Outline of Chapter 1

Chapter 1 takes the student on a direct route to the point where it is possible to study derivatives. Sections 1.1 through 1.3 are reviews of precalculus material and can be skipped in many calculus courses. Section 1.4 gives an intuitive explanation of the hyperreal numbers and how they can be used to find slopes of curves. This section has no problem set and is intended as the basis for an introductory lecture. The main content of Chapter 1 is in the last two sections, 1.5 and 1.6. In these sections, the student will learn how to work with the hyperreal numbers and in particular how to compute standard parts. Standard parts are used at the beginning of the next chapter to find derivatives of functions. Sections 1.5 and 1.6 take the place of the beginning chapter on limits found in traditional calculus texts.

For the benefit of the interested student, we have included an Epilogue at the end of the book that presents the theory underlying this chapter.