A very important graph

SinIx2sinEveryone studying mathematics needs to know some typical functions and some important characteristics. At the lowest level you need to know the trig functions and their (as applicable) domain, asymptotes, midlines, periods, etc and for exponentials and logarithms you should know the domains and asymptotes. At a slightly more advanced level, students should be aware of the  function $latex \frac{\sin(x)}{x}$ and how its limit exists at 0 (and is equal to 1) even though the function isn't defined there. Or that $latex x\sin(1/x)$ is a function whose limit at 0 exists and can be determined using the Squeeze Theorem. The graph show above is also a function that needs to filed away but, surprisingly, it isn't well known. I say that because I have trouble finding it in a typical calculus text (or the lesson it teaches). That's a shame; the function is $latex f(x)=x^2\sin(1/x^2)$ if $latex x \neq 0$ and $latex f(0)=0$.

Take the derivative and you'll get $latex f'(x)=x\sin(1/x^2)-(2/x)\cos(1/x^2)$. And the derivative at 0? Chances are overwhelming that your students will say the derivative doesn't exist at 0 because $latex f'(0)$ isn't defined. But that's wrong. That won't be apparent by looking at the graph of the derivative:DSinIx2sinFor the typical students who dive for a calculator so they can avoid thinking for themselves, they'll get to see that the calculator isn't going to help out here. They'll have to use the definition of the derivative (and the Squeeze Theorem). That's good practice. Ultimately this function shows that derivative can exist at a point even if the formula $f'(x)$ doesn't exist there. For all these reasons, this graph should appear in every calculus book.

The PDFs have been posted on the Graphics page.

Odds and Ends: Jan 22, 2014

Several issues to mention today:

  1. I've added 4 PDFs to the Graphics page. These illustrate how a graph can be discontinuous. More on that later.
  2. Magnus Carlsen now holds all world chess titles: World Chess Champion, World Rapid Champion, World Blitz Champion. The domination is complete.
  3. Common Core is becoming a conservative litmus test: The linked article notes, "Common Core standards initiative is now a “hot button issue within the GOP” that has even earned the nickname “ObamaCore,” one that “lays bare the political divide.”" and "Common Core standards will grow even more intense as their full impact will be experienced next fall when they become more widely implemented. For now, however, even U.S. Senate primary races in states such as Mississippi, Tennessee, and Louisiana have Common Core as a central issue."

The 4 PDFs on continuity deserves a comment. In teaching accelerated Precalculus I actually spend about 1/4 of the year covering 3 chapters in Calculus. As it was my first year teaching that in a US public school I "stuck to the script" and followed the book closely. The book mentions that points where a function is discontinuous is because

  • the function is not defined there (removable singularity)
  • the limit doesn't exist at that point
  • the limit exists there but differs from the function value

and goes on to introduce removable/nonremovable discontinuity and jump continuity. Those examples are shown in the PDFs I added (Discont1 through 3). But, strangely enough, the book never mentions essential discontinuity. If your book is the same, now you know why I have 4 pictures [Edit: It's $latex \sin\left(\frac{\pi}{x}\right)$]. My book then, would have to classify that as a nonremovable discontinuity which is not a jump discontinuity. Why not mention the term and use it to flesh out the idea of continuity: that essential discontinuities are the worst type of departure from continuity and that knowing information about the discontinuities of a function will be important in representation of functions as infinite series and Riemann integrability. I see no reason why the book should fail to mention the term essential discontinuity.

Sagetex: Combinatorics/Probability (8/9)

Comb8I've added 2 more problems to the Sagetex: Combinatorics/Probability page. The problem above has the form:

A high school committee of 6 is to be made from 32 boys and 41 girls. Within this set of students there are 2 senior boys and 3 senior girls. How many committees of 3 boys and 3 girls are there that contain at least one senior boy and one senior girl?

It's worth noting an interesting aspect of this problem: in order to properly count, the various cases need to be listed and this means the output of the solution will be dynamic in the sense that the number of cases to be listed changes in the solution will change depending on our specific number of senior boys and girls. In the case of 2 senior boys and 3 senior girls we need to throw out committees with

  • no senior boy and no senior girl
  • no senior boy and 1 senior girl
  • no senior boy and 2 senior girls
  • no senior boy and 3 senior girls
  • no senior girl and 1 senior boy
  • no senior girl and 2 senior boy

Change the number of senior girls or senior boys and that list given in the solution changes.

The output of the various cases is handled in the sagesilent block. We need to avoid counting no senior boy and no senior girl twice, so no senior boy and no senior girl is calculated here:

NoSeniors = binomial(girls-Sgirls,3)*binomial(boys-Sboys,3)

and 2 FOR loops create the rest of the cases. Here the committees with 0 senior girls are all created

for i in range(1,Sboys+1):
output += r"$0$ senior girls and $%s$ senior boys: $C(%s, 3)\cdot C(%s,%s)\cdot C(%s,%s)$\\"%(i,girls-Sgirls,Sboys,i,boys-Sboys,3-i)
remove += binomial(girls-Sgirls,3)*binomial(Sboys,i)*binomial(boys-Sboys,3-i)

A similar loop handles 0 senior boys. Note that 3 is hard coded. That's the number of girls (and boys on the committee).  Increasing that number will make too many cases to enumerate for a test. Likewise

Sboys = Integer(randint(2,3))
Sgirls = Integer(randint(2,3))

keeps the problem size more reasonable as well; 6 cases is enough and, depending on the ability of your class, it might be more appropriate to decrease the number of cases; i.e., make the hard coded 3 a two.

Odds and Ends: June 14, 2014

Just a few odds and ends to mention

  1. Norway Chess 2014 is over and it's Car---Karjakin winning it for his 2nd time in a row. Carlsen takes second and Grischuk third. Some interesting, spirited, fighting chess along the way but lots of blunders along the way. Aronian blundering a queen in round 3, Aronian, Caruana, and Topolov in Round 5, Giri's self demolition in Round 7, Carlsen's epic blunder in a completely winning position in Round 8; it makes you wonder how much rating inflation is out there. It doesn't seem like the Fischer, Karpov, Kasparov, Botvinnik, Capablanca, etc. made mistakes as often despite having lower ratings.
  2. China Smack had a Hong Kong elementary school admissions test math question posted a short while back. While it took me about 20 seconds to get it, there are apparently a lot of adults that struggled. From the sample comments below, I liked this one "I tried various numerical/mathematical sequences, and even used calculus,". D'oh! This might destroy your stereotypes of people who are good at math; my experience teaching in Asia meant I wasn't surprised...
  3. Gov Bobby Jindal vetoed Common Core legislation. The tide seems to be turning as "Common Core" evokes a pained response in more and more people. More states are having second thoughts. Leaving aside content issues, the implementation in my state has been very poor. Imagine knowing there's a hard deadline years in advance and (years after the deadline) we still don't have Common Core textbooks. If that happened in a private school someone would lose a job; in the public school system---well, nothing happens. No incentive or reward for doing your job well and no real punishment when you do a terrible job. Think of Michelle Rhee finding all the "lost" textbooks and supplies in that warehouse. Who lost a job for mismanagement?

Sagetex: Combinatorics problems (6/7)

Comb6I've added 2 more problems (along with the solution) to the collection of randomized problems on the Sagetex:  Combinatorics/Probability page. The two new problems have the following form:

Problem Type 6: A committee will be made from 5 English teachers, 3 math teachers, and 4 history teachers. If the committee must have 2 teachers from different disciplines then how many committees are there?

Problem Type 7: How many nonnegative integers less than 100000
contain the digit 5?

Handouts: Notes on Induction

InductionNotesI've added some notes on weak and strong induction; they're posted on the Handouts page. The notes are a stripped down version that you'll need to flesh out. While I've given a motivation, definitions, example problems, and then an assortment of problems, I've left out a lot that you'll probably want to fill in. A high school level induction to induction shouldn't be the clean sanitized version you'll find in the notes; it should include the process of getting to the answer. That is, rather than present the proof as given in the examples, I would explain the scratchwork that gets you to the answer which is then cleaned up into the final version given in the notes.

The notes are a hack of the LeGrand Orange book template to 1. Get an article version 2. change the color 3. add green and red boxes to highlight information.

The large assortment of problems contains problems of varying difficulty so you'll need to work them out to decide what level is best for your students.

TCEC Season 6: Game 18

A look at the brilliant game 18.

Odds and Ends: June 1, 2014

Several items of interest to report:

  1. Stockfish has won the TCEC Unofficial world computer chess championship by 35.5 to 28.5
  2. Game 18 of from the TCEC match showed brilliant play by Stockfish. You can watch the video here. That's a world class game that any grandmaster would be proud of.
  3. An interview with Kasparov regarding his running for the president of FIDE.
  4. Wired reports that somebody trademarked $latex \pi$. and is filing cease and desist orders for people using $latex \pi$ --- see the difference? Geeks of the world are up in arms.