URGENT Parametrics

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Dec 18th 2013, 07:22 PM
orange

URGENT Parametrics

Let http://data.artofproblemsolving.com/...1c4df4304f.gif be the graph of the parametric equations

http://data.artofproblemsolving.com/...d0ea6ee9f9.gif

What is the length of the smallest interval http://data.artofproblemsolving.com/...63c10b675d.gif such that the graph of these equations for all http://data.artofproblemsolving.com/...f9566ea488.gif produces the entire graph http://data.artofproblemsolving.com/...1c4df4304f.gif?
Dec 18th 2013, 08:25 PM
romsek

Re: URGENT Parametrics

Quote:

Originally Posted by orange

Let http://data.artofproblemsolving.com/...1c4df4304f.gif be the graph of the parametric equations

http://data.artofproblemsolving.com/...d0ea6ee9f9.gif

What is the length of the smallest interval http://data.artofproblemsolving.com/...63c10b675d.gif such that the graph of these equations for all http://data.artofproblemsolving.com/...f9566ea488.gif produces the entire graph http://data.artofproblemsolving.com/...1c4df4304f.gif?

Cos[4t] needs (2pi)/4 = pi/2

Sin[6t] needs (2pi)/6 = pi/3

what's the least common multiple of 1/2 and 1/3 ? So what's the minimum interval needed for a full cycle?
Dec 19th 2013, 02:04 PM
orange

Re: URGENT Parametrics

1 is their LCM. So I guess the minimum interval is just pi?
Dec 19th 2013, 02:29 PM
romsek

Re: URGENT Parametrics

give that man a kewpie doll!
Dec 19th 2013, 03:28 PM
Soroban

Re: URGENT Parametrics

Hello, orange!

I "eyeballed" the problem.
I think I have the solution.

Quote:

Let $G$ be the graph of the parametric equations: . $\begin{Bmatrix}x &=& \cos(4t) \\ y &=& \sin(6t)\end{Bmatrix}$

What is the length of the smallest interval $I$ such that the graph
of the equations for all $t \in I$ produces the entire graph of $G$ ?

When $t = 0\!:\;\begin{Bmatrix}x &=& \cos(0) &=& 1 \\ y &=& \sin(0) &=& 0\end{Bmatrix}$

At the "start", the graph is at $(1,0).$

Here is what I found:

. . $\begin{array}{c|c|c|} t & \cos(4t) & \sin(6t) \\ \hline 0 & 1 & 0 \\ \frac{\pi}{4} & \text{-}1 & \text{-}1 \\ \frac{\pi}{2} & 1 & 0 \\ \frac{3\pi}{4} & \text{-}1 & 1 \\ \hline \pi & 1 & 0 \\ \vdots & \vdots & \vdots\end{array}$

. . and the cycle repeats.

Therefore: . $I \,=\,[0,\,\pi]$
Dec 20th 2013, 11:09 AM
johng

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Re: URGENT Parametrics

Hi,
If a given parametric curve is periodic, finding its period is a mystery to me. Here's a problem that I tackled some time ago with its solution; you might try your hand at it (my proof was somewhat long and involved). I wish I knew some general procedure to find the period of such periodic functions.

Attachment 29944