viii Preface
The topics covered in the text are quite standard. Chapters 1 through 6 focus
on single variable calculus and are normally covered in the first semester of the
course. Chapters 7 through 11 are concerned with calculus in several variables and
are normally covered in the second semester.
Chapter 1 begins with a section on set theory. This is followed by the in-
troduction of the set of natural numbers as a set which satisfies Peano’s axioms.
Subsequent sections outline the construction, beginning with the natural numbers,
of the integers, the rational numbers, and finally the real numbers. This is only an
outline of the construction of the reals beginning with Peano’s axioms and not a
fully detailed development. Such a development would be much too time consuming
for a course of this nature. What is important is that, by the end of the chapter:
(1) students know that the real number system is a complete, Archimedean, or-
dered field; (2) they have some practice at using the axioms satisfied by such a
system; and (3) they understand that this system may be constructed beginning
with Peano’s axioms for the counting numbers.
Chapter 2 is devoted to sequences and limits of sequences. We feel sequences
provide the best context in which to first carry out a rigorous study of limits. The
study of limits of functions is complicated by issues concerning the domain of the
function. Furthermore, one has to struggle with the student’s tendency to think
that the limit of f(x) as x approaches a is just a pedantic way of describing f(a).
These complications don’t arise in the study of limits of sequences.
Chapter 3 provides a rigorous study of continuity for real-valued functions of
one variable. This includes proving the existence of minimum and maximum values
for a continuous function on a closed bounded interval as well as the Intermediate
Value Theorem and the existence of a continuous inverse function for a strictly
monotone continuous function. Uniform continuity is discussed, as is uniform con-
vergence for a sequence of functions.
The derivative is introduce in Chapter 4 and the main theorems concerning the
derivative are proved. These include the Chain Rule, the Mean Value Theorem,
existence of the derivative of an inverse function, the monotonicity theorem, and
L’Hˆ opital’s Rule.
In Chapter 5 the definite integral is defined using upper and lower Riemann
sums. The main properties of the integral are proved here along with the two
forms of the Fundamental Theorem of Calculus. The integral is used to define and
develop the properties of the natural logarithm. This leads to the definition of the
exponential function and the development of its properties.
Infinite sequences and series are discussed in Chapter 6 along with Taylor’s
series and Taylor’s formula.
The second half of the text begins in Chapter 7 with an introduction to d-
dimensional Euclidean space,
Rd,
as the vector space of d-tuples of real numbers.
We review the properties of this vector space while reminding the students of the
definition and properties of general vector spaces. We study convergence of se-
quences of vectors and prove the Bolzano-Weierstrass Theorem in this context. We
describe open and closed sets and discuss compactness and connectedness of sets
in Euclidean spaces. Throughout this chapter and subsequent chapters we follow
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