Day 49 in MAT229

Today we continue with parametric and start polar equations.

• Lab 11:

  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).

  • Divergence test: if the sequence of terms of the series aren't headed for zero in the limit, the series doesn't converge.
  • Geometric series: \[ \sum_{k=0}^\infty r^k = \frac{1}{1-r} \] provided $|r|<1$. More generally, for \[ \sum_{k=m}^\infty ar^k = \frac{ar^m}{1-r} = \frac{\textrm{first term}}{1-r} \]

    Some of you mistakenly took $r=\pi$, which would lead to a divergent series. You missed that \[ \pi^{-k} = \left(\frac{1}{\pi}\right)^k \]

  • Integral test: for positive terms $a_k=f(k)$:
    $\sum_{k=1}^\infty a_k$ converges if and only if $\int_{1}^\infty f(k) dk$ converges.

    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).

  • Comparisons:
    • Direct

      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.

    • Limit

      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...)

  • Alternating Series Test (AST):

    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: \[ |S-S_n| \le b_{n+1} \]

• Absolute convergence test (and conditional convergence)
• Ratio and root tests
• Power Series
• Functions as Power Series
• Taylor series
• Taylor polynomials (reprise)

In discussing a problem with Matthew, I made the point that parametric equations are useful for representing curves that cannot be represented as ordinary functions -- because the curves fail the vertical line test.

The most obvious example of this is a circle. It's so important, yet we can't represent it as a function $y=f(x)$. We have to write, for example (and somewhat ashamedly) \[ y(x)=\pm\sqrt{r^2-x^2} \]

But if we think of it as a parametric equation, it's easy to write as a function of time -- as a motion -- just the way that you might trace it out on the board or on your paper: \[ \begin{array}{c} {x(t)=\cos(t)}\cr {y(t)=\sin(t)} \end{array} \] as $t$ varies over $[0,2\pi)$. That's if you like to start at (1,0), and draw in a counter-clockwise fashion. If you like to start at the top and go in a clockwise fashion (like time on a clock), then you might prefer \[ \begin{array}{c} {x(t)=\cos(\frac{\pi}{2}-t)}\cr {y(t)=\sin(\frac{\pi}{2}-t)} \end{array} \] as $t$ varies over $[0,2\pi)$....

In today's materials you'll see some pretty wild curves -- and even learn how to compute their lengths!

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. If we were together 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....

• Calculus of Parametric Equations (.nb)
• Calculus of Parametric Equations (.pdf)
• Long talks you through Calculus of Parametric Equations.