This paper, More On Vertical Tangent Lines, proves a fact about vertical tangent lines. You need a solid understanding of the Mean Value Theorem as it is taught in undergraduate calculus courses. If you’re in the habit of reading proofs – in particular, the epsilon-delta proofs – then this should be manageable, given that you understand some of the more theoretical exercises about how to use the Mean Value Theorem.

To summarize the paper: Suppose f is continuous. In most calculus books, a function is said to have a vertical tangent line at x=a if and only if “f'(a)=+ or – infinity” (that is, [f(a+h)-f(a)]/h goes to  infinity or to minus infinity as h goes to 0). If f is differentiable away from x=a and f ‘(x) approaches + or – infinity as x approaches a, then it follows that f has a vertical tangent at x=a (this is the theorem). For our favorite functions, this is the most likely way for a vertical tangent to occur. However, sometimes a vertical tangent can exist at x=a even if f'(x) does NOT approach + or – infinity as x goes to a.

The paper proves the theorem and provides a function f such that f has a vertical tangent at 0, but f'(x) does not approach + or – infinity as x approaches zero.

UPDATE: See also this nice graph by fellow grad student Bob Willenbring. He gave me some TeX tips for doing these, and I’ll have to figure out how it all works sometime soon. I’d like to include this in the document itself next draft.


List of expository papers

October 8, 2010

Here is a list of my expository articles. These papers explain things that are already well-known, but at the most introductory level possible. The content is far from original, and sometimes the only innovation is an elaboration of some main ideas a more veteran reader would have recognized without further comment:

Convolutions and the Weierstrass Approximation Theorem – intended for an undergraduate math major audience; experience with epsilon-delta arguments (elementary real analysis) is almost necessary.

Kakeya sets: Unit Length Line Segments In Every Direction In The x-y Plane Cleverly Arranged To Take Up Zero Area – intended for an undergraduate audience. Strong technical and intuitive understanding of “calculus 1” material (integration) is required. Elementary real analysis is useful in places but not required to get the main idea.

Euler’s Formula and the Fundamental Theorem of Algebra – To understand how complex polynomials work, you must first understand what it looks like when you multiply two complex numbers. The deep but easy-to-use identity called Euler’s Formula allows us to do this. One common proof of the FToA follows from Euler’s Formula without the need for extra machinery.

Euler’s Formula and the Fundamental Theorem of Algebra (rough sketch of another popular proof) – At the moment, there is only an animation of what happens when you plug in a circle of z-values into a polynomial as you increase the radius. This is one version of the proof, but the reason the animation looks the way it does is Euler’s Formula. I am reasonably happy with the .pdf I have written, but the topological proof hinted at by the animation could warrant an additional essay. I am not sure whether such an article should be a .pdf (with no animation) or take the form of a web page.

More On Vertical Tangent Lines – Intended for someone who at least has a solid understanding of undergraduate calculus, especially formal definitions and epsilon-delta proofs. Also essential is a solid understanding of the more theoretical applications of the Mean Value Theorem that are sometimes included in problem sets. This paper isn’t too over the top, but someone new to the material should maybe read slowly, as the Mean Value Theorem can tend to be deceptively subtle – there is always more to it than meets the eye.

Non-standard Analysis (NSA) – If you’re good with epsilon-delta arguments and constructing the reals as Cauchy sequences of rationals, then hopefully this slide show is self-explanatory without need for a lecture. Also contains some spoofs. Contains a simple proof of Tychonoff’s product theorem using Robinson’s equivalent definition of compactness (proof of equivalence omitted). This should demonstrate that NSA might have some potential for making some proofs easier in the case where a proposition depends in a very deep way on the Axiom of Choice.