Calculus is a branch of mathematics that studies functions through the processes of limits, derivatives, and integrals. This text introduces the ideas of calculus through the context of mathematical models. This text is organized to review precalculus concepts necessary to be successful in learning calculus. The goal is not, however, to introduce these concepts to a beginner but to leverage some familiarity toward a deeper understanding.
The text begins with a basic review of numbers and variables with an aim of connecting these foundational concepts to the goals of modeling relationships between physically measured quantities. It then turns to the development of functions as the mathematical tool used for predictive relationships between variables. Intuitive ideas of continuity are introduced in relationship to making piecewise functions connected, with limits introduced as a way to characterize the behavior of a function at the break points.
Sequences are introduced early as a discrete illustration of many of the concepts relating to modeling with functions. Patterns found in sequences are familiar to students and these patterns can often be described both explicitly and recursively. This will serve as a prelude to the idea that functions can be defined explicitly with a formula or indirectly through differential equations. In addition, sequences foreshadow the ideas of limits, derivatives, and integrals. Describing the monotonicity and concavity of a sequence then provides a direct correspondence to describing the behavior of a function in terms of its first and second derivatives.
Summation rules and formulas naturally occur as part of the discussion of sequences. This motivates an early introduction of the definite integral as the generalization of the accumulation of increments of change. The rate of change is introduced in the context of accumulation and the reader is explicitly told to look forward to the Fundamental Theorem of Calculus as the formal connection between the rate of accumulation and the instantaneous rate of change as being equivalent. Properties of the definite integral as well as elementary formulas are introduced. The behavior of functions in terms of the first and second derivative are introduced using the integral representation of functions.
Next, we develop a more thorough investigation of limits and continuity of functions. Properties of limits are developed in the context of the limits of sequences. The epsilon-delta definition of a limit is given in an optional section. Continuity of functions is formalized and the major theorems for continuous functions are presented, namely the Intermediate Value Theorem and the Extreme Value Theorem.
This is followed by the formal development of the derivative and the rules of differentiation, including the first part of the Fundamental Theorem of Calculus. Applications of derivatives and antiderivatives are included.
You can download a PDF copy of the book with the links below. There are differently formatted options, depending on how you intend to use the PDF.
http://educ.jmu.edu/~waltondb/MA2C/model-calculus.pdf (Margins formatted to print 2-sided)
http://educ.jmu.edu/~waltondb/MA2C/model-calculus-1page.pdf (Margins formatted to print 1-sided)
In order to keep track of changes during the semester, the following list describes the changes that have taken place in the text since August 2018.