This case study is presented here as four modules that appear at four different points in calculus. Used in this way it both unifies the course and illustrates the idea that as we learn more mathematics we learn more about the real world. Because this idea is key, we begin the first contact with this case study by an ambitious list of phenomena and questions and then make only a little bit of progress. By the end of the year we will have answered all the questions in the list below. You may want to add questions that will not be answered in the first year. Students should see questions that motivate future courses and future research. For example, this case study appears again as motivation for the heat equation where we look at how heat diffuses through the walls of an adobe house.
We begin by establishing the most important theme of this course -- the interplay of mathematics and the real world. We start with data -- both experimental data and everyday experience -- and, more importantly, we start with questions.
We begin with the first question. We heat two cups of water to the boiling point and then we use three temperature probes to record the temperature in the cups and the ambient temperature while they cool. As they cool, one of the two cups is placed under a fan to simulate wind and the other is left in calm air. The data from this experiment enables us to answer the first question above and it provides good experience in modeling. Newton's Model of Cooling does a good job describing cooling in calm air at moderate temperatures but it fails in windy air and at high temperatures because it does not take into account evaporative cooling.
In the remaining three modules we look at what happens when the ambient temperature is changing. This work is motivated by several observations -- the sun is strongest each day near noon and each year at the time of the summer solstice but the warmest part of the day is about 2 or 3 PM and the hottest time of the year is usually in August.
In this module we use slope fields to study Newton's Model of Cooling. We begin with the simplest situation involving Newton's Model of Cooling
where the ambient temperature A is constant. Then we look at what happens when the constant A is replaced by a function A(t) -- for example,
We observe that this mathematics models at least qualitatively the time lag we see, for example, in daily temperature lagging behind the intensity of sun light. One of the themes we want to emphasize is the complementarity of graphic, numeric, and symbolic methods. This first qualitative look helps us understand why there is a time lag between the temperature of an object and the ambient temperature.
In this module we use numeric methods via Maple, Mathematica, or the TI-92 to study Newton's Model of Cooling.
and we see more quantitatively the time lag that we expected both from data and our first qualitative look at the differential equation.
Students experiment in the lab to see numerically the effects of the constants k and omega.
In this module we use the standard procedure for solving first order linear differential equations to study Newton's Model of Cooling. We begin by finding closed form solutions for the specific examples
using the standard technique for solving first order linear differential equations and integration by parts. This is a nice use of these techniques and also requires using the usual trig identities to express the solution nicely.
The closed form solution makes plain the effect of the constant k on the time lag and the amplitude of the oscillations of T(t).
Since this is the last appearance of this thread in the first year of calculus, it is important to summarize what we have done and to emphasize the ways in which the different approaches -- numeric, graphic, and analytic -- complement each other.
In particular, we note that looking at the algebraic solution alone one might think the effects of k and omega were consequences of trigonometric identities and algebra but looking at slope fields and doing numerical experiments with other ambient temperature functions, A(t), one can see that these effects are quite general.
The four modules described above are described in more detail in instructors' versions and in student modules in the Calculus "book."