Calculus: A Rigorous First Course

929 Pages·2017·66.07 MB·English

by  Daniel J. Velleman

CALCULUS
A RIGOROUS FIRST COURSE
CALCULUS
A RIGOROUS FIRST COURSE
Daniel J. Velleman
Amherst College
University of Vermont
DOVER PUBLICATIONS, INC.
Mineola, New York
Copyright
Copyright © 2016 by Daniel J. Velleman All rights reserved.
Bibliographical Note
Calculus: A Rigorous First Course is a new work, first published by Dover Publications, Inc., in 2016.
A Solutions Manual for this book is available to instructors only. If you are an instructor and would like
to receive it, please send an email to: calculus@ doverpublications.com, and include the name of your
institution.
Library of Congress Cataloging-in-Publication Data
Names: Velleman, Daniel J.
Title: Calculus : a rigorous first course / Daniel J. Velleman.
Description: Mineola, New York : Dover Publications, [2016] | Series: Aurora: Dover modern math
originals | Includes bibliographical references and index.
Identifiers: LCCN 2016034302| ISBN 9780486809366 | ISBN 0486809366
Subjects: LCSH: Calculus—Textbooks.
Classification: LCC QA303.2 .V45 2016 | DDC 515—dc23 LC record available at
https://lccn.loc.gov/2016034302
Manufactured in the United States by LSC Communications
80936601 2016
www.doverpublications.com
Contents
Preface
1 Preliminaries
1.1 Numbers and Sets
1.2 Graphs in the Plane
1.3 Functions
1.4 Combining Functions
2 Limits
2.1 What is Calculus About?
2.2 What Does “Limit” Mean?
2.3 Limits by the Definition
2.4 Limit Theorems
2.5 Variations on Limits
2.6 Limits of Compositions
2.7 Continuity
2.8 Sequences and the Nested Interval Theorem
2.9 Monotone Sequences and the Completeness of the Real Numbers
3 Derivatives
3.1 Rates of Change and Slopes
3.2 Derivatives
3.3 Derivative Rules
3.4 The Chain Rule
3.5 Implicit Differentiation
4 Applications of Differentiation
4.1 Related Rates
4.2 The Mean Value Theorem
4.3 Increasing and Decreasing Functions
4.4 Concavity
4.5 Sophisticated Graphing
4.6 Optimization Problems
4.7 Maxima and Minima on Finite Closed Intervals
4.8 L’Hôpital’s Rule
4.9 Antiderivatives
5 Integrals
5.1 Summations
5.2 Accumulation and Area
5.3 Definite Integrals
5.4 The Fundamental Theorems of Calculus
5.5 Integration by Substitution
5.6 Proofs of Theorems
6 Applications of Integration
6.1 Area Between Curves
6.2 Volume by Disks, Washers, and Slices
6.3 Volume by Cylindrical Shells
6.4 Work
6.5 Center of Mass
7 Inverse Functions, the Natural Logarithm, and the Exponential
Function
7.1 Inverse Functions
7.2 Calculus with Inverse Functions
7.3 The Natural Logarithm
7.4 The Exponential Function
7.5 The Inverse Trigonometric Functions
7.6 L’Hôpital’s Rule Again
8 Techniques of Integration
8.1 Partial Fractions
8.2 Integration by Parts
8.3 Trigonometric Integrals
8.4 Substitution with Inverse Functions
8.5 Trigonometric Substitutions
8.6 Numerical Integration
8.7 Improper Integrals
9 Parametric Equations and Polar Coordinates
9.1 Parametric Equations
9.2 Arc Length
9.3 Surface Area
9.4 Polar Coordinates
9.5 Areas in Polar Coordinates
10 Infinite Series and Power Series
10.1 Infinite Series
10.2 Convergence Tests
10.3 The Comparison and Limit Comparison Tests
10.4 The Ratio and Root Tests
10.5 Absolute Convergence and the Alternating Series Test
10.6 Power Series
10.7 Calculus with Power Series
10.8 Taylor Series
10.9 The Binomial Series
10.10 Taylor Polynomials and the Taylor Remainder
10.11 Proof of Theorem 10.7.1
Appendix Answers to Odd-Numbered Exercises
Index
Preface
To the Student
My advice to students reading this book is very simple: Don’t believe anything
you read in this book.
Perhaps I should explain further. You may be used to studying mathematics
by memorizing formulas and procedures for solving different types of problems.
This method can be successful in high school math, but when you get to more
advanced subjects like calculus, it doesn’t work well. There are just too many
different types of problems in calculus to memorize a procedure for solving each
one. And for some types of calculus problems, there is no step-by-step procedure
you can follow to get the answer.
An approach that works better is to grasp the concepts of calculus, so that you
can understand why problems are done the way they are. With that
understanding, when you come to a problem that is a little different from ones
you have seen before, you can figure out how to solve it, rather than trying to
apply a memorized procedure for solving it. When calculus is approached in this
way, solving problems is not just a matter of calculation; it involves reasoning.
For reasoning in mathematics to be effective, it must be held to a very high
standard. Our standard will be certainty: when reasoning about a problem, our
goal will be not just to determine the answer, but to become certain of the
answer. This pursuit of certainty should be evident throughout this book.
Whenever we solve a problem, we will present a solution that is intended not
merely to find the answer, but to convince you, with complete certainty, that the
answer is correct. You should read this book with a skeptical attitude, refusing to
believe that an answer is correct unless the solution is completely convincing.
(That’s why you shouldn’t believe anything you read in this book.) And you
should take a similar skeptical attitude toward your own solutions to problems:
your goal is not merely to get the answer, but to be certain of the answer.
This skeptical attitude is important for success in calculus. In calculus there
are often many different approaches that could be taken to a problem, some of
which work and some of which don’t. It is sometimes impossible to know in
advance which approach will work. When trying to solve a problem, you may
have to try one approach, recognize that it doesn’t work, and then switch to a
different approach. Thus, success in calculus requires not only the ability to find
correct solutions, but also the ability to reject incorrect ones.
In your previous study of mathematics, you may have focused mainly on
learning how to find correct solutions. Learning to recognize, and reject,
incorrect solutions may be a new skill for you. How do you know when to reject
a proposed solution? The best answer we can give is that you must insist on
certainty. If your reasoning on a problem does not completely convince you of
the answer, then it is insufficient and must be either improved or rejected. Your
best defense against incorrect solutions is the skeptical attitude that we try to
encourage in this book.
For reasoning to achieve certainty, it must be expressed with precision. We
introduce many technical terms in this book, and when a term is introduced we
always provide a precise definition. It is important to understand that
terminology and notation in mathematics are always used to mean exactly what
the definitions say—no more and no less. You should pay close attention to
definitions, referring back to them if necessary. In many cases, the best way to
get started on a problem is to be guided by the definitions of the words and
notation appearing in the statement of the problem.
Often the methods we use to solve problems are based on general principles
that are stated in the form of theorems. We have provided proofs of almost all of
the theorems stated in this book. You may choose to skip some of these proofs,
especially on a first reading. But we hope that your skeptical attitude will make
you want to read them, so that you can be convinced that the theorems are true,
rather than merely accepting them. These proofs demonstrate one of the
fundamental principles of mathematics, without which the skeptical approach
would be impossible: everything in math has a reason. Reading the proof of a
theorem can not only help you understand why the theorem is true, it can also
deepen your understanding of the meaning of the theorem. The proofs also
provide you with good models of how mathematical reasoning should be carried
out and expressed in writing.
This book requires no previous knowledge of calculus, but it does require a
good background in algebra and trigonometry. Chapter 1 gives a brief review of
the ideas from algebra and trigonometry that will be most important to us. If this
review is not sufficient for you, then you may need to refer back to other
resources on algebra and trigonometry.
To the Instructor
The topics covered in this book are the usual topics of first-year calculus: limits,
derivatives, integrals, and infinite series. We have tried to give a mathematically
rigorous treatment of these topics, while keeping the focus on the use of calculus
to solve problems, rather than on the theoretical foundations of the subject. This
is a rigorous calculus book, not an analysis book.
Some instructors may believe that rigor is an advanced topic that shouldn’t be
introduced until a student is taking analysis. This book is an attempt to justify
the opposite point of view: rigor is a useful guide to beginning students that can
help them learn to distinguish between correct mathematical reasoning and
reasoning that is plausible but flawed. Calculus is full of such tempting but
incorrect approaches to problems, and success requires learning to recognize
them. Doing so when reasoning is kept at an intuitive level can be extremely
difficult. Our point of view is that rigor is not an advanced topic; it is intuitive
reasoning that is an advanced topic.
One of the most difficult ideas in calculus is the precise definition of limits,
but a rigorous approach to the subject is not possible without such a definition.
We devote an entire section to motivating and explaining the definition, using
words, pictures, and formulas. A second section gives practice using the
definition to prove limit statements. Reasoning based on this definition occurs at
various points throughout the book.
Another essential theoretical idea is the completeness of the real numbers. Our
fundamental completeness statement is the nested interval theorem, which says
that for any nested sequence of closed intervals whose lengths approach 0, there
is a unique number that is in all of the intervals. We state the nested interval
theorem without proof, but a diagram makes it very plausible. We then use the
theorem in a number of proofs throughout the book, including a proof of another
version of completeness, the existence of least upper bounds and greatest lower
bounds.
Our discussion of the nested interval theorem uses sequences and their limits,
so these concepts are introduced early, in the last two sections of Chapter 2. The
study of sequences also provides an opportunity to introduce the method of
mathematical induction, which is used in a number of places later. For example,
we use induction to prove that for every positive integer n, the derivative of xn is
nxn−1.
We have chosen to restrict our discussion of definite integrals to continuous
functions, and to use only uniform partitions in our definition of the integral.
This is sufficient for everything we do with integrals in first-year calculus. Our
definition of definite integrals also makes use of sequence limits: the definite