Understanding The Fundamental Theorem Of Calculus Through Writing

Lew Ludwig, Denison University

I often use writing assignments in my classes to help students better understand new topics and concepts. In my calculus classes, students are often “hired” by individuals in the business community as consultants to help solve real world problems that directly or indirectly relate to a new concept they are learning. The students must communicate their solutions to the problems to the businessperson in language that the layperson can understand and apply. For example, to develop the concept of area under the curve, I used the following scenario:

A golf course was recently wracked by high winds. Much of the protective netting (see Picture 1) used to deflect arrant golf balls was ripped to shreds. The management wants to replace the old black netting with a newly developed clear netting that is more aesthetically pleasing. The new netting, sold by Clear Vu Inc., is quite expensive. So the management of the golf course must find a way to accurately measure the area underneath a freely hanging chain suspended by two vertical post of a given height in order to know how much netting to purchase.

Picture 1

Students used a physical model in the lab (see Picture 2) and developed a series of three to four reports directed toward a final solution for the businessperson. The first one or two reports dealt with an attempt to approximate the area under the curve through various types of geometry and/or Riemann sums. The businessperson responded to these reports and explained that although the approximation was good in theory, it took too many measurements. In the next report, the students described how they used a parabola to approximate the curve. Next, they explained how the fundamental theorem of calculus was used to find the area beneath the approximating parabola. The businessperson liked the new, quicker method, but questioned the validity of the curve. In the final report, the students discussed the catenary and the difficulties of approximating the curve with a closed form of the hyperbolic cosine. The students then argued either in favor of using software to approximate the catenary or why it was good enough to approximate the curve with a parabola. In either case, it was clear that there was more than one possible solution involved and the students had to weigh the pros and cons of each before making their final recommendation.

Picture 2