I used to use MuPAD a lot in the (good old) days when it was available for free, and came to be very impressed with it. And I don’t know if my university’s Matlab license includes the symbolic toolbox – my own private Matlab licence certainly doesn’t!

Given my druthers I don’t think I’d choose Matlab for teaching calculus, but for my engineering students, who use Matlab extensively through their courses, it’s probably the best bet.

(1) I wouldn’t say that MATLAB is “less well equipped” for calculus, but rather “differently equipped”. (I cringe at the political correctness there, but anyway.) MATLAB does numerical and graphical stuff extremely well, better IMO than Maple. Maple does symbolic stuff extremely well, better quite obviously than straight MATLAB. Which leads me to the second point:

(2) I wouldn’t say, given the development of mathematics over the last 10-15 years, that calculus anymore is “primarily analytical and symbolic”. To have a modern understanding of calculus, I really do think that we have to put the numerical and graphical approaches to calculus on equal, if not greater, footing with the symbolic.

But that’s good news for using MATLAB in calculus. Because so many actual applications of calculus tend more toward the graphical and numerical, so much the better for those wanting to use MATLAB, whose strengths lie in exactly those approaches.

Also, although we have Maple on our network at my college, I think a combination of MATLAB for numerical and graphical problems and Wolfram|Alpha for symbolic calculations is a powerful combination. With W|A the expense of doing automatic symbolic calculations just went down several orders of magnitude.

Finally, it’s worth noting that although (as you point out) the Symbolic Toolbox is an absurd way to do calculations from the MATLAB command line, using the MuPad notebook interface is actually pretty natural. It has a nice look-and-feel similar to Maple (although not as nice as Maple), gives you pretty-print output, and all that. All the symbolic calculations you showed above from the command line would be done in MuPad with much less fuss.

So the prognosis for using MATLAB in calculus is even better than you indicated! That’s good news, right? :)

This is a nice animation example! I liked it, so I implemented it in Mathematica as well.

Here is the Mathematica code

r = 1; step = 0.1;
backgroundAxes = Plot[0, {x, -Pi, 3*Pi}, PlotRange -> {Automatic, {-r/2, 2*r + 0.5}},
AspectRatio -> Automatic];

Animate[Show[{backgroundAxes, ListPlot[Table[{x - Sin[x], 1 – Cos[x]}, {x, 0, t, step}], Joined -> True],
Graphics[{PointSize[Large], Red, Point[{t - Sin[t], 1 – Cos[t]}]}], Graphics[Circle[{t, 1}, r]]}],
{t, 0, 2*Pi, step}, AnimationRate -> 5]

I exported the above to an animated GIF, here it is

http://12000.org/my_notes/simulation/animating_moving_circle/test.GIF

–Nasser

On my own initiative, I have learned how to use sage and have found it to be very useful. I would love to see a program like sage (my preferred math program) be taught to younger children so that when they have a problem, they can quickly animate it themselves to see what’s “going wrong” or what an expression “actually means.”

I remember teaching sorting techniques (bubble sort, insertion sort, quick sort etc) and one of my higher level students showed me a lovely animation of the various types of sort. But apart from the most obvious aspects (i.e. that quicksort is quick) it turned out that the animation, even running fairly slowly, only made sense to the students who had already grasped the ideas behind that particular sorting technique. To the others, it was just a blur of moving bars.

Gordon

you could do this:

modulus:p,
a:rat(b),

and then a^b will do the job. And inverse_mod would be 1/a or a^(-1).

The representatives of the field used by this setup are balanced around zero — does that matter? E.g. mod 5 you have numbers -2 -1 0 1 2, not 0 1 2 3 4.

Your blog was pointed out to me as an example of a Sage vs Maxima comparison, because it shows Sage commands to be easier to read.
Maxima commands can be shortened using the rational function representation.