One of the chapter test questions we use in our course, toward the end of calculus A, is:
5. Find the max and min of the function f(x) = 2/3 sin (pi*x) on the interval [1,2]. Sketch its graph.
6. Find the equation which x solves if x is the location of the minimum value of g(x) = 1/x sin(pi*x) on the interval [1,2]. Is the minimum value larger or smaller than the minimum of f in the previous problem? Hint: consider g(3/2) and g'(3/2).
#5 is a problem which can be done just fine by algebraic means. The idea of doing it with a calculator seems absurd to me -- students should be able to look at this function and immediately know what it looks like. They shouldn't even need to do any algebra.
#6, then, is a problem that could be done with a graphing calculator. But it seems to me that doing it without one builds more understanding. Of course, if you wanted to find the ACTUAL minimum of g, a calculator would be a fine way to go. But doing the algebra that gives the equation the minimum would satisfy, and then seeing that the minimum must be smaller because g(3/2) = f(3/2) = min of f, but g'(3/2) is nonzero, seems much more revealing than just finding something on the calculator.
What do you think about this kind of problem? Are these good test questions? What is the role of the calculator (if any) in your opinion?