Computers

I think the best way to use a computer in learning Calculus is as a sort of solutions manual, but better. Do a problem first by hand. Then verify correctness of your solution. This is way better than what you get by using a solutions manual!

• You can try similar problems (not in the homework) and also verify your answers. This is like playing solitaire, but is much more creative.
• You can verify key steps of what you did by hand using the computer. E.g., if you're confused about one of part of your approach to computing an integral, you can compare what you get with the computer. Solution manuals either give you only the solution or a particular sequence of steps to get there, which might have little to do with the brilliantly original strategy you invented.

For this course its most useful to have a program that does symbolic integration. I recommend maxima, which is a fairly simple completely free and open source program written (initially) in the 1960s at MIT. Download it for free from

http://maxima.sourceforge.net

It's not insanely powerful, but it'll instantly do (something with) pretty much any integral in this class, and a lot more. Plus if you know lisp you can read the source code. (You could also buy Maple or Mathematica, or use a TI-89 calculator.)

Here are some maxima examples:

(%i2) integrate(x^2 + 1 + 1/(x^2+1), x);
                                          3
                                         x
(%o2)                          atan(x) + -- + x
                                         3

(%i3) integrate(sqrt(5/x), x);
(%o3)                          2 sqrt(5) sqrt(x)

(%i4) integrate(sin(2*x)/sin(x), x);
(%o4)                              2 sin(x)

(%i5) integrate(sin(2*x)/sin(x), x, 0, %pi);
(%o5)                                  0

(%i6) integrate(sin(2*x)/sin(x), x, 0, %pi/2);
(%o6)                                  2