math

Plotting with TikZ, Part III: Plotting a function defined by a formula, and plotting functions that go through certain points

In the last post, I described how I use TikZ to create a graph that might otherwise be created freehand: In this post, I will describe one method for using TikZ to to plot a function defined by a formula, such as $$y = \left(2x^2 - x - 1\right)e^{-x}.$$ Then I will show two ways to make a function go through a certain point. Plotting from a formula Simple example There are a number of ways to achieve this, and PGF actually includes the functionality to perform calculations in TeX.

Plotting with TikZ, Part II: Functions without formulæ

In the last post, I explained why TikZ is awesome for making plots. One good use case for handmade TikZ plots is to typeset a question like this: (2 points each) A plot of the graph of the function $f$ is below: For each of the following expressions, either evaluate the expression or state that it is undefined: a. $\lim_{x \rightarrow 2^{-}} f(x)$ b. $\lim_{x \rightarrow 2^{+}} f(x)$

Plotting with TikZ, Part I: Why?

This is the first part of a three-part series of posts on generating plots of graphs with TikZ. Last year, I left my position as a mathematics professor, after teaching mathematics at the college level for 15 years (nine years as a faculty member, and six years as a graduate teaching fellow). In that time, I picked up a lot of tricks using LaTeX to produce teaching materials (handouts and slides).

Adventures in TikZ: tkz-graph

The other day, I was writing some lecture notes for my linear algebra class, and wanted to create the following diagram (to illustrate the concept of a Markov chain): I had a very limited time in which to finish these notes. Fortunately, I found the tkz-graph package, which made this a snap: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 \documentclass{standalone} \usepackage{tikz} \usepackage{fouriernc} \usepackage{tkz-graph} \begin{document} \begin{tikzpicture} \SetGraphUnit{5} \Vertex[x=0, y=10]{0 points}; \Vertex[x=0, y=5]{1 point}; \Vertex[x=0, y=0]{Win}; \Vertex[x=5, y=5]{Lose}; \Edge[style ={->}, label={$1/3$}]({0 points})({1 point}); \Edge[style ={->}, label={$1/3$}]({1 point})({Win}); \Edge[style ={->}, label={$1/6$}]({0 points})({Lose}); \Edge[style ={->}, label={$1/6$}]({1 point})({Lose}); \Loop[style ={->}, label={$1/2$}, labelstyle={fill=white}]({0 points}); \Loop[style ={->}, label={$1/2$}, labelstyle={fill=white}]({1 point}); \Loop[style ={->}, label={$1$}, dir=EA, labelstyle={fill=white}]({Lose}); \Loop[style ={->}, label={$1$}, labelstyle={fill=white}]({Win}); \end{tikzpicture} \end{document} You don’t even have to specify the locations of the vertices; you can throw caution to the wind and have LaTeX decide where to place them!

Book recommendation: How Not to Be Wrong

Today, I finished reading How Not to Be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg. This is a very enjoyable, very well-written, general-audience book about mathematics, which I recommend whole-heartedly. Ellenberg, a math professor at the University of Wisconsin, does a great job weaving together a plethora of mathematical topics, including non-Euclidean geometry, probability, statistics, and mathematical analysis of voting systems. He writes in a way that someone who only vaguely remembers—or never really understood—high school algebra would be able to follow and enjoy.

Sum question!

The American Mathematical Society made the following amusing post to their Facebook page today: It’s National Punctuation Day (at least in the U,S,)! To honor the occasion, for positive integers n, let n? = 1 + 2 + … + n. What is 3???? (Hint: < 1000) You can calculate 3???? = 26796, which of course is greater than 1000. The joke is that they aren’t asking you to calculate 3?

This joke is awesome and deserves more than just a retweet

This joke is awesome and deserves more than just a retweet: I'd wager that right now, Eric Cantor’s sorrows are… (•_•) ( •_•)⌐■-■ (⌐■_■) uncountable. — “patrick” (@importantshock) June 11, 2014 (Georg Cantor’s diagonal argument shows that the real numbers are uncountable.)

Brains over calculators, part 2.

MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], ['\\(','\\)']], displayMath: [['$$','$$'], ['\\[','\\]']], processEscapes: true, processEnvironments: true, skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'], TeX: { equationNumbers: { autoNumber: "AMS" }, extensions: ["AMSmath.js", "AMSsymbols.js"] } } }); I recently found another calculation that I would consider reasonable for my students to perform that their calculators cannot. On my Calc II students’ final exam, I asked them to evaluate the infinite series $\displaystyle \sum_{n = 1}^\infty \frac{4}{(n + 3)(n+5)}$.

Brains over calculators!

I enjoy finding calculations that I would consider reasonable for my students to perform that their calculators cannot. MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], ['\\(','\\)']], displayMath: [['$$','$$'], ['\\[','\\]']], processEscapes: true, processEnvironments: true, skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'], TeX: { equationNumbers: { autoNumber: "AMS" }, extensions: ["AMSmath.js", "AMSsymbols.js"] } } }); Using a double-angle identity and the pythagorean identity, it’s pretty straightforward to show that $\sin\left(2\sin^{-1}\left(-\frac{4}{5}\right)\right) = -\frac{24}{25}.$ However, my TI-89 returns an unhelpful $-\sin(2\sin^{-1}(4/5))$ instead.

Arizona State Senator opposes Common Core, in part, because ALGEBRA!

According to the Arizona Daily Star, the Arizona Senate Education Committee passed a bill that would prohibit the state from implementing the Common Core standards. Quoth the Daily Star (emphasis mine): [Gubernatorial candidate and state Senator Al] Melvin said he understands “some of the reading material is borderline pornographic.” And he said the program uses “fuzzy math,” substituting letters for numbers in some examples. Later on in the same article, Senator Melvin is quoted as expressing concern about the rigor of American academic standards, arguing that “We have cheated several generations of Americans out of a decent education.