Show HN: Interactive Calculus via LaTeX (opens in new tab)

I'm one of the developers of Ximera.

Expression equality is handled by https://github.com/kisonecat/math-expressions using a few heuristics, which is necessary because Richardson's theorem says this problem is undecidable. The main trick is to regard the expressions as complex-valued functions, find a point where the value is (nearly) equal, and then check for numerical equality in a neighborhood. (Doing this over the complex plane addresses the problem that many common expressions like sqrt(small number-x^2) have a restricted domain over the reals).

We rely on unit tests from Chris Sangwin (who wrote the textbook _Computer Aided Assessment of Mathematics_) to ensure that we're adjudicating expression equality mostly correctly.

If you goto:

https://ximera.osu.edu/introduction

you can see how to make your own materials.

There is a set of example environments here:

https://ximera.osu.edu/examples/exampleActivityCollection

One way to do this is to install Mathematica and then run

  $ wolframscript --code "6 == 5 + 1"
  True

Suppose you know that a function is equal to "(x - 1)(x + 1) - 42". You want to check whether the student has given an equivalent function, like "x^2 - 43":

  $ wolframscript --code "f[x_] := (x - 1)(x + 1) - 42;
  Simplify[f[x]] == Simplify[x^2 - 43]"
  True

But it should also return True for any other form, like "-43 + x^2". And indeed it does:

  $ wolframscript --code "f[x_] := (x - 1)(x + 1) - 42; 
  Simplify[f[x]] == Simplify[-43 + x^2]"
  True

So in general, you can write a script which inserts the student's answer into the right-hand "Simplify[...]" block. If it prints True, they're equivalent.

It's also possible to feed a natural language expression like "the integral of Sin(x) from 0 to pi is equal to 2" into wolfram alpha, and check whether it returns True or False:

http://www.wolframalpha.com/input/?i=the+integral+of+Sin(x)+...

(To be clear, I don't think this is how this project does it. I was just giving a way to do it.)

There are some R packages that allow for development of e-learning materials, these are Rmarkdown based so similar to knitr rather than sweave, but basically the same idea. The downside is the lack of symbolic support in R. Ximera does look interesting however. A second question I would have is how this would compare with Jupyter interactive notebooks which have LaTeX support and automated grading as well as symbolic algebra capabilities? Nevertheless additional tools are always welcome.