Fun Topics in Calculus II
Some of the things we cover in calculus are related to much richer topics. I have a selection of these here, together with some links for the curious. You might also be interested in Fun Topics in Calculus I.
Convergence for Sequences ¶
Consider the sequence whose terms are defined by the formula at right.
Use a calculator to compute the first 8 or 10 terms, and make a guess about the number to which this sequence converges. Can you think of a proof that your guess is right? After you've thought about it a little, check out this cautionary tale (XKCD).
Gabriel's Horn ¶
Infinite surface area but finite volume! This was one of our first hints that when you're talking about the infinite, some strange things can happen. The MathWorld page has a very nice model of Gabriel's Horn, together with the *correct* calculations for the surface area and volume. The Wikipedia page also has correct calculations, together with some discussion of the "painter's paradox", which arises if you try to imagine using this horn as a paintbucket (not a good idea).
Elsewhere on the web, watch out for bad calculations of the surface area of this object--it's not so simple as you might think. We don't cover arc length or surface area in class because they require an extra subtlety not necessary for volume. If you want to know what's going on though, it's in chapter 10 of our textbook (Salas-Hille-Etgen, 10th edition).
The Cantor Set ¶
The Cantor set is created by removing the middle third from the unit interval, and then removing the middle thirds from the remaining intervals, and repeating this removal infinitely many times. The set of points left over is the Cantor set. We showed in class that the total length of the intervals which are removed is 1, but the Cantor set still has infinitely many points in it. In fact, the Cantor set has just as many points as the entire unit interval does, even though the total length of all those points taken together is zero. An explanation of these facts is given on the Wikipedia page for the Cantor Set. Some more technical facts about the Cantor set can be found on the MathWorld Cantor Set page.
These facts are the motivation for the Cantor Function, a function which is continuous and monotonic increasing (non-decreasing), but whose derivative is zero *almost* everywhere. You can see a good picture of the graph of this function at the MathWorld Cantor Function page, or a slightly less good one at the Wikipedia Cantor Function page.
The Banach-Tarski Paradox ¶
The Cantor Set is an example showing that the unit interval can be divided into an infinite number of pieces: the "middle-third" intervals and the "leftovers", in such a way that the two groups have the same number of points, but one has length 1 and the other has length 0. Banach and Tarski showed that something even more surprising can be done in three dimensions. A solid ball can be divided into a *finite* number of pieces, and those pieces can be rearranged to make two complete solid balls, each of which is the same size and shape as the original!
How many pieces does it take? According to the MathWorld page, just 5! But they're *really* complicated. Harvey-Mudd has a nice introduction to the Banach-Tarski Paradox, including some reflections on what this means about our intuitive notions of things like volume (or length). The Wikipedia page on the Banach-Tarski Paradox has a formal statement of their results, and an extensive explanation of why the result is true.
The Gaussian and Its Integral ¶
We saw in class that the indefinite integral of the Gaussian function (Wikipedia) can be computed by taking the Taylor series for the exponential function, plugging in x^2, and integrating term by term. The Gaussian integral (Wikipedia) is the improper integral formed by integrading the Gaussian over the entire real line. One way to calculate this might be to use the series expansion for the indefinite integral, and try to take a limit. Another way uses the following trick: The square of the area under the Gaussian curve is the volume shown below. It is the same as the volume of the solid formed by *rotating* the Gaussian function around the y-axis! We've computed this volume many many times, and it's easy as π.
The "brief proof" section of the Wikipedia page gives a more detailed version of this computation (more symbols, fewer words). The key to this trick is the following riddle: when is a circle the same as a square? Answer: when they both have infinite width. The plane which forms the base of the solid below can be interpreted as a square with infinite base, or a disk with infinite radius.
Relationship of the Fibonacci Sequence to the Golden Ratio ¶
In class, we considered the exponential generating function (Wikipedia) for the Fibonacci sequence. The recursion formula for the terms in the Fibonacci sequence means that this generating function is a solution to the differential equation y'' - y' - y = 0. This is a differential equation we can solve, and the solution gives the first hint that the Golden Ratio is relevant to the Fibonacci sequence. Calculating the derivatives of this function at 0 gives a formula for the terms in the Fibonacci sequence!
The golden ratio appears in lots of fun mathematics and artwork, and also in lots of bogus claims. You can try the Wikipedia page, or for some nice pictures, this article from +plus magazine about the Golden Ratio is a good start. Or just browse through google images.
And the old version of this webpage shows an experiment with Golden ratio rectangles in web design.
Regular Trig Functions with Hyperbolic Trig Functions ¶
The relationship between the regular trig functions and the hyperbolic trig functions is easiest to see by considering the Taylor series for each of these functions. More precisely, one should compare the Taylor series for the exponential function with that for the regular trig functions. This yields an explanation for Euler's formula (Wikipedia).
The math department at Fullerton has a nice module explaining complex trig functions, via Euler's formula. They spell out the details of the relationship between the regular and hyperbolic trig functions, in case you missed them in class. The relationship between the two graphs shown here? These are formed by intersecting a plane with different parts of the 'graphs' of the complex trig functions. For something even more *crazy*, here's a YouTube movie of some fractals generated by the complex cosine (the first part is a little dull, but it gets better).