Expert: Paul Klarreich - 3/4/2011

Question
Hello, i was wondering how i could sketch the graph of (cos(2*PI*t/T)^3) Where i believe T=2*PI. I am not allowed to use a graphical calculator so i was wondering how i could obtain the graph by only using brain power!

Answer
Questioner: jay
Country: United Kingdom
Category: Advanced Math
Private: No
Subject: integration
Question: Hello, i was wondering how i could sketch the graph of (cos(2*PI*t/T)^3) Where i believe T=2*PI. I am not allowed to use a graphical calculator so i was wondering how i could obtain the graph by only using brain power!


If  T = 2pi, then you just have:

y = (cos t)^3

and you do the standard stuff, limited to [0,2pi]:

1. Symmetry (even)
2. Asympotes (none)
3. Intercepts: solve  cos t = 0
4. Critical points:  dy/dx = -3 sin t cos^2(t),  then set it = 0 and solve.
5. Inflection points. y'' = (use product rule), set it = 0 and solve.

Naturally, the G.C. can be used while nobody is watching, to give you some hints.  The above is the standard calculus graph-sketching routine.  It will take time, and I assume you have pencil-power to go with your brain-power.