Q. 8 - Calculus

Question

Most of the parametric equations and vector-valued functions we have studied have component functions that are continuous. What happens when one of the component functions is discontinuous at a point? For example, the “floor” function $z\left(t\right)=⌊t⌋$ has a jump discontinuity for every integer $t$. What is the graph of the equations $x = \cos 2\pi t, y = \sin 2\pi t,z = t, t \in R$?

Step-by-Step Solution

Answer

The image of the graph has been added below.

Step 1. Given Information

The floor function is $z\left(t\right)=⌊t⌋$, it has jump discontinuity for every integer $t$.

Step 2. Taking example of circular helix

As we know, the objective is to construct a graph when one component is discontinuous at a point
Take an example of circular helix

 
$x\left(t\right)=\cos 2\mathrm{\pi }t......\left(1\right)$
$y\left(t\right)=\sin 2\mathrm{\pi t}......\left(2\right)$
$z\left(t\right)=⌊t⌋......\left(3\right)$

Step 3. Tabulating different values of t

A table of different values of $t$

$t$	$x\left(t\right)=\cos 2\mathrm{\pi }t$	$y\left(t\right)=\sin 2\mathrm{\pi t}$	$z\left(t\right)=⌊t⌋$	$(x,y,z)$
$0$	$1$	$0$	$0$	$(1,0,0)$
$1$	$1$	$0$	$1$	$(1,0,1)$
$2$	$1$	$0$	$2$	$(1,0,2)$
$3$	$1$	$0$	$3$	$(1,0,3)$
$4$	$1$	$0$	$4$	$(1,0,4)$

Step 4. Making the graph

Image of the graph.