2nd Fundamental Theorem of Calculus

Our last calculus class looked into the 2nd Fundamental Theorem of Calculus (FTOC). We talked through the first FTOC last week, focusing on position velocity and acceleration to make sense of the result. Our interpretation was that the FTOC-1 finds the area by using the anti-derivative. How are those connected?

This week we wanted to peek at the 2nd part. (OK, it was me.) We looked up the result on Mathworld, and talked about barriers to understanding. The teachers identified that the conflation between the antiderivative and the integral (meaning area under a curve) is almost total, so that the theorem is just restating what we already think. Using this completely confusing notation and totally new way to define a function.  Perfect situation for a GeoGebra sketch to allow students to explore.

This sketch uses several GGB4 features.  It uses the integral[ ] command to find the area under a curve, which I would have had to cheat before, the input boxes to allow real freedom of entering a function, and buttons so that students don't have to know GeoGebra commands to refresh the view. This is my first sketch uploaded to GeoGebraTube, which is a huge improvement over the old webhosting at geogebra.org. You can link to a teacher page or a student page, there's a link to download the file,  and there's easy to find embed code.  Best of all, the front page has a search, and shows recent uploads so it's fun just to check in.  And everything uploaded is CC3.0; darn near ideal resource.

If the embed code worked on blogger, I would have put it right here. (That's why only darn near ideal.)

Click here to go to the sketch on GeoGebraTube.

The teachers noticed all sorts of interesting things, recognizing anti-derivatives, seeing the +C (constant of integration) in action, and seemed to make sense of the integral definition of a function. They picked interesting functions to try, like increasing degrees of polynomials, trigonometric functions, functions without an analytic antiderivative (like cos(x^2)) and the fabulous e^x.

I've added the grid in to help students see the area more clearly, and set the grid to distance 1 to keep it unit sized.  Linda Fahlberg and John Scammell helped me with the right script for the button with quick Twitter responses. ZoomIn[1] to get a CTRL-F (refresh view) effect, and UpdateConstruction[] to get a CTRL-R effect (recompute all objects).  (Written down so I will never forget again.)

Photo credit: ajlvi @ Flickr. He (?) said he tried to capture everything he needed to know for the GRE on the board and then take a picture, but it was illegible on his phone. If he's that clever, I'm sure he did fine.

The Euler spiral

Students of literature read Shakespeare; students of music listen to Bach. In mathematics such a tradition is, if not entirely absent, at least fairly uncommon. This book is meant to address that situation. Although it is not intended as a history of the calculus, I have come to regard it as a gallery of the calculus. - (William Dunham)

I am reading Dunham's book. Calculus Gallery. While reading I realized that I couldn't do the integral $$\int \cos{(x^2)}\ dx ,$$ anyway to make a long story short I discovered yet another beautiful curve ( most beautiful ever? maybe ): the Euler spiral, it is parametric plot of two Fresner integrals.

An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin.

Exercise ( read: question ) How can we rotate the spiral?

Peek preview of Dunham's book.

Calculus video lectures at the Worldwide Center of Mathematics

The Worldwide Center of Mathematics publishes ( modern... ) calculus textbooks ( not a bad idea if you ask James Stewart ) and has a section with freely available video lectures on calculus and multi-variable calculus as well as a section with talks on the research level. There are also talks about research which are meant for undergraduates. Sort of to give you an idea of the field these researchers operate in.

Well, the Calculus lectures can be of help if you are on MST121, MS221 or MST209. There are a lot of options nowadays if you are looking for calculus videos.

Video lectures on multivariable calculus ( and more ) ...

I read a commentary on the future of universities.

Education and knowledge have become more accessible in the internet age. Students can do a lot of their work online or at home. But does that mean that universities will become obsolete? Of course not. - Books have to be written, exams have to be prepared and marked, research has to be done and published and students will always want some form of live interaction with a professor or tutor. Even if all lectures would be given in the form of video lectures then these lectures have to produced and as is the case with textbooks, redone every few years. Things have changed and will continue to change. Let's hope for the better.

Some video lectures, I recommended:
- [ NEW ] Multivariable calculus ( Berkeley )
- Video lectures number theory
- [News] - Video Lectures Algebraic Topology ( for Undergraduates )
- video lectures complex analysis

No set of video lectures is better than a well written text-book with lots of examples and exercises. Like this magnificent book on Complex Analysis for example ( I'll start M337 october next year ):

Infinite series ( and Euler's identity revisited )

The importance of infinite series and sequences can not be underestimated in my opinion, they literally pop up everywhere. To my surprise however, there isn't a single course at the Open University ( or any other university to my knowledge ) that deals exclusively with the topic of 'Infinite Series'. Instead the theory is stuffed away somewhere else, as if it is not really important. Like in M208 for example. - I have been searching for books on the subject and there -is- a ( recent ) book on the subject. It is called 'Real Infinite Series' by D. Bonar / M. Khoury published my TMAA. Tests not in M208 but in this book are for example Raabe's Test, Rummer's Test. Cauchy's Condensation Test, Abel's Test, and Dirichlet's Test as well as Bertrand's Test. It includes an entire chapter on the harmonic series with different divergent proofs. In the appendix there is an overview of the literature on infinite series.

P.S.
This video shows the proof of $e^{i\pi}+1=0$ using infinite series.

Watched MIT 18.02 - lecture 11

In 18.02 lecture 11 Prof. Denis Aroux talks about differentials and the Chain Rule. Two of the examples used to illustrate the main topic are of particularly interesting: a new proof for the differentiation of products and quotients, and the conversion between rectangular and polar coordinates.

A main result of this lecture is $$df = f_u \frac{du}{dx} + f_v \frac{dv}{dx},$$ where $f$ is a function of two variables $u,v$ which are both dependent on $x$, and $f_u, f_v$ are partial derivatives. The quotient rule can be derived from this result as follows. Let $g(x) = \frac{u}{v}$, with $u,v$ both dependent on $x$ :
$\begin{align*}

df &=f_u \frac{du}{dx}+f_v \frac{dv}{dx} \\
&=  \frac{1}{v}\frac{du}{dx}-\frac{u}{v^2}\frac{dv}{dx} \\
&= \frac{ v \frac{du}{dx}-u \frac{dv}{dx} }{v^2}
\end{align*} $
The last expression is the quotient rule for differentiation.

This lecture inspired me to some experimentation ( play ) with Mathematica's PolarPlot function. A polar coordinate is in fact a function of two variables $x,y$ which are both dependent on $r$ and $\theta$ with $x=r \cos(\theta)$, $y=r \sin(\theta)$. By applying the theory above one suddenly gets control over geometric objects like this:

Click to enlarge

Finally the concept of a gradient was mentioned which is merely a vector of partial derivatives. Gradients are the topic of lecture 12. I designed some problems and exercises ( and other experiments ) for functions in polar coordinates. I am delighted I feel more able in that regard.

More on MST209 in relation to MIT video lectures

MIT has a video lecture series on multivariable calculus: 18.02, which is in fact a prerequisite for 18.03. I looked further into the topics of MST209 and I now think that 18.02 is a much better preparation for MST209 than 18.03.

It's more or less like this:
MST209 = 18.02 + ( part of ) 18.03
MST209 + MST326 = 18.02 + 18.03

18.02 has lectures on
Lecture 15: Partial Differential Equations
Lecture 16: Double Integrals
Lecture 19: Vector Fields
Lecture 21: Gradient Fields
Lecture 25: Triple Integrals
Lecture 27: Vector Fields in 3D
Lecture 30: Line Integrals
which are topics in MST209.

18.02 Multivariable Calculus

Khan Academy

I was listening to one of those great talks by Lew Rockwell. He talked about some 530 million dollar school building in Los Angeles built to house 4,000 (!) high school students and which looked like a prison. Probably because it is a prison for these kids, he added. Lew mentioned that teaching like that is not at all like it should be with the ( technological ) possibilities we have today. Or if you like, with our ( read: the US's ) financial situation. He pointed out that it can be done differently. That it in fact -is- done differently by The Khan Academy, a not-for-profit 501(c)(3) with the mission of providing a world-class education to anyone, anywhere.

For me too? I thought, immediately. And yes, he recently added video lectures for a course in Differential Equations to the approximately 1800 video's already stored on their website. http://www.khanacademy.org/ Besides differential equations they also have an impressive set of calculus video's.

Since I already started the 18.03 lecture program ( competition breeds innovation ! ) I probably continue to do so but I'll be making comparisons.

For your listening and viewing pleasure

One of the many interesting ceiling panels.

Goals of Math Ed

Arthur Benjamin, self-proclaimed mathemagician and a very popular mathematician among teaching mathematicians, at TED on why calculus is not an appropriate pinnacle for math education. To be replaced by: discrete mathematics (statistics and probability).

Teacher Props
Taylor Mali, teacher/slam poet on What Teachers Make. One of my calc students shared this. As Mike warned me, I'll warn you: profanity.

Interesting glass floor and 6 or 7 story dome of the main rotunda.

Planning
An audio link from the new teacher resource center: an audio interview with Suzanne Lieurance, a teacher trainer and children's author, about planning in threes. There's a lot here that's compatible with workshop teaching, emphasis on assessment and evaluation and teaching for engagement. A form for note taking is in the post here.

The graphics are from the beautifully restored Michigan State Capitol that we visited this weekend, which include a couple of nice mathematical designs . Lots of nice 4-, 8- and 16-fold rotational symmetry. And much nice frieze-type translational symmetry, too. The architect was Elijah Meyers, and this was his crowning work.

Mobile Math

For the Math in Art Festival I did with Susan Walborn (an amazing teacher who's moved on to becoming an amazing retailer - must just be amazing, eh?), one of my favorite lessons was a mobile lesson (link leads to a pdf of a verrry complete 3rd grade lesson plan) based on the art of Alexander Calder and the math of area and average.

The emphasis is on the average as a balance. In calculus today, we covered center of mass, and built the connections among the moment, the weighted average and the idea of balance. Students used the ideas to create cardboard cutouts of curves and find the balancing point. They did a great job. Mine is the unimpressive cubic.

Trig Rummy

My calc students have really been struggling with trig substitution, mostly, i think, due to lack of fluency with trigonometric functions. I'm a big believer in games for skill practice, as they are an engaging context that encourages more practice and ideally offer a chance for forging new connections.

So today we played Trig Rummy. Many of my card games are based on concentration, rummy, or go fish. And occasionally Euchre. (I still think the partially ordered quadrilateral Euchre has a future.) Links for printables are after the rules. I think it would adapt to high school trig pretty easily. I'd replace the calculus relations with graphs and triangle identifications.

Trig Rummy

Objective: the player will practice and gain facility with trigonometric identities and calculus relations.

Players: 2-4

Goal: get the most cards in play.

Set up: randomize cards and deal 7 cards to each player. (You may want to introduce the game with only 4 cards as 7 is overwhelming.)

Play: On your turn you may do one of:
• take any card from the discard pile (not just the top card).
• draw a new card.
After that you may do either or both of:
• pick up any sets you have.
• play any new sets you have.
At the end of your turn, if you did not play a set, discard a card.

Special:
• if at any point the discard pile contains a complete set, the first player to otice can cal “Rummy!” and take the set out to play for themselves.
• you can not play matching operator cards, like a pair of d/dx cards.
• There is a wild card, which you can choose to be -1× (times -1), or d/dx, or ∫ ⋅ dx.

End: After the last card is drawn from the deck, each player gets one more turn from the discard pile. Then the cards are counted up and the player with the most cards played wins. No penalty for unplayed cards.

Variation: Play until one player is out.

Example plays:
• Play sin2(x) matching 1 − cos2(x).
• Play d/dx with sin(x) matching cos(x).
• Play ∫ ⋅ dx with sec(x)tan(x) matching d/dx with ln|sec(x) +tan(x)| (since both are equal to sec(x).)

If a player lays a mistaken combination and no one catches it before the next player’s turn, they stay in play but turned face down. If someone does catch it, that player has to pick up the pair.

Feel free to use a trig cheat sheet.

Rules PDF - includes a trig cheat sheet
Trig Cards PDF - for 54 2-sided cards of trig identities and calc relations. Could use for flash cards or other games also.