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Collection Highlights

Calculus Single and Multivariable 7th Edition Deborah

Hughes-Hallett

Multivariable calculus 8th Edition Stewart

Multivariable advanced calculus Kuttler K

Multivariable Calculus 8th Edition James Stewart

Multivariable Advanced Calculus Kenneth L. Kuttler

Teaching Writing: Balancing Process and Product, with

Enhanced Pearson eText -- Access Card Package (What's New
in Literacy) 7th Edition Tompkins

Calculus single variable Bretscher

Single Variable Calculus Robert Lopez

Solutions Stewart s Multivariable Calculus 8th 8th Edition

James Stewart
 CALCULUS
Seventh Edition

Produced by the Calculus Consortium and initially funded by a

National Science Foundation Grant.

Deborah Hughes- William G. McCallum Andrew M.

Hallett Gleason
University of University of Arizona Harvard
Arizona University

Eric Connally David Lovelock Douglas Quinney

Harvard University University of Arizona University of Keele
Extension

Daniel E. Flath Guadalupe I. Lozano Karen Rhea

Macalester College University of Arizona University of
Michigan

Selin Kalaycıoğlu Jerry Morris Ayşe şahin

New York University Sonoma State University Wright State
University

Brigitte Lahme David Mumford Adam H. Spiegler

Sonoma State Brown University Loyola University
University Chicago
Patti Frazer Lock Brad G. Osgood Jeff Tecosky-
Feldman
St. Lawrence Stanford University Haverford College
University

David O. Lomen Cody L. Patterson Thomas W. Tucker

University of University of Texas at Colgate University
Arizona San Antonio

Aaron D. Wootton
University of Portland

with the assistance of

Otto K. Bretscher Adrian Iovita David E. Sloane,
MD
Colby College University of Harvard Medical
Washington School

Coordinated by
Elliot J. Marks
We dedicate this book to Andrew M. Gleason.
His brilliance and the extraordinary kindness and dignity with
which he treated others made an enormous difference to us, and to
many, many people. Andy brought out the best in everyone.
Deb Hughes Hallett
for the Calculus Consortium
ACQUISITIONS EDITOR Shannon Corliss
VICE PRESIDENT AND Laurie Rosatone
DIRECTOR
DEVELOPMENT EDITOR Adria Giattino
FREELANCE Anne Scanlan-Rohrer/Two Ravens
DEVELOPMENTAL EDITOR Editorial
MARKETING MANAGER John LaVacca
SENIOR PRODUCT David Dietz
DESIGNER
SENIOR PRODUCTION Laura Abrams
EDITOR
COVER DESIGNER Maureen Eide
COVER AND CHAPTER © Patrick Zephyr/Patrick Zephyr
OPENING PHOTO Nature Photography

Problems from Calculus: The Analysis of Functions, by Peter D.

Taylor (Toronto: Wall & Emerson, Inc., 1992). Reprinted with
permission of the publisher.

This book was set in Times Roman by the Consortium using TEX,
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 PREFACE
Calculus is one of the greatest achievements of the human intellect.
Inspired by problems in astronomy, Newton and Leibniz developed
the ideas of calculus 300 years ago. Since then, each century has
demonstrated the power of calculus to illuminate questions in
mathematics, the physical sciences, engineering, and the social and
biological sciences.
Calculus has been so successful both because its central theme—
change—is pivotal to an analysis of the natural world and because of
its extraordinary power to reduce complicated problems to simple
procedures. Therein lies the danger in teaching calculus: it is
possible to teach the subject as nothing but procedures—thereby
losing sight of both the mathematics and of its practical value. This
edition of Calculus continues our effort to promote courses in which
understanding and computation reinforce each other. It reflects the
input of users at research universities, four-year colleges, community
colleges, and secondary schools, as well as of professionals in partner
disciplines such as engineering and the natural and social sciences.

Mathematical Thinking Supported by Theory

and Modeling
The first stage in the development of mathematical thinking is the
acquisition of a clear intuitive picture of the central ideas. In the next
stage, the student learns to reason with the intuitive ideas in plain
English. After this foundation has been laid, there is a choice of
direction. All students benefit from both theory and modeling, but
the balance may differ for different groups. Some students, such as
mathematics majors, may prefer more theory, while others may
prefer more modeling. For instructors wishing to emphasize the
connection between calculus and other fields, the text includes:

A variety of problems from the physical sciences and

engineering.
 Examples from the biological sciences and economics.
Models from the health sciences and of population growth.
Problems on sustainability.
Case studies on medicine by David E. Sloane, MD.

Active Learning: Good Problems

As instructors ourselves, we know that interactive classrooms and
well-crafted problems promote student learning. Since its inception,
the hallmark of our text has been its innovative and engaging
problems. These problems probe student understanding in ways
often taken for granted. Praised for their creativity and variety, these
problems have had influence far beyond the users of our textbook.
The Seventh Edition continues this tradition. Under our approach,
which we call the “Rule of Four,” ideas are presented graphically,
numerically, symbolically, and verbally, thereby encouraging
students to deepen their understanding. Graphs and tables in this
text are assumed to show all necessary information about the
functions they represent, including direction of change, local
extrema, and discontinuities.
Problems in this text include:

Strengthen Your Understanding problems at the end of

every section. These problems ask students to reflect on what
they have learned by deciding “What is wrong?” with a statement
and to “Give an example” of an idea.
ConcepTests promote active learning in the classroom. These
can be used with or without personal response systems (e.g.,
clickers), and have been shown to dramatically improve student
learning. Available in a book or on the web at
www.wiley.com/college/hughes-hallett.
Class Worksheets allow instructors to engage students in
individual or group class-work. Samples are available in the
Instructor's Manual, and all are on the web at
www.wiley.com/college/hughes-hallett.
 Data and Models Many examples and problems throughout the
text involve data-driven models. For example, Section 11.7 has a
series of problems studying the spread of the chikungunya virus
that arrived in the US in 2013. Projects at the end of each chapter
of the E-Text (at www.wiley.com/college/hughes-hallett) provide
opportunities for sustained investigation of real-world situations
that can be modeled using calculus.
Drill Exercises build student skill and confidence.

Enhancing Learning Online

This Seventh Edition provides opportunities for students to
experience the concepts of calculus in ways that would not be
possible in a traditional textbook. The E-Text of Calculus, powered
by VitalSource, provides interactive demonstrations of concepts,
embedded videos that illustrate problem-solving techniques, and
built-in assessments that allow students to check their
understanding as they read. The E-Text also contains additional
content not found in the print edition:

Worked example videos by Donna Krawczyk at the University of

Arizona, which provide students the opportunity to see and hear
hundreds of the book's examples being explained and worked out
in detail
Embedded Interactive Explorations, applets that present and
explore key ideas graphically and dynamically—especially useful
for display of three-dimensional graphs
Material that reviews and extends the major ideas of each
chapter: Chapter Summary, Review Exercises and Problems, CAS
Challenge Problems, and Projects
Challenging problems that involve further exploration and
application of the mathematics in many sections
Section on the ε, δ definition of limit (1.10)
Appendices that include preliminary ideas useful in this course
Problems Available in WileyPLUS
Students and instructors can access a wide variety of problems
through WileyPLUS with ORION, Wiley's digital learning
environment. ORION Learning provides an adaptive, personalized
learning experience that delivers easy-to-use analytics so instructors
and students can see exactly where they're excelling and where they
need help. WileyPLUS with ORION features the following resources:

Online version of the text, featuring hyperlinks to referenced

content, applets, videos, and supplements.
Homework management tools, which enable the instructor to
assign questions easily and grade them automatically, using a
rich set of options and controls.
QuickStart pre-designed reading and homework assignments.
Use them as-is or customize them to fit the needs of your
classroom.
Intelligent Tutoring questions, in which students are prompted
for responses as they step through a problem solution and receive
targeted feedback based on those responses.
Algebra & Trigonometry Refresher material, delivered through
ORION, Wiley's personalized, adaptive learning environment
that assesses students' readiness and provides students with an
opportunity to brush up on material necessary to master
Calculus, as well as to determine areas that require further
review.

Flexibility and Adaptability: Varied Approaches

The Seventh Edition of Calculus is designed to provide flexibility for
instructors who have a range of preferences regarding inclusion of
topics and applications and the use of computational technology. For
those who prefer the lean topic list of earlier editions, we have kept
clear the main conceptual paths. For example,

The Key Concept chapters on the derivative and the definite

integral (Chapters 2 and 5) can be covered at the outset of the
 course, right after Chapter 1.
Limits and continuity (Sections 1.7, 1.8, and 1.9) can be covered in
depth before the introduction of the derivative (Sections 2.1 and
2.2), or after.
Approximating Functions Using Series (Chapter 10) can be
covered before, or without, Chapter 9.
In Chapter 4 (Using the Derivative), instructors can select freely
from Sections 4.3–4.8.
Chapter 8 (Using the Definite Integral) contains a wide range of
applications. Instructors can select one or two to do in detail.

To use calculus effectively, students need skill in both symbolic

manipulation and the use of technology. The balance between the
two may vary, depending on the needs of the students and the wishes
of the instructor. The book is adaptable to many different
combinations.
The book does not require any specific software or technology. It has
been used with graphing calculators, graphing software, and
computer algebra systems. Any technology with the ability to graph
functions and perform numerical integration will suffice. Students
are expected to use their own judgment to determine where
technology is useful.

Content
This content represents our vision of how calculus can be taught. It is
flexible enough to accommodate individual course needs and
requirements. Topics can easily be added or deleted, or the order
changed.
Changes to the text in the Seventh Edition are in italics. In all
chapters, problems were added and others were updated. In total,
there are more than 1300 new problems.
Chapter 1: A Library of Functions
This chapter introduces all the elementary functions to be used in the
book. Although the functions are probably familiar, the graphical,
numerical, verbal, and modeling approach to them may be new. We
introduce exponential functions at the earliest possible stage, since
they are fundamental to the understanding of real-world processes.
The content on limits and continuity in this chapter has been revised
and expanded to emphasize the limit as a central idea of calculus.
Section 1.7 gives an intuitive introduction to the ideas of limit and
continuity. Section 1.8 introduces one-sided limits and limits at
infinity and presents properties of limits of combinations of
functions, such as sums and products. The new Section 1.9 gives a
variety of algebraic techniques for computing limits, together with
many new exercises and problems applying those techniques, and
introduces the Squeeze Theorem. The new online Section 1.10
contains the ε, δ definition of limit, previously in Section 1.8.

Chapter 2: Key Concept: The Derivative

The purpose of this chapter is to give the student a practical
understanding of the definition of the derivative and its
interpretation as an instantaneous rate of change. The power rule is
introduced; other rules are introduced in Chapter 3.

Chapter 3: Short-Cuts to Differentiation

The derivatives of all the functions in Chapter 1 are introduced, as
well as the rules for differentiating products; quotients; and
composite, inverse, hyperbolic, and implicitly defined functions.

Chapter 4: Using the Derivative

The aim of this chapter is to enable the student to use the derivative
in solving problems, including optimization, graphing, rates,
parametric equations, and indeterminate forms. It is not necessary to
cover all the sections in this chapter.
Chapter 5: Key Concept: The Definite Integral
The purpose of this chapter is to give the student a practical
understanding of the definite integral as a limit of Riemann sums
and to bring out the connection between the derivative and the
definite integral in the Fundamental Theorem of Calculus.
The difference between total distance traveled during a time
interval is contrasted with the change in position.

Chapter 6: Constructing Antiderivatives

This chapter focuses on going backward from a derivative to the
original function, first graphically and numerically, then analytically.
It introduces the Second Fundamental Theorem of Calculus and the
concept of a differential equation.

Chapter 7: Integration
This chapter includes several techniques of integration, including
substitution, parts, partial fractions, and trigonometric substitutions;
others are included in the table of integrals. There are discussions of
numerical methods and of improper integrals.

Chapter 8: Using the Definite Integral

This chapter emphasizes the idea of subdividing a quantity to
produce Riemann sums which, in the limit, yield a definite integral.
It shows how the integral is used in geometry, physics, economics,
and probability; polar coordinates are introduced. It is not necessary
to cover all the sections in this chapter.
Distance traveled along a parametrically defined curve during a
time interval is contrasted with arc length.

Chapter 9: Sequences and Series

This chapter focuses on sequences, series of constants, and
convergence. It includes the integral, ratio, comparison, limit
comparison, and alternating series tests. It also introduces geometric
series and general power series, including their intervals of
convergence.
Rearrangement of the terms of a conditionally convergent series is
discussed.

Chapter 10: Approximating Functions

This chapter introduces Taylor Series and Fourier Series using the
idea of approximating functions by simpler functions.
The term Maclaurin series is introduced for a Taylor series centered
at 0. Term-by-term differentiation of a Taylor series within its
interval of convergence is introduced without proof. This term-by-
term differentiation allows us to show that a power series is its own
Taylor series.

Chapter 11: Differential Equations

This chapter introduces differential equations. The emphasis is on
qualitative solutions, modeling, and interpretation.

Chapter 12: Functions of Several Variables

This chapter introduces functions of many variables from several
points of view, using surface graphs, contour diagrams, and tables.
We assume throughout that functions of two or more variables are
defined on regions with piecewise smooth boundaries. We conclude
with a section on continuity.

Chapter 13: A Fundamental Tool: Vectors

This chapter introduces vectors geometrically and algebraically and
discusses the dot and cross product.
An application of the cross product to angular velocity is given.
Chapter 14: Differentiating Functions of
Several Variables
Partial derivatives, directional derivatives, gradients, and local
linearity are introduced. The chapter also discusses higher order
partial derivatives, quadratic Taylor approximations, and
differentiability.

Chapter 15: Optimization

The ideas of the previous chapter are applied to optimization
problems, both constrained and unconstrained.

Chapter 16: Integrating Functions of Several

Variables
This chapter discusses double and triple integrals in Cartesian, polar,
cylindrical, and spherical coordinates.

Chapter 17: Parameterization and Vector

Fields
This chapter discusses parameterized curves and motion, vector
fields and flowlines.
Additional problems are provided on parameterizing curves in 3-
space that are not contained in a coordinate plane.

Chapter 18: Line Integrals

This chapter introduces line integrals and shows how to calculate
them using parameterizations. Conservative fields, gradient fields,
the Fundamental Theorem of Calculus for Line Integrals, and
Green's Theorem are discussed.

Chapter 19: Flux Integrals and Divergence

This chapter introduces flux integrals and shows how to calculate
them over surface graphs, portions of cylinders, and portions of
spheres. The divergence is introduced and its relationship to flux
integrals discussed in the Divergence Theorem.
We calculate the surface area of the graph of a function using flux.

Chapter 20: The Curl and Stokes' Theorem

The purpose of this chapter is to give students a practical
understanding of the curl and of Stokes' Theorem and to lay out the
relationship between the theorems of vector calculus.

Chapter 21: Parameters, Coordinates, and

Integrals
This chapter covers parameterized surfaces, the change of variable
formula in a double or triple integral, and flux though a
parameterized surface.

Appendices
There are online appendices on roots, accuracy, and bounds;
complex numbers; Newton's method; and vectors in the plane. The
appendix on vectors can be covered at any time, but may be
particularly useful in the conjunction with Section 4.8 on parametric
equations.

Supplementary Materials and Additional

Resources
Supplements for the instructor can be obtained online at the book
companion site or by contacting your Wiley representative. The
following supplementary materials are available for this edition:

Instructor's Manual containing teaching tips, calculator

programs, overhead transparency masters, sample worksheets,
and sample syllabi.
Computerized Test Bank, comprised of nearly 7,000
questions, mostly algorithmically-generated,which allows for
 multiple versions of a single test or quiz.
Instructor's Solution Manual with complete solutions to all
problems.
Student Solution Manual with complete solutions to half the
odd-numbered problems.
Graphing Calculator Manual, to help students get the most
out of their graphing calculators, and to show how they can apply
the numerical and graphing functions of their calculators to their
study of calculus.
Additional Material, elaborating specially marked points in
the text and password-protected electronic versions of the
instructor ancillaries, can be found on the web at
www.wiley.com/college/hughes-hallett.

ConcepTests
ConcepTests, modeled on the pioneering work of Harvard physicist
Eric Mazur, are questions designed to promote active learning
during class, particularly (but not exclusively) in large lectures. Our
evaluation data show students taught with ConcepTests
outperformed students taught by traditional lecture methods 73%
versus 17% on conceptual questions, and 63% versus 54% on
computational problems.

Advanced Placement (AP) Teacher's Guide

The AP Guide, written by a team of experienced AP teachers,
provides tips, multiple-choice questions, and free-response questions
that align to each chapter of the text. It also features a collection of
labs designed to complement the teaching of key AP Calculus
concepts.
New material has been added to reflect recent changes in the
learning objectives for AB and BC Calculus, including extended
coverage of limits, continuity, sequences, and series. Also new to
this edition are grids that align multiple choice and free-response
questions to the College Board's Enduring Understandings,
Learning Objectives, and Essential Knowledge.
Acknowledgements
First and foremost, we want to express our appreciation to the
National Science Foundation for their faith in our ability to produce
a revitalized calculus curriculum and, in particular, to our program
officers, Louise Raphael, John Kenelly, John Bradley, and James
Lightbourne. We also want to thank the members of our Advisory
Board, Benita Albert, Lida Barrett, Simon Bernau, Robert Davis, M.
Lavinia DeConge-Watson, John Dossey, Ron Douglas, Eli Fromm,
William Haver, Seymour Parter, John Prados, and Stephen Rodi.
In addition, a host of other people around the country and abroad
deserve our thanks for their contributions to shaping this edition.
They include: Huriye Arikan, Pau Atela, Ruth Baruth, Paul
Blanchard, Lewis Blake, David Bressoud, Stephen Boyd, Lucille
Buonocore, Matthew Michael Campbell, Jo Cannon, Ray Cannon,
Phil Cheifetz, Scott Clark, Jailing Dai, Ann Davidian, Tom Dick,
Srdjan Divac, Tevian Dray, Steven Dunbar, Penny Dunham, David
Durlach, John Eggers, Wade Ellis, Johann Engelbrecht, Brad Ernst,
Sunny Fawcett, Paul Feehan, Sol Friedberg, Melanie Fulton, Tom
Gearhart, David Glickenstein, Chris Goff, Sheldon P. Gordon, Salim
Haïdar, Elizabeth Hentges, Rob Indik, Adrian Iovita, David Jackson,
Sue Jensen, Alex Kasman, Matthias Kawski, Christopher Kennedy,
Mike Klucznik, Donna Krawczyk, Stephane Lafortune, Andrew
Lawrence, Carl Leinert, Daniel Look, Andrew Looms, Bin Lu, Alex
Mallozzi, Corinne Manogue, Jay Martin, Eric Mazur, Abby
McCallum, Dan McGee, Ansie Meiring, Lang Moore, Jerry Morris,
Hideo Nagahashi, Kartikeya Nagendra, Alan Newell, Steve Olson,
John Orr, Arnie Ostebee, Andrew Pasquale, Scott Pilzer, Wayne
Raskind, Maria Robinson, Laurie Rosatone, Ayse Sahin, Nataliya
Sandler, Ken Santor, Anne Scanlan-Rohrer, Ellen Schmierer,
Michael Sherman, Pat Shure, David Smith, Ernie Solheid, Misha
Stepanov, Steve Strogatz, Carl Swenson, Peter Taylor, Dinesh
Thakur, Sally Thomas, Joe Thrash, Alan Tucker, Doug Ulmer,
Ignatios Vakalis, Bill Vélez, Joe Vignolini, Stan Wagon, Hannah
Winkler, Debra Wood, Deane Yang, Bruce Yoshiwara, Kathy
Yoshiwara, and Paul Zorn.
Reports from the following reviewers were most helpful for the sixth
edition:
Barbara Armenta, James Baglama, Jon Clauss, Ann Darke, Marcel
Finan, Dana Fine, Michael Huber, Greg Marks, Wes Ostertag, Ben
Smith, Mark Turner, Aaron Weinberg, and Jianying Zhang.
Reports from the following reviewers were most helpful for the
seventh edition:
Scott Adamson, Janet Beery, Tim Biehler, Lewis Blake, Mark Booth,
Tambi Boyle, David Brown, Jeremy Case, Phil Clark, Patrice
Conrath, Pam Crawford, Roman J. Dial, Rebecca Dibbs, Marcel B.
Finan, Vauhn Foster-Grahler, Jill Guerra, Salim M. Haidar, Ryan A.
Hass, Firas Hindeleh, Todd King, Mary Koshar, Dick Lane, Glenn
Ledder, Oscar Levin, Tom Linton, Erich McAlister, Osvaldo Mendez,
Cindy Moss, Victor Padron, Michael Prophet, Ahmad Rajabzadeh,
Catherine A. Roberts, Kari Rothi, Edward J. Soares, Diana Staats,
Robert Talbert, James Vicich, Wendy Weber, Mina Yavari, and
Xinyun Zhu.
Finally, we extend our particular thanks to Jon Christensen for his
creativity with our three-dimensional figures.
Deborah Hughes- David O. Lomen Douglas Quinney
Hallett
Andrew M. Gleason David Lovelock Karen Rhea
William G. McCallum Guadalupe I. Ayşe şahin
Lozano
Eric Connally Jerry Morris Adam Spiegler
Daniel E. Flath David O. Mumford Jeff Tecosky-
Feldman
Selin Kalaycıoğlu Brad G. Osgood Thomas W. Tucker
Brigitte Lahme Cody L. Patterson Aaron D. Wootton
Patti Frazer Lock

To Students: How to Learn from this Book

This book may be different from other math textbooks that you
have used, so it may be helpful to know about some of the
differences in advance. This book emphasizes at every stage the
meaning (in practical, graphical or numerical terms) of the
symbols you are using. There is much less emphasis on “plug-
and-chug” and using formulas, and much more emphasis on the
interpretation of these formulas than you may expect. You will
often be asked to explain your ideas in words or to explain an
answer using graphs.
The book contains the main ideas of calculus in plain English.
Your success in using this book will depend on your reading,
questioning, and thinking hard about the ideas presented.
Although you may not have done this with other books, you
should plan on reading the text in detail, not just the worked
examples.
There are very few examples in the text that are exactly like the
homework problems. This means that you can't just look at a
homework problem and search for a similar–looking “worked
out” example. Success with the homework will come by grappling
with the ideas of calculus.
Many of the problems that we have included in the book are
open-ended. This means that there may be more than one
approach and more than one solution, depending on your
analysis. Many times, solving a problem relies on common sense
ideas that are not stated in the problem but which you will know
from everyday life.
Some problems in this book assume that you have access to a
graphing calculator or computer. There are many situations
where you may not be able to find an exact solution to a problem,
but you can use a calculator or computer to get a reasonable
approximation.
This book attempts to give equal weight to four methods for
describing functions: graphical (a picture), numerical (a table of
values) algebraic (a formula), and verbal. Sometimes you may
find it easier to translate a problem given in one form into
another. The best idea is to be flexible about your approach: if
one way of looking at a problem doesn't work, try another.
Students using this book have found discussing these problems in
small groups very helpful. There are a great many problems
which are not cut-and-dried; it can help to attack them with the
other perspectives your colleagues can provide. If group work is
not feasible, see if your instructor can organize a discussion
session in which additional problems can be worked on.
You are probably wondering what you'll get from the book. The
answer is, if you put in a solid effort, you will get a real
understanding of one of the most important accomplishments of
the millennium—calculus—as well as a real sense of the power of
mathematics in the age of technology.
CONTENTS
Cover Page
Title Page
Dedication
Copyright
Preface
Acknowledgments
Chapter 1: Foundation for Calculus: Functions and Limits
1.1 Functions and change
1.2 Exponential functions
1.3 New functions from old
1.4 Logarithmic functions
1.5 Trigonometric functions
1.6 Powers, Polynomials, and Rational functions
1.7 Introduction to limits and continuity
1.8 Extending the idea of a limit
1.9 Further limit calculations using Algebra
1.10 Optional preview of the formal definition of a limit
Chapter 2: Key Concept: The Derivative
2.1 How do we measure speed?
2.2 The Derivative at a point
2.3 The Derivative function
2.4 Interpretations of the derivative
2.5 The second derivative
2.6 Differentiability
Chapter 3: Short-Cuts to Differentiation
3.1 Powers and Polynomials
3.2 The Exponential Function
 3.3 The Product and Quotient Rules
3.4 The Chain Rule
3.5 The Trigonometric functions
3.6 The chain rule and inverse functions
3.7 Implicit functions
3.8 Hyperbolic functions
3.9 Linear approximation and the derivative
3.10 Theorems about differentiable functions
Chapter 4 Using the Derivative
4.1 Using first and second derivatives
4.2 Optimization
4.3 Optimization and Modeling
4.4 Families of functions and Modeling
4.5 Applications to marginality
4.6 Rates and related rates
4.7 L'Hopital's rule, growth, and dominance
4.8 Parametric Equations
Chapter 5: Key Concept: The Definite Integral
5.1 How do we measure distance traveled?
5.2 The definite integral
5.3 The fundamental theorem and interpretations
5.4 Theorems about definite integrals
Chapter 6: Constructing Antiderivatives
6.1 Antiderivatives graphically and numerically
6.2 Constructing antiderivatives analytically
6.3 Differential equations and motion
6.4 Second fundamental theorem of calculus
Chapter 7: Integration
7.1 Integration by substitution
7.2 Integration by parts
 7.3 Tables of integrals
7.4 Algebraic identities and trigonometric substitutions
7.5 Numerical methods for definite integrals
7.6 Improper integrals
7.7 Comparison of improper integrals
Chapter 8: Using the Definite Integral
8.1 Areas and volumes
8.2 Applications to geometry
8.3 Area and ARC length in polar coordinates
8.4 Density and center of mass
8.5 Applications to physics
8.6 Applications to economics
8.7 Distribution Functions
8.8 Probability, mean, and median
Chapter 9: Sequences and Series
9.1 Sequences
9.2 Geometric series
9.3 Convergence of series
9.4 Tests for convergence
9.5 Power series and interval of convergence
Chapter 10: Approximating Functions using Series
10.1 Taylor polynomials
10.2 Taylor series
10.3 Finding and using taylor series
10.4 The error in taylor polynomial approximations
10.5 Fourier Series
Chapter 11: Differential Equations
11.1 What is a differential equation?
11.2 Slope fields
11.3 Euler's method
 11.4 Separation of variables
11.5 Growth and decay
11.6 Applications and modeling
11.7 The Logistic model
11.8 Systems of differential equations
11.9 Analyzing the phase plane
11.10 Second-order differential equations: Oscillations
11.11 Linear second-order differential equations
Chapter 12: Functions of Several Variables
12.1 Functions of two variables
12.2 Graphs and surfaces
12.3 Contour diagrams
12.4 Linear functions
12.5 Functions of three variables
12.6 Limits and continuity
Chapter 13: A Fundamental Tool: Vectors
13.1 Displacement vectors
13.2 Vectors in general
13.3 The Dot product
13.4 The Cross product
Chapter 14: Differentiating Functions of Several Variables
14.1 The Partial derivative
14.2 Computing partial derivatives algebraically
14.3 Local linearity and the differential
14.4 Gradients and directional derivatives in the plane
14.5 Gradients and directional derivatives in space
14.6 The Chain Rule
14.7 Second-order partial derivatives
14.8 Differentiability
Chapter 15: Optimization: Local and Global Extrema
 15.1 Critical Points: Local extrema and saddle points
15.2 Optimization
15.3 Constrained optimization: Lagrange multipliers
Chapter 16: Integrating Functions of Several Variables
16.1 The Definite integral of a function of two variables
16.2 Iterated integrals
16.3 Triple integrals
16.4 Double integrals in polar coordinates
16.5 Integrals in cylindrical and spherical coordinates
16.6 Applications of integration to probability
Chapter 17: Parameterization and Vector Fields
17.1 Parameterized curves
17.2 Motion, velocity, and acceleration
17.3 Vector fields
17.4 The Flow of a vector field
Chapter 18: Line Integrals
18.1 The Idea of a line integral
18.2 Computing line integrals over parameterized curves
18.3 Gradient fields and path-independent fields
18.4 Path-dependent vector fields and green's theorem
Chapter 19: Flux Integrals and Divergence
19.1 The Idea of a flux integral
19.2 Flux integrals for graphs, cylinders, and spheres
19.3 The Divergence of a vector field
19.4 The Divergence theorem
Chapter 20: The Curl and Stokes' Theorem
20.1 The Curl of a vector field
20.2 Stokes' theorem
20.3 The Three fundamental theorems
Chapter 21: Parameters, Coordinates, and Integrals
 21.1 Coordinates and parameterized surfaces
21.2 Change of coordinates in a multiple integral
21.3 Flux integrals over parameterized surfaces
Appendices
A Roots, Accuracy, and Bounds
B Complex Numbers
C Newton's Method
D Vectors in the Plane
Ready Reference
Index
EULA
CHAPTER 1
FOUNDATION FOR CALCULUS: FUNCTIONS AND LIMITS
1.1 FUNCTIONS AND CHANGE
In mathematics, a function is used to represent the dependence of one quantity upon another.
Let's look at an example. In 2015, Boston, Massachusetts, had the highest annual snowfall, 110.6
inches, since recording started in 1872. Table 1.1 shows one 14-day period in which the city broke
another record with a total of 64.4 inches.1
Day 1 2 3 4 5 67 8 9 10 11 12 13 14
Snowfall 22.1 0.2 0 0.7 1.3 0 16.2 0 0 0.8 0 0.9 7.4 14.8

Table 1.1 Daily snowfall in inches for Boston, January 27 to February 9, 2015
You may not have thought of something so unpredictable as daily snowfall as being a function, but
it is a function of day, because each day gives rise to one snowfall total. There is no formula for the
daily snowfall (otherwise we would not need a weather bureau), but nevertheless the daily snowfall
in Boston does satisfy the definition of a function: Each day, t, has a unique snowfall, S, associated
with it.
We define a function as follows:

A function is a rule that takes certain numbers as inputs and assigns to each a definite output
number. The set of all input numbers is called the domain of the function and the set of
resulting output numbers is called the range of the function.

The input is called the independent variable and the output is called the dependent variable. In the
snowfall example, the domain is the set of days {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} and the
range is the set of daily snowfalls {0, 0.2, 0.7, 0.8, 0.9, 1.3, 7.4, 14.8, 16.2, 22.1}. We call the function
f and write S = f(t). Notice that a function may have identical outputs for different inputs (Days 8
and 9, for example).
Some quantities, such as a day or date, are discrete, meaning they take only certain isolated values
(days must be integers). Other quantities, such as time, are continuous as they can be any number.
For a continuous variable, domains and ranges are often written using interval notation:

The set of numbers t such that a ≤ t ≤ b is called a closed interval and written [a, b].

The set of numbers t such that a < t < b is called an open interval and written (a, b).

The Rule of Four: Tables, Graphs, Formulas, and Words

Functions can be represented by tables, graphs, formulas, and descriptions in words. For example,
the function giving the daily snowfall in Boston can be represented by the graph in Figure 1.1, as
well as by Table 1.1.
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