclength and surface area of revolution

Consider the cycloid given parametrically by $c(t)=(t-\sin t,1-\cos t)$.

(a) Find the length of one arch of the cycloid.

(b) Suppose one arch of the cycloid is revolved around the $x$-axis. Find the area of this surface of revolution.