An Interesting Problem From Calculus and Much More

Robert Kreczner

University of Wisconsin Stevens Point
Stevens Point, WI 54481
USA
rkreczne@uwspmail.uwsp.edu

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ABSTRACT

In the latest, 3rd edition of Calculus by Stewart, early transcendentals, we can encounter many new and challenging problems whose solutions can be obtained, in an enjoyable way, by using modern technologies. Further, these problems can lead to a sophisticate mathematical thinking and discovering new Mathematics. One of such problems is the problem mentioned in our title. Shortly, this problem is about two circles: C, xy1; and C2, (x1)yr. And the problem is to find a limit of x intercept of the line passing through the point (O, r) and the intersection point of C1 and C2, as r approaches 0. We will discuss not only solution to this problem, but more interestingly, its many generalizations. For example, when the circle C2 is replaced by any curve passing through the origin, which will lead to surprising results. In the process of seeking solutions, we will make frequent references to the Geometer's Sketchpad and Mathematica which turn out to be indispensable tools for handling complicate formulas. All the discussion is easily accessible to Calculus students, and might be used for students' research projects.

Keyword(s): calculus, geometry, Mathematica, Geometer's Sketchpad