"Spot on"! Of course the key here is what "simple" means. Where one draws the line as to what must be done with paper and pencil and what may be done by computer depends on who's drawing the line.
Here's an example of using Mathematica, due to Frank Wattenberg, that I may have mentioned in other threads earlier: It's the standard Calculus I problem of minimizing the total travel time when you are going to rescue a drowning person first run along the edge of an ocean beach and then swim out towards him. There are many variants (some involving a dog, others concerning a utility line).
Once the student sets up the model and calculates the derivative of the target function, finding critical points may be easy or it may be hard -- Depending on what the parameters are.
A particularly interesting version of the problem, the one Frank included in his calculus textbook, is to minimize the travel time of a light ray going at an angle first through air and then water, or vice versa. By solving the problem, you see what the refraction is as the light passes from one medium to the other.
It so happens that the velocities of light in air and water, even when rounded, are not too nice to deal with: you get a 4th degree polynomial. There is a formula/method for solving quartic equations exactly, but trying to implement it, without error, with paper and pencil is not especially to be recommended.
That's where Mathematica can come into play: either get the exact critical points symbolically in terms of coordinates of the two ends of the light ray, then for particular pairs of coordinates take N; or else use FindRoot.
By avoiding the distraction of solving the quartic, whether symbolically or numerically and instead relegating that to the computer, one may now ask and quickly answer many questions about the behavior of light. For example, where do your toes appear to be as you soak in a tub of water?
On May 29, 2013, at 3:58 AM, Andrzej Kozlowski <email@example.com> wrote:
> . . .I just wanted to make a final comment on > one other matter. It's nothing new, I have written this before and it is > something that I differ about with some of the persons who have posted > in this thread. Briefly, I am not an enthusiast of using Mathematica for > "simple" symbolic manipulation. I am sure that may view of this is the > same as that of the great majority of mathematicians and that the reason > for this is not "conservatism". Generally we believe that in mathematics > computers should be used to do things that are too hard or too time > consuming for humans and human beings should do the things that humans > find easy. The reasons for this are practical (this is the only way to > really remain "in control" of your work - which is essential to avoid > producing nonsense) and educational (doing symbolic and even numerical > computations by hand is an essential activity that is needed to acquire > understanding of concepts). > Of course I do not object to Mathematica having capabilities I do not > need or intend to use or teach. If other people find them useful, they > could be a selling point for Mathematica and help to pay for the > development of more features that I want (of course, as long as the > issue of excessive complexity of interface is kept under control).
--- Murray Eisenberg firstname.lastname@example.org Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2838 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305