http://calc3.askbot.com/questions/Open source question and answer forum written in Python and DjangoenCopyright Askbot, 2010-2011.Tue, 01 Jul 2014 11:52:40 -0500http://calc3.askbot.com/question/35/matching-functions-to-pictures/Does anyone have a particular way of matching some of the more complicated circular functions to the pictures in the in-class sheet and test review? When the function combines linear and circular motion I find it easier to break them apart and visualize them, but I have more trouble picturing what is going on with the functions that compose a lot of circular motion, such as $\vec{p}(t) = 2\langle\cos(t), \sin(t), 0 \rangle + \langle \cos(t), \sin(t), 0\rangle + \langle 0, 0, 1\rangle \cos(22t)$. I am probably going to try to graph a lot of these in Mathematica or Woflram Alpha, but I was wondering if anyone had a particular way of visualizing these functions or determining what is going on with one.Tue, 01 Jul 2014 07:26:16 -0500http://calc3.askbot.com/question/35/matching-functions-to-pictures/http://calc3.askbot.com/question/35/matching-functions-to-pictures/?answer=38#post-id-38The easiest way for me to picture these, is to break them up individually. For instance with the equation you are looking at, we have $\vec p(t) = 2\langle\cos(t), \sin(t), 0 \rangle + \langle \cos(t), \sin(t), 0\rangle\sin(22t) + \langle 0,0,1\rangle \cos (22t)$ so if we break that up and multiply out , our first portion is $\langle 2\cos(t), 2\sin(t), 0 \rangle$ from this we can tell it is a circle in the xy plane that has a radius of 2. next we have $\langle \sin(22t)\cos(t), \sin(t)\sin(22t), 0\rangle$ which is a ellipse in the x coordinate, with a $sin^2$ wave in the y coordinate. This forms that slinky look you see in the picture. The last portion is the $\langle0,0,\cos(22t)\rangle$ this takes the above two and extends them into the z coordinate following a sped up cosine wave. By combining them, we see that we start with a circle in the xy plane, we add the slinky feature from part two, and then make it three dimensional following the cosine wave from part three. which all adds up to the top right picture from the in class assignment. Tue, 01 Jul 2014 11:52:40 -0500http://calc3.askbot.com/question/35/matching-functions-to-pictures/?answer=38#post-id-38