Printable View
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/2Quote:
Originally Posted by orange
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?
1 is their LCM. So I guess the minimum interval is just pi?
give that man a kewpie doll!
Hello, orange!
I "eyeballed" the problem.
I think I have the solution.
Quote:
Letbe the graph of the parametric equations: .
What is the length of the smallest intervalsuch that the graph
of the equations for allproduces the entire graph of
When
At the "start", the graph is at
Here is what I found:
. .
. . and the cycle repeats.
Therefore: .
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
