Lession 1: Equations, Functions and Graphs

To begin the study of calculus, we begin by having the student review some important properties of equations and functions, and how to graph different kinds of functions. A solid understanding of analytic geometry is essential to developing the techniques of differentiation and integration presented in this book.

In this lesson, the idea of a function being a table of numbers, where one column is the input variable (x) and the other comumn is the output variable (y). An important technique students will need to understand throughout their study of calculus is how to evaluate functions by either plugging in values for the input variable, or in more complicated cases like in developing the equations for derivatives and integrals, substituting algebraic expressions in for the input variable of a function.

For instance, one definition of the derivative of a function will have students evaluating functions for the value x + h, i.e. f(x+h), and then finding the value of the evaluated function in the limit as h approaches zero. Making certain that students understand how to do this, particularly the technique of substituting the entire expression in the function's argument for the input variable in the function expression, is essential for them to develop facility with understanding many topics throughout this course.

Although much of the notation that is being introduced in this and subsequent lessons is very formal, it is important to stress that functions are so important because they enable us to effectively model a number of real world phenomena. In the exercises for this chapter, the relationship between the independent variable x, and the example of modeling costs is provided through both linear and nonlinear examples.