Mark's Calc III - Individual question feed

Finding the distance traveled over a length of time (of an object)

Kyouko — Wed, 02 Jul 2014 07:44:06 -0500

Questions 1c and 2 on the review test as us to find an integral that could be solved in order to find the distance traveled by an object over a parametric plot. I can visualize this process, however, I am not quite sure how to go about setting up an integral for this using both \vec{x} = t + cos(4t) and \vec{y} = -sin(4t). Here question 1c is asking us to find the integral described above for the following vector $$ \vec{p}(t) = \langle t + cos(4t), -sin(4t) \rangle $$ and $$ \vec{p'}(t) = \langle 1 - 4sin(4t), -4cos(4t) \rangle $$ I believe setting up the integral would set up using the derivative of the position function integrated over the time interval giving the integral $$ \int_0^{2\pi} \ <1 - 4sin(4t), -4cos(4t)> \mathrm{d}t $$ Any help would be appreciated

The bug on the wheel goes round and round

SpaceManSpiff — Mon, 30 Jun 2014 09:26:18 -0500

I'm pretty confident that I understand how to parametrize the bug's movement. But I don't understand what kind of movement I need to parametrize. can someone clarify the bugs movement in non-mathematical terms. Is the wheel moving in any dimension or just spinning?

Spheres/Will this work?/In Class #3

SpaceManSpiff — Tue, 01 Jul 2014 09:43:05 -0500

So if you set the equations for the two spheres equal to each other you get a plane. $$x^{2}+y^{2}+z^{2}=(x-1)^{2}+(y-1)^2+(z-1)^{2}$$ $$-2x-2y-2z=3$$ I believe that simplification is correct. The question is, if I parametrize the circle created by the intersection of the unit sphere with this plane, will that describe the intersection of the two spheres?

Matching functions to pictures

Dylan — Tue, 01 Jul 2014 07:26:16 -0500

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.

How do you find a parametric equation for a hypercycloid?

Anonymous — Mon, 30 Jun 2014 13:26:15 -0500

In section 10.4, problem number 7: A wheel of radius 1 rolls around the outside of a circle of radius 3. A point P on the rim of the wheel traces out a curve called a hypercycloid. Assuming P starts at the point (3,0), find the parametric equations for the curve. This isn't an assigned problem, I am just curious and I am not sure how to geometrically break this problem apart to

How do I visualize space curves?

Wes — Sun, 29 Jun 2014 19:02:51 -0500

I'm having a bit of trouble visualizing some of the space curves on the homework (section 13.1, #1-4). I can draw sketches of what they look like in the x-y and the x-z planes, but when it comes to "combining" the two, I'm struggling. Do any of you have any suggestions? Or is this not even necessary for the homework as it just says to "investigate?"

How do I make a parametric plot?

Mark — Sun, 22 Jun 2014 18:19:49 -0500

How can I generate a parametric plot with Mathematica? For example, the following function should supposedly generate a spiral: $$\vec{p}(t) = \langle t\cos(t), t\sin(t) \rangle.$$ How can I generate this spiral and show it on AskBot? Also, what about parametric plots in 3D? For example, the following function supposedly generates a helix: $$\vec{p}(t) = \langle t, \cos(t), \sin(t) \rangle.$$ What about other tools like [Desmos](https://www.desmos.com/) or [WolframAlpha](http://www.wolframalpha.com/)?