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Anyone Have Interesting Graphing Calculator Problems?

From David Lederman, "David Lederman's Math Club":
If you are a teacher of calculus and are using graphing calculators in teaching calculus, I would like to hear from you. Also, if you have any interesting ways of presenting any topic(s) in calculus or have interesting graphing-calculator problems, please communicate with me at lederman@nwdc.com.

Editor's Note: If you have such ideas, please also post them here so that everybody can benefit from them and contribute to the discussion.

From Alex Pintilie (teacher), Bayview Glen School:

I think that it would be quite neat to illustrate Newton's Method of finding the roots of an "ugly" equation using graphing calculators. Take a point on the curve, draw the tangent, cut the x-axis. Zoom in, take the tangent at the new point and so on. Students will (hopefully) understand better what hides behind the numerical iterations. This procedure can also illustrate why chosing some "bad" starting point, the method does not converge.

From bassman@recorder.ca, Grenville Christian College, Brockville, ON:

As an OAC Algebra student,I have occasional use for a graphing calculator. I own a TI-85 by Texas Instruments and I am wondering if it is possible to get it to reduce function equations to a form in which the y variable is expressed in terms of x since that is the only format that it accepts to graph a function. Apart from doing it by hand, I haven't been able to make it do this. Any ideas would really be appreciated.
Reply by Philip Spencer, University of Toronto:
There is no general procedure to solve an equation for one of the variables. There are two issues involved: first, does such a function even exist, and secondly, if it does, can it be expressed in terms of "familiar" functions like polynomials, exponentials, trig functions, and so on.

First of all, an equation in x and y may not always define y as a function of x. For instance, the equation x - y^2 = 0 does not define y as a function of x: for x = 1, there are two y-values (1 and -1) which satisfy the equation, so y is not uniquely determined as a function of x.

There is a way to tell in most cases whether or not an equation defines y as a function of x, but it involves notions from multi-variable calculus so I won't get into it in detail here, unless someone wants me to.

However, even if you know that an equation does define y as a function of x, there is no guarantee that this function can be expressed in terms of familiar functions like polynomials, exponentials, trig functions, and so on. The best you can do is try various "by hand" techniques of solving the equation for y, and see if any of them work.

Because this involves some intelligent guesswork and choosing of appropriate manipulations, it is beyond the scope of a graphing calculator. A more sophisticated computer algebra program like Mathematica or Maple will be able to do it in many cases, but even then it's not guaranteed to work.

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