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Today we continue with parametric equations for Lab 14.
Suppose we want to compute \[ \frac{d^2y}{dx^2} \]
when faced with parametric equations $x=f(t)$ and $y=g(t)$.
It turns out to be just another example of the chain rule, just as
\[ \frac{dy}{dx} = \frac{d}{dt}\left(y\right)\frac{dt}{dx} \] or \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}} {\frac{dx}{dt}} \equiv h(t) \] For the second derivative, we simply do it again -- but now we already know $\frac{dy}{dx}$ as a function of time (I called it $h(t)$ above), so \[ \frac{d^2y}{dx^2} = \frac{d}{dt} \left(\frac{dy}{dx}\right) \frac{dt}{dx} = \frac{\frac{d}{dt} \left(h(t)\right)} {\frac{dx}{dt}} = \frac{h'(t)} {\frac{dx}{dt}} \] It's a two-stage process. And obviously we could continue, computing third, fourth, etc. derivatives for the path in space.
Recall that this lab was supposed to get you ramped up for the last exam. It featured a whole bunch of different tests, and summarized them while requiring you to demonstrate that you know how to use them.
Note: Those tests remain useful for the upcoming exam. In addition, we focus on the ratio test and the Absolute convergence test (handy when the sign of terms is not alternating, but perhaps irregular).
Some of you mistakenly took $r=\pi$, which would lead to a divergent series. You missed that \[ \pi^{-k} = \left(\frac{1}{\pi}\right)^k \]
Furthermore, we have an error estimate: the absolute error in using the $n^{th}$ partial sum to approximate the series is bounded by \[ |S-S_n| \le \int_{n}^\infty f(k) dk \]
Ideally you should use the function $f$ (not a bounding function, which gives a cruder result).
In this case, you need to find a series which is larger than or smaller than yours, depending on whether you're trying to show convergence or divergence.
In this case, you're looking for a series which, in the limit, is proportional to the one you have. And you need to compute a limit! (hence the name...)
If the terms $a_k$ of a series alternate in sign, and the positive $b_k=|a_k|$ decrease in size (eventually), and have limit 0 as $k\to\infty$, then the series \[ \sum_{k=0}^\infty a_k \]
converges by the AST.
Furthermore there is a simple error formula: the absolute error is bounded by the first neglected term: \[ |S-S_n| \le b_{n+1} \]
But hey, that happens. And we just pick ourselves up, dust ourselves off, and find another test.
A couple of comments:
Some of you are entranced by Mathematica. I understand that. But you can't decide the number of terms required for a partial sum to approximate a series by subtracting partial sums from Mathematica's sum of the series.
The point is that we're to imagine that we don't know (or Mathematica doesn't know) the value of the series -- otherwise, we'd just use that! We wouldn't need to approximate, right?
So we learn these techniques, and it's nice when we can actually check that they're behaving as we think that they should. But imagine that you're in a situation where you DON'T KNOW the actual value of the series. In that case you're interested in two things:
We'll also compute tangent lines to curves. In terms of motions, the tangent line is the path a particle would travel (initially) if whatever force is keeping the particle in its trajectory were to release.
I mentioned in yesterday's materials that, if we were in class, I'd be swinging a ball attached to a string over my head, and imagining along with you what would happen if one were to cut the string....
Well guess what: some of us are in class! So here we go... Uniform circular motion.
Both feature an unusual curve, which you may have never heard of: the cycloid.