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The links of the chapters below take you to these same section pages. The advantage of accessing them through the Table of Contents is that you can see how the entire book fits together. The links below let you focus on one chapter at a time.
|1 Modeling Discrete Data
The first chapter deals with many of the Elementary Functions. We study polynomials, trigonometric functions, logarithms, and exponentials (postponing the inverse trig functions until Chapter 4). These functions have been very useful in modeling real-world phenomena. Probably for this reason, we will rely upon them so heavily as we develop and test the ideas of calculus starting in the next chapter. Thus, we give a thorough description of their properties here. The chapter concludes with a real-world modeling issue involving the AIDS virus.
|2 Modeling Rates of Change
The study of calculus truly begins with rates of change. After discussing the concepts of function and limit, and the related notion of continuity, we introduce the definition of the derivative of a function. Then we develop properties of the derivative, including some calculational rules and consequences of the definition. Following a discussion of some applications of the derivative, we conclude with a longer real-world modeling problem involving water flowing out of a cylindrical tank.
|3 Modeling with Differential Equations
Chapter 3 is devoted to first-order differential equations. We discuss slope fields, and the related technique of Euler's Method. We study exponential growth and decay, and more generally, separable differential equations. We also consider how to obtain a sketch of a function f from knowledge of its first and second derivative. Finally, we conclude with a real-world case study involving population modeling.
|4 Modeling Accumulations
So far we have discussed the derivative as the first big idea of calculus. The second, and accompanying, grand notion of calculus is that of integration. We will see that integration involves summations, or accumulations, that are part of a process of passing to the limit. After developing properties of integrals, we study the Fundamental Theorem of Calculus that relates derivatives and integrals, thereby connecting the two parts of calculus. We conclude the chapter with a case study that looks at the real-world problem of flooding.
|5 Culminating Experience
This chapter develops a case study involving the discrete data points recorded by Galileo in one of his rolling ball experiments. Without the aid of calculus, which had not been invented yet, Galileo was able to find a function that modeled the data. Our task is to use the modern ideas of calculus which we have been studying to see if we can make sense of Galileo's original data, and to see if we can draw any conclusions about the acceleration due to gravity.