This question arose in an earlier post and I find it quite interesting. I've been using graphing calculators in my calculus classes and have found that my assessment procedures must change. My classroom discussions are different, too.
For example, while discussing properties of the natural log function recently I asked my students (In class) to use their calculators to graph the two functions
f(x) = ln(x^2) and g(x) = 2 ln x
I made certain that they cleared the first function before graphing the second. This led to a discussion of why the calculator gave two "different" graphs even though many (most) of the students expected to get the same graph. Questions like this utilize the power of the technology without losing sight of some of the "pencil & paper" work needed.
Another example from an exam :
Use your graphing calculator to approximate the x-coordinate of the relative minimum of the function
f(x) = x^5 - x^4 + x^2 - x
Merely zooming in on the point does not give very satisfactory results. My suggestion with a similar example in class was to simultaneously graph f'(x) and determine where its graph crosses the x-axis (going from negative to positive) .
I feel that this type of example also combines the use of technology and calculus analysis.
This is a VERY interesting thread that I hope we continue.
Herb Kasube Department of Mathematics Bradley University Peoria, IL 61625 firstname.lastname@example.org