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![2 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
Table 1.1 Daily snowfall in inches for Boston, January 27 to February 9, 2015
Day 1 2 3 4 5 6 7 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
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.
2 4 6 8 10 12 14
0
5
10
15
20
25
day
snowfall (inches)
Figure 1.1: Boston snowfall, starting January 27, 2015
As another example of a function, consider the snowy tree cricket. Surprisingly enough, all such
crickets chirp at essentially the same rate if they are at the same temperature. That means that the
chirp rate is a function of temperature. In other words, if we know the temperature, we can determine
1http://w2.weather.gov/climate/xmacis.php?wfo=box. Accessed June 2015.](https://image.slidesharecdn.com/64463-250501203018-223f0589/75/Calculus-Single-and-Multivariable-7th-Edition-Deborah-Hughes-Hallett-29-2048.jpg)
![1.1 FUNCTIONS AND CHANGE 3
100 140
40
100
200
300
400
𝑇 (◦
F)
𝐶 (chirps per minute)
𝐶 = 4𝑇 − 160
Figure 1.2: Cricket chirp rate versus temperature
the chirp rate. Even more surprisingly, the chirp rate, 𝐶, in chirps per minute, increases steadily with
the temperature, 𝑇 , in degrees Fahrenheit, and can be computed by the formula
𝐶 = 4𝑇 − 160
to a fair level of accuracy. We write 𝐶 = 𝑓(𝑇 ) to express the fact that we think of 𝐶 as a function of
𝑇 and that we have named this function 𝑓. The graph of this function is in Figure 1.2.
Notice that the graph of 𝐶 = 𝑓(𝑇 ) in Figure 1.2 is a solid line. This is because 𝐶 = 𝑓(𝑇 ) is
a continuous function. Roughly speaking, a continuous function is one whose graph has no breaks,
jumps, or holes. This means that the independent variable must be continuous. (We give a more
precise definition of continuity of a function in Section 1.7.)
Examples of Domain and Range
If the domain of a function is not specified, we usually take it to be the largest possible set of real
numbers. For example, we usually think of the domain of the function 𝑓(𝑥) = 𝑥2 as all real numbers.
However, the domain of the function 𝑔(𝑥) = 1∕𝑥 is all real numbers except zero, since we cannot
divide by zero.
Sometimes we restrict the domain to be smaller than the largest possible set of real numbers.
For example, if the function 𝑓(𝑥) = 𝑥2 is used to represent the area of a square of side 𝑥, we restrict
the domain to nonnegative values of 𝑥.
Example1 The function 𝐶 = 𝑓(𝑇 ) gives chirp rate as a function of temperature. We restrict this function to
temperatures for which the predicted chirp rate is positive, and up to the highest temperature ever
recorded at a weather station, 134◦F. What is the domain of this function 𝑓?
Solution If we consider the equation
𝐶 = 4𝑇 − 160
simply as a mathematical relationship between two variables 𝐶 and 𝑇 , any 𝑇 value is possible.
However, if we think of it as a relationship between cricket chirps and temperature, then 𝐶 cannot
be less than 0. Since 𝐶 = 0 leads to 0 = 4𝑇 − 160, and so 𝑇 = 40◦F, we see that 𝑇 cannot be less
than 40◦F. (See Figure 1.2.) In addition, we are told that the function is not defined for temperatures
above 134◦. Thus, for the function 𝐶 = 𝑓(𝑇 ) we have
Domain = All 𝑇 values between 40◦F and 134◦F
= All 𝑇 values with 40 ≤ 𝑇 ≤ 134
= [40, 134].
Example2 Find the range of the function 𝑓, given the domain from Example 1. In other words, find all possible
values of the chirp rate, 𝐶, in the equation 𝐶 = 𝑓(𝑇 ).
Solution Again, if we consider 𝐶 = 4𝑇 − 160 simply as a mathematical relationship, its range is all real 𝐶
values. However, when thinking of the meaning of 𝐶 = 𝑓(𝑇 ) for crickets, we see that the function
predicts cricket chirps per minute between 0 (at 𝑇 = 40◦F) and 376 (at 𝑇 = 134◦F). Hence,
Range = All 𝐶 values from 0 to 376
= All 𝐶 values with 0 ≤ 𝐶 ≤ 376
= [0, 376].](https://image.slidesharecdn.com/64463-250501203018-223f0589/75/Calculus-Single-and-Multivariable-7th-Edition-Deborah-Hughes-Hallett-30-2048.jpg)
Calculus Single and Multivariable 7th Edition Deborah Hughes-Hallett
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- 6. Lines Slope of linethrough (𝑥1, 𝑦1) and (𝑥2, 𝑦2): 𝑚 = 𝑦2 − 𝑦1 𝑥2 − 𝑥1 Point-slope equation of line through (𝑥1, 𝑦1) with slope 𝑚: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) Slope-intercept equation of line with slope 𝑚 and 𝑦-intercept 𝑏: 𝑦 = 𝑏 + 𝑚𝑥 Rules of Exponents 𝑎𝑥 𝑎𝑡 = 𝑎𝑥+𝑡 𝑎𝑥 𝑎𝑡 = 𝑎𝑥−𝑡 (𝑎𝑥 )𝑡 = 𝑎𝑥𝑡 Definition of Natural Log 𝑦 = ln 𝑥 means 𝑒𝑦 = 𝑥 ex: ln 1 = 0 since 𝑒0 = 1 1 1 𝑥 𝑦 𝑦 = ln 𝑥 𝑦 = 𝑒𝑥 Identities ln 𝑒𝑥 = 𝑥 𝑒ln 𝑥 = 𝑥 Rules of Natural Logarithms ln(𝐴𝐵) = ln 𝐴 + ln 𝐵 ln ( 𝐴 𝐵 ) = ln 𝐴 − ln 𝐵 ln 𝐴𝑝 = 𝑝 ln 𝐴 Distance and Midpoint Formulas Distance 𝐷 between (𝑥1, 𝑦1) and (𝑥2, 𝑦2): 𝐷 = √ (𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 Midpoint of (𝑥1, 𝑦1) and (𝑥2, 𝑦2): ( 𝑥1 + 𝑥2 2 , 𝑦1 + 𝑦2 2 ) Quadratic Formula If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, then 𝑥 = −𝑏 ± √ 𝑏2 − 4𝑎𝑐 2𝑎 Factoring Special Polynomials 𝑥2 − 𝑦2 = (𝑥 + 𝑦)(𝑥 − 𝑦) 𝑥3 + 𝑦3 = (𝑥 + 𝑦)(𝑥2 − 𝑥𝑦 + 𝑦2 ) 𝑥3 − 𝑦3 = (𝑥 − 𝑦)(𝑥2 + 𝑥𝑦 + 𝑦2 ) Circles Center (ℎ, 𝑘) and radius 𝑟: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2 Ellipse 𝑥2 𝑎2 + 𝑦2 𝑏2 = 1 −𝑎 𝑎 −𝑏 𝑏 𝑥 𝑦 Hyperbola 𝑥2 𝑎2 − 𝑦2 𝑏2 = 1 𝑎 𝑥 𝑦 𝑦 = 𝑏𝑥∕𝑎 𝑦 = −𝑏𝑥∕𝑎
- 7. Geometric Formulas Conversion BetweenRadians and Degrees: 𝜋 radians = 180◦ Triangle 𝐴 = 1 2 𝑏ℎ = 1 2 𝑎𝑏 sin 𝜃 𝜃 ✲ ✛ 𝑏 𝑎 ℎ Circle 𝐴 = 𝜋𝑟2 𝐶 = 2𝜋𝑟 𝑟 Sector of Circle 𝐴 = 1 2 𝑟2𝜃 (𝜃 in radians) 𝑠 = 𝑟𝜃 (𝜃 in radians) 𝜃 𝑟 𝑟 𝑠 Sphere 𝑉 = 4 3 𝜋𝑟3 𝐴 = 4𝜋𝑟2 Cylinder 𝑉 = 𝜋𝑟2ℎ Cone 𝑉 = 1 3 𝜋𝑟2ℎ Trigonometric Functions sin 𝜃 = 𝑦 𝑟 cos 𝜃 = 𝑥 𝑟 tan 𝜃 = 𝑦 𝑥 tan 𝜃 = sin 𝜃 cos 𝜃 cos2 𝜃 + sin2 𝜃 = 1 ✛ ✛ 𝑟 (𝑥, 𝑦) 𝜃 ✻ ❄ 𝑦 ✲ ✛ 𝑥 sin(𝐴±𝐵) = sin 𝐴 cos 𝐵±cos 𝐴 sin 𝐵 cos(𝐴±𝐵) = cos 𝐴 cos 𝐵∓sin 𝐴 sin 𝐵 sin(2𝐴) = 2 sin 𝐴 cos 𝐴 cos(2𝐴) = 2 cos2 𝐴−1 = 1−2 sin2 𝐴 𝜋 2𝜋 −1 1 𝑦 = sin 𝑥 𝑥 𝑦 𝜋 2𝜋 −1 1 𝑦 = cos 𝑥 𝑥 𝑦 −𝜋 𝜋 𝑦 = tan 𝑥 𝑥 𝑦 The Binomial Theorem (𝑥 + 𝑦)𝑛 = 𝑥𝑛 + 𝑛𝑥𝑛−1 𝑦 + 𝑛(𝑛 − 1) 1 ⋅ 2 𝑥𝑛−2 𝑦2 + 𝑛(𝑛 − 1)(𝑛 − 2) 1 ⋅ 2 ⋅ 3 𝑥𝑛−3 𝑦3 + ⋯ + 𝑛𝑥𝑦𝑛−1 + 𝑦𝑛 (𝑥 − 𝑦)𝑛 = 𝑥𝑛 − 𝑛𝑥𝑛−1 𝑦 + 𝑛(𝑛 − 1) 1 ⋅ 2 𝑥𝑛−2 𝑦2 − 𝑛(𝑛 − 1)(𝑛 − 2) 1 ⋅ 2 ⋅ 3 𝑥𝑛−3 𝑦3 + ⋯ ± 𝑛𝑥𝑦𝑛−1 ∓ 𝑦𝑛
- 8.
- 9. We dedicate thisbook 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
- 10. CALCULUS Seventh Edition Produced bythe Calculus Consortium and initially funded by a National Science Foundation Grant. Deborah Hughes-Hallett William G. McCallum Andrew M. Gleason University of Arizona University of Arizona Harvard University Eric Connally David Lovelock Douglas Quinney Harvard University Extension University of Arizona University of Keele 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 University Brown University Loyola University Chicago Patti Frazer Lock Brad G. Osgood Jeff Tecosky-Feldman St. Lawrence University Stanford University Haverford College David O. Lomen Cody L. Patterson Thomas W. Tucker University of Arizona University of Texas at San Antonio Colgate University Aaron D. Wootton University of Portland with the assistance of Otto K. Bretscher Adrian Iovita David E. Sloane, MD Colby College University of Washington Harvard Medical School Coordinated by Elliot J. Marks
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- 12. PREFACE Calculus is oneof 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 re- sponse 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 v
- 13. vi Preface in theUS 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 in- teractive 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 sec- tions • 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 prob- lem 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 prefer- ences 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.
- 14. Preface vii • Limitsand 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 indi- vidual 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 prob- ably 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 deriva- tive 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 differentiatingproducts; 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 opti- mization, graphing, rates, parametric equations, and indeterminate forms. It is not necessary to cover all the sections in this chapter.
- 15. viii Preface 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 nu- merically, 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, com- parison, 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 approximatingfunctions 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 convergenceis introducedwithout 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 inter- pretation. 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.
- 16. Preface ix Chapter 14:Differentiating Functions of Several Variables Partial derivatives, directional derivatives, gradients, and local linearity are introduced. The chapter also dis- cusses 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 coor- dinate 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 cylin- ders, 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,overheadtransparencymasters, sam- ple worksheets, and sample syllabi. • ComputerizedTest Bank, comprisedof nearly 7,000questions,mostly algorithmically-generated,which allows for multiple versions of a single test or quiz.
- 17. x Preface • Instructor’sSolution 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 contribu- tions 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, Shel- don 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 Na- gahashi, 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
- 18. Preface xi Foster-Grahler, JillGuerra, 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-Hallett David O. Lomen Douglas Quinney Andrew M. Gleason David Lovelock Karen Rhea William G. McCallum Guadalupe I. Lozano Ayşe Şahin 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.
- 19. xii Preface CONTENTS 1 FOUNDATIONFOR CALCULUS: FUNCTIONS AND LIMITS 1 1.1 FUNCTIONS AND CHANGE 2 1.2 EXPONENTIAL FUNCTIONS 13 1.3 NEW FUNCTIONS FROM OLD 23 1.4 LOGARITHMIC FUNCTIONS 32 1.5 TRIGONOMETRIC FUNCTIONS 39 1.6 POWERS, POLYNOMIALS, AND RATIONAL FUNCTIONS 49 1.7 INTRODUCTION TO LIMITS AND CONTINUITY 58 1.8 EXTENDING THE IDEA OF A LIMIT 67 1.9 FURTHER LIMIT CALCULATIONS USING ALGEBRA 75 1.10 OPTIONAL PREVIEW OF THE FORMAL DEFINITION OF A LIMIT ONLINE REVIEW PROBLEMS ONLINE PROJECTS ONLINE 2 KEY CONCEPT: THE DERIVATIVE 83 2.1 HOW DO WE MEASURE SPEED? 84 2.2 THE DERIVATIVE AT A POINT 91 2.3 THE DERIVATIVE FUNCTION 99 2.4 INTERPRETATIONS OF THE DERIVATIVE 108 2.5 THE SECOND DERIVATIVE 115 2.6 DIFFERENTIABILITY 123 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 3 SHORT-CUTS TO DIFFERENTIATION 129 3.1 POWERS AND POLYNOMIALS 130 3.2 THE EXPONENTIAL FUNCTION 140 3.3 THE PRODUCT AND QUOTIENT RULES 144 For online material, see www.wiley.com/college/hughes-hallett.
- 20. Preface xiii 3.4 THECHAIN RULE 151 3.5 THE TRIGONOMETRIC FUNCTIONS 158 3.6 THE CHAIN RULE AND INVERSE FUNCTIONS 164 3.7 IMPLICIT FUNCTIONS 171 3.8 HYPERBOLIC FUNCTIONS 174 3.9 LINEAR APPROXIMATION AND THE DERIVATIVE 178 3.10 THEOREMS ABOUT DIFFERENTIABLE FUNCTIONS 186 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 4 USING THE DERIVATIVE 191 4.1 USING FIRST AND SECOND DERIVATIVES 192 4.2 OPTIMIZATION 203 4.3 OPTIMIZATION AND MODELING 212 4.4 FAMILIES OF FUNCTIONS AND MODELING 224 4.5 APPLICATIONS TO MARGINALITY 233 4.6 RATES AND RELATED RATES 243 4.7 L’HOPITAL’S RULE, GROWTH, AND DOMINANCE 252 4.8 PARAMETRIC EQUATIONS 259 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 5 KEY CONCEPT: THE DEFINITE INTEGRAL 271 5.1 HOW DO WE MEASURE DISTANCE TRAVELED? 272 5.2 THE DEFINITE INTEGRAL 283 5.3 THE FUNDAMENTAL THEOREM AND INTERPRETATIONS 292 5.4 THEOREMS ABOUT DEFINITE INTEGRALS 302 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 6 CONSTRUCTING ANTIDERIVATIVES 315 6.1 ANTIDERIVATIVES GRAPHICALLY AND NUMERICALLY 316 6.2 CONSTRUCTING ANTIDERIVATIVES ANALYTICALLY 322
- 21. xiv Preface 6.3 DIFFERENTIALEQUATIONS AND MOTION 329 6.4 SECOND FUNDAMENTAL THEOREM OF CALCULUS 335 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 7 INTEGRATION 341 7.1 INTEGRATION BY SUBSTITUTION 342 7.2 INTEGRATION BY PARTS 353 7.3 TABLES OF INTEGRALS 360 7.4 ALGEBRAIC IDENTITIES AND TRIGONOMETRIC SUBSTITUTIONS 366 7.5 NUMERICAL METHODS FOR DEFINITE INTEGRALS 376 7.6 IMPROPER INTEGRALS 385 7.7 COMPARISON OF IMPROPER INTEGRALS 394 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 8 USING THE DEFINITE INTEGRAL 401 8.1 AREAS AND VOLUMES 402 8.2 APPLICATIONS TO GEOMETRY 410 8.3 AREA AND ARC LENGTH IN POLAR COORDINATES 420 8.4 DENSITY AND CENTER OF MASS 429 8.5 APPLICATIONS TO PHYSICS 439 8.6 APPLICATIONS TO ECONOMICS 450 8.7 DISTRIBUTION FUNCTIONS 457 8.8 PROBABILITY, MEAN, AND MEDIAN 464 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 9 SEQUENCES AND SERIES 473 9.1 SEQUENCES 474 9.2 GEOMETRIC SERIES 480 9.3 CONVERGENCE OF SERIES 488 9.4 TESTS FOR CONVERGENCE 494 9.5 POWER SERIES AND INTERVAL OF CONVERGENCE 504 REVIEW PROBLEMS ONLINE PROJECTS ONLINE
- 22. Preface xv 10 APPROXIMATINGFUNCTIONS USING SERIES 513 10.1 TAYLOR POLYNOMIALS 514 10.2 TAYLOR SERIES 523 10.3 FINDING AND USING TAYLOR SERIES 530 10.4 THE ERROR IN TAYLOR POLYNOMIAL APPROXIMATIONS 539 10.5 FOURIER SERIES 546 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 11 DIFFERENTIAL EQUATIONS 561 11.1 WHAT IS A DIFFERENTIAL EQUATION? 562 11.2 SLOPE FIELDS 567 11.3 EULER’S METHOD 575 11.4 SEPARATION OF VARIABLES 580 11.5 GROWTH AND DECAY 586 11.6 APPLICATIONS AND MODELING 597 11.7 THE LOGISTIC MODEL 606 11.8 SYSTEMS OF DIFFERENTIAL EQUATIONS 616 11.9 ANALYZING THE PHASE PLANE 626 11.10 SECOND-ORDER DIFFERENTIAL EQUATIONS: OSCILLATIONS 632 11.11 LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS 640 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 12 FUNCTIONS OF SEVERAL VARIABLES 651 12.1 FUNCTIONS OF TWO VARIABLES 652 12.2 GRAPHS AND SURFACES 660 12.3 CONTOUR DIAGRAMS 668 12.4 LINEAR FUNCTIONS 682 12.5 FUNCTIONS OF THREE VARIABLES 689 12.6 LIMITS AND CONTINUITY 695 REVIEW PROBLEMS ONLINE PROJECTS ONLINE
- 23. xvi Preface 13 AFUNDAMENTAL TOOL: VECTORS 701 13.1 DISPLACEMENT VECTORS 702 13.2 VECTORS IN GENERAL 710 13.3 THE DOT PRODUCT 718 13.4 THE CROSS PRODUCT 728 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 14 DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES 739 14.1 THE PARTIAL DERIVATIVE 740 14.2 COMPUTING PARTIAL DERIVATIVES ALGEBRAICALLY 748 14.3 LOCAL LINEARITY AND THE DIFFERENTIAL 753 14.4 GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE PLANE 762 14.5 GRADIENTS AND DIRECTIONAL DERIVATIVES IN SPACE 772 14.6 THE CHAIN RULE 780 14.7 SECOND-ORDER PARTIAL DERIVATIVES 790 14.8 DIFFERENTIABILITY 799 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 15 OPTIMIZATION: LOCAL AND GLOBAL EXTREMA 805 15.1 CRITICAL POINTS: LOCAL EXTREMA AND SADDLE POINTS 806 15.2 OPTIMIZATION 815 15.3 CONSTRAINED OPTIMIZATION: LAGRANGE MULTIPLIERS 825 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 16 INTEGRATING FUNCTIONS OF SEVERAL VARIABLES 839 16.1 THE DEFINITE INTEGRAL OF A FUNCTION OF TWO VARIABLES 840 16.2 ITERATED INTEGRALS 847 16.3 TRIPLE INTEGRALS 857 16.4 DOUBLE INTEGRALS IN POLAR COORDINATES 864
- 24. Preface xvii 16.5 INTEGRALSIN CYLINDRICAL AND SPHERICAL COORDINATES 869 16.6 APPLICATIONS OF INTEGRATION TO PROBABILITY 878 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 17 PARAMETERIZATION AND VECTOR FIELDS 885 17.1 PARAMETERIZED CURVES 886 17.2 MOTION, VELOCITY, AND ACCELERATION 896 17.3 VECTOR FIELDS 905 17.4 THE FLOW OF A VECTOR FIELD 913 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 18 LINE INTEGRALS 921 18.1 THE IDEA OF A LINE INTEGRAL 922 18.2 COMPUTING LINE INTEGRALS OVER PARAMETERIZED CURVES 931 18.3 GRADIENT FIELDS AND PATH-INDEPENDENT FIELDS 939 18.4 PATH-DEPENDENT VECTOR FIELDS AND GREEN’S THEOREM 949 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 19 FLUX INTEGRALS AND DIVERGENCE 961 19.1 THE IDEA OF A FLUX INTEGRAL 962 19.2 FLUX INTEGRALS FOR GRAPHS, CYLINDERS, AND SPHERES 973 19.3 THE DIVERGENCE OF A VECTOR FIELD 982 19.4 THE DIVERGENCE THEOREM 991 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 20 THE CURL AND STOKES’ THEOREM 999 20.1 THE CURL OF A VECTOR FIELD 1000
- 25. xviii Preface 20.2 STOKES’THEOREM 1008 20.3 THE THREE FUNDAMENTAL THEOREMS 1015 REVIEW PROBLEMS ONLINE PROJECTS ONLINE 21 PARAMETERS, COORDINATES, AND INTEGRALS 1021 21.1 COORDINATES AND PARAMETERIZED SURFACES 1022 21.2 CHANGE OF COORDINATES IN A MULTIPLE INTEGRAL 1033 21.3 FLUX INTEGRALS OVER PARAMETERIZED SURFACES 1038 REVIEW PROBLEMS ONLINE PROJECTS ONLINE APPENDICES Online A ROOTS, ACCURACY, AND BOUNDS ONLINE B COMPLEX NUMBERS ONLINE C NEWTON’S METHOD ONLINE D VECTORS IN THE PLANE ONLINE E DETERMINANTS ONLINE READY REFERENCE 1043 ANSWERS TO ODD-NUMBERED PROBLEMS 1061 INDEX 1131
- 26. WileyPLUS with ORIONdelivers easy-to-use analytics that help educators and students see strengths and weaknesses to give learners the best chance of succeeding in the course. www.ORION.wileyplus.com Identify which students are struggling early in the semester. Educators assess the real-time engagement and performance of each student to inform teaching decisions. Students always know what they need to work on. Help students organize their learning and get the practice they need. With ORION’s adaptive practice, students quickly understand what they know and don’t know. They can then decide to study or practice based on their proficiency. Measure outcomes to promote continuous improvement. With visual reports, it’s easy for both students and educators to gauge problem areas and act on what’s most important. A personalized, adaptive learning experience.
- 28. Contents 1.1 Functions andChange. . . . . . . . . . . . . . . . . . . . . . . . 2 The Rule of Four . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Examples of Domain and Range . . . . . . . . . . . . . . . 3 Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Families of Linear Functions . . . . . . . . . . . . . . . . . . 5 Increasing versus Decreasing Functions . . . . . . . . . 6 Proportionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Exponential Functions. . . . . . . . . . . . . . . . . . . . . . . 13 Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Elimination of a Drug from the Body . . . . . . . . . . 15 The General Exponential Function . . . . . . . . . . . . 15 Half-Life and Doubling Time . . . . . . . . . . . . . . . . 16 The Family of Exponential Functions . . . . . . . . . . 16 Exponential Functions with Base e . . . . . . . . . . . . 17 1.3 New Functions from Old. . . . . . . . . . . . . . . . . . . . . 23 Shifts and Stretches . . . . . . . . . . . . . . . . . . . . . . . . 23 Composite Functions . . . . . . . . . . . . . . . . . . . . . . . 24 Odd and Even Functions: Symmetry . . . . . . . . . . . 25 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.4 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 32 Logarithms to Base 10 and to Base e. . . . . . . . . . . 32 Solving Equations Using Logarithms . . . . . . . . . . 33 1.5 Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . 39 Radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 The Sine and Cosine Functions . . . . . . . . . . . . . . . 40 The Tangent Function . . . . . . . . . . . . . . . . . . . . . . 43 The Inverse Trigonometric Functions . . . . . . . . . . 44 1.6 Powers, Polynomials, and Rational Functions . . . 49 Power Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Dominance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . 53 1.7 Introduction to Limits and Continuity . . . . . . . . . 58 The Idea of Continuity. . . . . . . . . . . . . . . . . . . . . . 58 The Idea of a Limit . . . . . . . . . . . . . . . . . . . . . . . . 59 Definition of Limit. . . . . . . . . . . . . . . . . . . . . . . . . 60 Definition of Continuity. . . . . . . . . . . . . . . . . . . . . 60 The Intermediate Value Theorem. . . . . . . . . . . . . . . 60 Finding Limits Exactly Using Continuity and Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 1.8 Extending the Idea of a Limit. . . . . . . . . . . . . . . . . 67 One-Sided Limits. . . . . . . . . . . . . . . . . . . . . . . . . . 67 Limits and Asymptotes . . . . . . . . . . . . . . . . . . . . . 68 1.9 Further Limit Calculations using Algebra . . . . . . 75 Limits of Quotients . . . . . . . . . . . . . . . . . . . . . . . . 75 Calculating Limits at Infinity. . . . . . . . . . . . . . . . . 78 The Squeeze Theorem . . . . . . . . . . . . . . . . . . . . . . 79 Chapter One FOUNDATION FOR CALCULUS: FUNCTIONS AND LIMITS
- 29. 2 Chapter 1FOUNDATION 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 Table 1.1 Daily snowfall in inches for Boston, January 27 to February 9, 2015 Day 1 2 3 4 5 6 7 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 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. 2 4 6 8 10 12 14 0 5 10 15 20 25 day snowfall (inches) Figure 1.1: Boston snowfall, starting January 27, 2015 As another example of a function, consider the snowy tree cricket. Surprisingly enough, all such crickets chirp at essentially the same rate if they are at the same temperature. That means that the chirp rate is a function of temperature. In other words, if we know the temperature, we can determine 1http://w2.weather.gov/climate/xmacis.php?wfo=box. Accessed June 2015.
- 30. 1.1 FUNCTIONS ANDCHANGE 3 100 140 40 100 200 300 400 𝑇 (◦ F) 𝐶 (chirps per minute) 𝐶 = 4𝑇 − 160 Figure 1.2: Cricket chirp rate versus temperature the chirp rate. Even more surprisingly, the chirp rate, 𝐶, in chirps per minute, increases steadily with the temperature, 𝑇 , in degrees Fahrenheit, and can be computed by the formula 𝐶 = 4𝑇 − 160 to a fair level of accuracy. We write 𝐶 = 𝑓(𝑇 ) to express the fact that we think of 𝐶 as a function of 𝑇 and that we have named this function 𝑓. The graph of this function is in Figure 1.2. Notice that the graph of 𝐶 = 𝑓(𝑇 ) in Figure 1.2 is a solid line. This is because 𝐶 = 𝑓(𝑇 ) is a continuous function. Roughly speaking, a continuous function is one whose graph has no breaks, jumps, or holes. This means that the independent variable must be continuous. (We give a more precise definition of continuity of a function in Section 1.7.) Examples of Domain and Range If the domain of a function is not specified, we usually take it to be the largest possible set of real numbers. For example, we usually think of the domain of the function 𝑓(𝑥) = 𝑥2 as all real numbers. However, the domain of the function 𝑔(𝑥) = 1∕𝑥 is all real numbers except zero, since we cannot divide by zero. Sometimes we restrict the domain to be smaller than the largest possible set of real numbers. For example, if the function 𝑓(𝑥) = 𝑥2 is used to represent the area of a square of side 𝑥, we restrict the domain to nonnegative values of 𝑥. Example1 The function 𝐶 = 𝑓(𝑇 ) gives chirp rate as a function of temperature. We restrict this function to temperatures for which the predicted chirp rate is positive, and up to the highest temperature ever recorded at a weather station, 134◦F. What is the domain of this function 𝑓? Solution If we consider the equation 𝐶 = 4𝑇 − 160 simply as a mathematical relationship between two variables 𝐶 and 𝑇 , any 𝑇 value is possible. However, if we think of it as a relationship between cricket chirps and temperature, then 𝐶 cannot be less than 0. Since 𝐶 = 0 leads to 0 = 4𝑇 − 160, and so 𝑇 = 40◦F, we see that 𝑇 cannot be less than 40◦F. (See Figure 1.2.) In addition, we are told that the function is not defined for temperatures above 134◦. Thus, for the function 𝐶 = 𝑓(𝑇 ) we have Domain = All 𝑇 values between 40◦F and 134◦F = All 𝑇 values with 40 ≤ 𝑇 ≤ 134 = [40, 134]. Example2 Find the range of the function 𝑓, given the domain from Example 1. In other words, find all possible values of the chirp rate, 𝐶, in the equation 𝐶 = 𝑓(𝑇 ). Solution Again, if we consider 𝐶 = 4𝑇 − 160 simply as a mathematical relationship, its range is all real 𝐶 values. However, when thinking of the meaning of 𝐶 = 𝑓(𝑇 ) for crickets, we see that the function predicts cricket chirps per minute between 0 (at 𝑇 = 40◦F) and 376 (at 𝑇 = 134◦F). Hence, Range = All 𝐶 values from 0 to 376 = All 𝐶 values with 0 ≤ 𝐶 ≤ 376 = [0, 376].
- 31. 4 Chapter 1FOUNDATION FOR CALCULUS: FUNCTIONS AND LIMITS In using the temperature to predict the chirp rate, we thought of the temperature as the indepen- dent variable and the chirp rate as the dependent variable. However, we could do this backward, and calculate the temperature from the chirp rate. From this point of view, the temperature is dependent on the chirp rate. Thus, which variable is dependent and which is independent may depend on your viewpoint. Linear Functions The chirp-rate function, 𝐶 = 𝑓(𝑇 ), is an example of a linear function. A function is linear if its slope, or rate of change, is the same at every point. The rate of change of a function that is not linear may vary from point to point. Olympic and World Records During the early years of the Olympics, the height of the men’s winning pole vault increased approx- imately 8 inches every four years. Table 1.2 shows that the height started at 130 inches in 1900, and increased by the equivalent of 2 inches a year. So the height was a linear function of time from 1900 to 1912. If 𝑦 is the winning height in inches and 𝑡 is the number of years since 1900, we can write 𝑦 = 𝑓(𝑡) = 130 + 2𝑡. Since 𝑦 = 𝑓(𝑡) increases with 𝑡, we say that 𝑓 is an increasing function. The coefficient 2 tells us the rate, in inches per year, at which the height increases. Table 1.2 Men’s Olympic pole vault winning height (approximate) Year 1900 1904 1908 1912 Height (inches) 130 138 146 154 This rate of increase is the slope of the line in Figure 1.3. The slope is given by the ratio Slope = Rise Run = 146 − 138 8 − 4 = 8 4 = 2 inches/year. Calculating the slope (rise/run) using any other two points on the line gives the same value. What about the constant 130? This represents the initial height in 1900, when 𝑡 = 0. Geometri- cally, 130 is the intercept on the vertical axis. 4 8 12 130 140 150 𝑦 (height in inches) 𝑡 (years since 1900) ✲ ✛ Run = 4 ✻ ❄ Rise = 8 𝑦 = 130 + 2𝑡 Figure 1.3: Olympic pole vault records You may wonder whether the linear trend continues beyond 1912. Not surprisingly, it does not exactly. The formula 𝑦 = 130+2𝑡 predicts that the height in the 2012 Olympics would be 354 inches or 29 feet 6 inches, which is considerably higher than the actual value of 19 feet 7.05 inches. There is clearly a danger in extrapolating too far from the given data. You should also observe that the data in Table 1.2 is discrete, because it is given only at specific points (every four years). However, we have treated the variable 𝑡 as though it were continuous, because the function 𝑦 = 130 + 2𝑡 makes
- 32. 1.1 FUNCTIONS ANDCHANGE 5 sense for all values of 𝑡. The graph in Figure 1.3 is of the continuous function because it is a solid line, rather than four separate points representing the years in which the Olympics were held. As the pole vault heights have increased over the years, the time to run the mile has decreased. If 𝑦 is the world record time to run the mile, in seconds, and 𝑡 is the number of years since 1900, then records show that, approximately, 𝑦 = 𝑔(𝑡) = 260 − 0.39𝑡. The 260 tells us that the world record was 260 seconds in 1900 (at 𝑡 = 0). The slope, −0.39, tells us that the world record decreased by about 0.39 seconds per year. We say that 𝑔 is a decreasing function. Difference Quotients and Delta Notation We use the symbol Δ (the Greek letter capital delta) to mean “change in,” so Δ𝑥 means change in 𝑥 and Δ𝑦 means change in 𝑦. The slope of a linear function 𝑦 = 𝑓(𝑥) can be calculated from values of the function at two points, given by 𝑥1 and 𝑥2, using the formula 𝑚 = Rise Run = Δ𝑦 Δ𝑥 = 𝑓(𝑥2) − 𝑓(𝑥1) 𝑥2 − 𝑥1 . The quantity (𝑓(𝑥2) − 𝑓(𝑥1))∕(𝑥2 − 𝑥1) is called a difference quotient because it is the quotient of two differences. (See Figure 1.4.) Since 𝑚 = Δ𝑦∕Δ𝑥, the units of 𝑚 are 𝑦-units over 𝑥-units. 𝑥1 𝑥2 𝑦 = 𝑓(𝑥) ✲ ✛ Run = 𝑥2 − 𝑥1 ✻ ❄ Rise = 𝑓(𝑥2) − 𝑓(𝑥1) 𝑥 𝑦 (𝑥2, 𝑓(𝑥2)) (𝑥1, 𝑓(𝑥1)) Figure 1.4: Difference quotient = 𝑓(𝑥2) − 𝑓(𝑥1) 𝑥2 − 𝑥1 Families of Linear Functions A linear function has the form 𝑦 = 𝑓(𝑥) = 𝑏 + 𝑚𝑥. Its graph is a line such that • 𝑚 is the slope, or rate of change of 𝑦 with respect to 𝑥. • 𝑏 is the vertical intercept, or value of 𝑦 when 𝑥 is zero. Notice that if the slope, 𝑚, is zero, we have 𝑦 = 𝑏, a horizontal line. To recognize that a table of 𝑥 and 𝑦 values comes from a linear function, 𝑦 = 𝑏 + 𝑚𝑥, look for differences in 𝑦-values that are constant for equally spaced 𝑥-values. Formulas such as 𝑓(𝑥) = 𝑏 + 𝑚𝑥, in which the constants 𝑚 and 𝑏 can take on various values, give a family of functions. All the functions in a family share certain properties—in this case, all the
- 33. Discovering Diverse ContentThrough Random Scribd Documents
- 34. And then wegot out the car and went to town. I drove, at her request, and between bumps and mud holes watched her out of one corner of my eye for any signs of a breakdown. But none came, either then or later in the long sheds where the sweated fruit roared down the channel of the separator, falling into the bins like golden hail, which the wives and daughters of the neighboring ranchers stood swiftly packing; a most competent lot of females, very swift and precise and earning a good bit of pin money thus every year. Peaches stood outside all day, checking up the lugs as they arrived, arranging about freight rates, overseeing the allotment of box cars to the various growers, and generally doing a man's job. And never once during the twelve months which followed did I know her to fail in her work—her magnificent constitution helping, no doubt, to pull her through. But I could see that a permanent change had taken place in her from the day of Abby's letter. She was no longer the madcap, and though she was even more beautiful she was different —and through love, the great tamer—as Blake would have it. This was the first incident to which I have referred as punctuating the monotony of the war for us. The second occurred more than a year later, in November, 1918, when we, like many another group of ranchers throughout the country, thought the town hall was on fire when all the time it was only the armistice. Mr. Markheim, Pinto and Alicia and myself were indoors, an unusually cold snap having offered us the treat of an open fire, a not unmixed pleasure by reason of our being under some anxiety about the trees. But on the whole it was what some modern poet whose name I cannot at the moment recall has termed the end of a perfect day. To begin with, I had dispatched three pounds of wool to Euphemia, whom Galadia, my only source of information about my sister, had written was doing great work for the Red Cross; her chief natural gift, that of knitting, had suddenly become of immense importance since the outbreak of the war, and she had to her credit and the honor of the family three hundred pair of socks. The achievement
- 35. appeared almost foreignto me, inasmuch as I had not knitted any socks since that momentous pair at Monte Carlo, a surprising faculty for a more active existence having developed in me during my sojourn on the ranch. At any rate I had sent out the wool, finished my last jar of marmalade, of which I had made an experimental thousand for a market which Mr. Pegg intended the development of, and Mr. Markheim had returned from a visit East in company with Pinto. Peaches had that day succeeded in breaking a pony she had long desired as a saddle horse and had hitherto been unsuccessful with. Mr. Pegg had a special design for the marmalade jars—a crystal orange, of the natural size and shape, the preserved fruit to furnish the color, and he and I were most enthusiastic over it. Mr. Markheim also credited himself with a successful trip, though from a wholly different cause. It appeared that he had at length contrived to install in his house a picture which he had long coveted, and this picture was none other than the Madonna of the Lamp, for which he had paid five hundred thousand dollars. Since his purchase of it the picture had been stored, and it seemed to me a strange time to trouble with getting it out. But Sebastian Markheim, with the fervor of the true collector and the madness which seems the hall- mark of his kind, was apparently oblivious of this circumstance and became wrapt in his description of it. You must have seen it in Vienna, he said. Good heavens, don't say you have seen photographs of it! You cannot imagine the beauty of the thing itself. I have given directions for the remodeling of the south wall of my library in the Ossining house for its occupancy. It will hang all alone on that wall—it's only a small picture, you know, so I have had Hasbrock, the architect, design some panels to encircle it I hope it is going to please you, Alicia. What? said Mr. Pegg twirling round suddenly from the bowl of ripe olives with which he was occupied. What's that? Why should Alicia be pleased? She's going to live there with it! said Markheim. She promised this afternoon!
- 36. Oh, no! Isaid getting to my feet. But nobody seemed to hear me. Yes, father, said Alicia. Then Pinto's face broke into a sort of crooked smile and he held out his hands to both of them. Well, I'll be damned! he said. Think of my Peaches picking out a friend of her father's! Why, Markheim, you must be somewhere near my own age! Why, pa, how rude! said Alicia. Aren't you going to kiss me? And you too, Free! Stop standing there like a dummy! People get married all the time—there's nothing unusual about it, you poor nuts! Come on, congratulate us! Well, of course, I recovered myself as best I could, and pecked her on the cheek. But I didn't feel my congratulations—I simply couldn't feel them. To marry that old man. And a foreigner! And a German Swiss! And everything! It was too dreadful! Nothing could make me feel that she was doing it for any reason except pity and because he had nagged her into it with his ceaseless attentions. Of course we had nothing against him, absolutely nothing, because after all being a millionaire art collector is not in itself strictly criminal. But with the memory of that beautiful romance in Italy still fresh in my own mind I could not understand it—I simply could not; and every fiber of my being resented it. Youth and age! It was all wrong. She had a silly notion that her heart was dead, and that it didn't matter what she did. That if it gave Sebastian happiness to marry her—why, he was good and kind and rich and cultured and famous, and why not give joy since one could no longer experience it? I could see in a flash what had gone on in her simple, honest, generous mind, and it nearly drove me wild, while all the time I had to stand there grinning and patting her on the shoulder, and saying how wonderful it all was, when in reality I wanted to drag her out of the room and shake her for being such a great silly fool, and force her to stop it before anyone else heard of her folly and she found herself in the complications of public knowledge of her engagement.
- 37. Instead of whichI stood round and admired the wonderful five-carat diamond ring which Markheim produced, and behaved like an idiot generally. Well, well, when is it to be? Mr. Pegg wanted to know. Alicia turned her big eyes slowly from her marvelous jewel to her father's puzzled face. I have promised Sebastian, she said slowly, to marry him as soon as the war is over! Her tone had, to my ears, the expectancy of a long reprieve. And it was at that minute that the fire bells began to ring. You can be sure we all rushed out at that, crying, Where is it? What is the matter? and many other similar exclamations natural to the situation. But at first nobody seemed to know. The Chinese cook came out, frying pan in hand, and began running round in circles. The hands were soon straggling in from their camp in the gulch by the river. Somebody, Mr. Pegg, I think, tried the telephone, but could get no answer. By this time almost everybody on the ranch had assembled before the house, shivering with the frost and searching the sky for signs of the incendiary glare, but in vain. An automobile dashed by down the Letterbox road with two prospectors in it. One was firing a gun like mad and he yelled something unintelligible at us in passing but ignored our invitation to stop. Then from the direction of the town a flivver emerged out of the swiftly falling dusk, and as it stopped in front of our gate a man in the uniform of an American captain jumped down with the aid of his uninjured arm, the other being supported by a sling, and came running toward us, flinging his cap into the air, the lights from our porch gleaming upon his excited face and upon the decorations on his breast. Victory! he shouted. Victory! Schoolhouse fire? Hell! The armistice was signed at two o'clock to-day!
- 38. It was Richard,the chauffeur, and I assure you that it was at that moment that I recognized the strong family resemblance and decided that he might after all be a Talbot—one of our Talbots. You can imagine the wild riot into which the news and the bearer of it threw us. I cannot describe it. Everyone went crazy and I have a blurred recollection of kissing several persons, the Chinaman among them. But only one thing remains clearly in my mind—Alicia standing like a stone in a corner of the veranda, her white face lifted to the rising moon, and Markheim running toward her with burning words which seemed to fall upon deaf ears. Alicia, Alicia, it's the end of the war! he was shouting.
- 39. X I recall uponone occasion my dear father having said that love in a cottage was better than politeness in a mansion, and this came at once to mind upon the occasion of our visit to Sebastian Markheim's palace on the banks of the upper section of the Hudson River. This took place just six months after that wonderful night when my dear nephew, as I was now convinced he was, returned, so to speak, with the armistice in his pocket. Sebastian, as I was now instructed to call Mr. Markheim, had desired us to come sooner, in order that Peaches might herself assist in selecting the plans and furnishings incident to the remodeling of what was to be her home. But Peaches was reluctant to go. Of course there was a good deal of readjustment to be done on all her father's ranches, and while he was in the south, where the big orchards were, we set in order the home ranch, which had been practically in our charge for a year and a half, and she gave as excuse for the delay the necessity for making these readjustments herself. Richard was to be left in complete charge and she busied herself quite unnecessarily in showing him a thousand details. Every week she would promise to be ready, and when the time came she would have discovered something that nobody else could take care of, which was all nonsense, because a citrus ranch practically takes care of itself during the winter months. But by hook and crook she held us off until April, and then at last we were ready to go. I will state that I for one was unreservedly eager to go home—to go East. I was, in point of fact, so excited at the prospect that on the night before our departure I found myself unable to compose myself to slumber, and rising from my uneasy couch I donned a robe and ventured forth from my bedchamber, which was upon the ground floor.
- 40. The moonlight, whichflooded the garden, gave it an uncanny distorted aspect, and all at once as I sat there, huddled upon a bench close to the wall of the house, I seemed to see the ranch and its surroundings with the same eyes which envisioned it upon my arrival so long ago. This sudden clarity of vision was doubtless due to the subconscious influence of my impending departure. At any rate the place, which I had grown so accustomed to that I beheld it only with the blindness of familiarity, seemed once more the impossibly crude wilderness that it appeared to be upon my arrival. For in the northern part of California there is little of the induced luxuriance of the South. There is something of the Eastern farmer's fight with the elements and a Nature that is not always overly kind or utterly dependable, and our garden was not a thing of lovely lawns, dense shrubs and misty glades. Far from it. Our flower beds were as practically irrigated as our orchards, standing deep in mud and lifting their wonderful blossoms from the mire we so religiously provided for them. There was none of the trimness of an Eastern estate about our more than practical, enterprising organization. Rather it bore the general aspect of Boston Common after an August holiday. It was, in plain truth, shockingly untidy, and I was horrified to realize that even I, who had been so carefully reared by the immaculate Euphemia, had made only the most feeble sort of effort to tidy up. I had been unable to see the molehills for the mountains, as one might say. But now, with the thought of the concentrated, condensed East before me, I perceived the unevenness of our paths, the forgotten bundle of old papers outside the storehouse, the broken gate which everyone cursed at but forgot to mend; and the olive and orange clad hills beyond grew dim in my mind's eye even as they formed but indistinguishable black patches in the cloud- changing moonlight. A deep longing for my own kind of living swept over me, and I even went so far as to experience a desire for Euphemia's breakfast room on Chestnut Street, and the mended table linen—the careful little things of life grown dear through years of painstakingly careful usage.
- 41. Moved by thisoverwhelming impulse I was on the verge of rising and gathering up that disgracefully untidy bundle of papers and carrying it to the trash bin where it belonged, thus at once satisfying a normal impulse and proving to myself that my upbringing had not been in vain, when I became aware that the window above my head had been opened softly and that someone—Peaches, without a doubt, since that was her chamber—was standing there, crying softly. My first impulse was to speak—to go to her with what comfort I was capable of offering, but having for an instant refrained I could not do so. Since the announcement of her betrothal to Markheim a wall had sprung up between us as far as her intimate life was concerned. Indeed she seemed to have withdrawn into herself curiously, though I doubt that anyone realized it as keenly as did I. And then having failed to speak immediately I found myself in an awkward predicament. Should I move or not? I had no desire to eavesdrop for the confidence she withheld, and yet I felt it my bounden duty as her chaperon and guardian and older woman generally to know all about her by one means or another, for her own good, and not out of mere female curiosity. And so allowing my sense of responsibility to conquer my delicacy I kept very still, and before long my diligence was rewarded. A clean sweep! whispered Peaches at her window. No use kidding myself. I'll make the break clean. It's the only thing to do! There was a short silence punctuated only by a few sniffs, and then an object flew through the air over my head and landed in the pool with a splash. The window above was closed with a snap. Whatever ritual she had been at was over. But not so the fulfillment of my duty as her protectress. No sooner had I made sure that she was not going to change her mind and come down after it, than I crept stealthily to the water's edge, having carefully noted the very spot where the object fell, and kneeling on the concrete basin's brim, greatly to the detriment of
- 42. that portion ofmy anatomy which bore the weight, being clad only for private life, I fished determinedly for the best part of half an hour, my sleeves rolled up but not escaping the effects of my earnest endeavor, and my curls getting thoroughly soaked. Fortunately Peaches' aim, usually so accurate and far reaching in the pursuit of the national sport of baseball, or in any other emergency such as reaching a high-hung apple, had fallen a little short this time, her secret having hit the shallow end of the pond. And so it was that after a very considerable period of effort I did retrieve the object, and retreated with it to the seclusion of my room. Once there I lit the lamp, drew the curtains, locked the door and proceeded with my duty still further. It was a terribly moist little bundle, done up in a silk handkerchief and weighted with the bronze paper-weight I had given Peaches for Christmas. But I was too much interested to mind this slight. For inside the bundle were two letters, already a mere pulpy mass from the soaking they had sustained, a brittle something which might once have been a rose, and the duke's wallet! The latter was still intact, but before examining it I made a little fire on the hearth, and by diligent coaxing managed to consume the remnants of the other souvenirs. They were no one's affairs except that of the lovers and no other eyes should behold them unbidden. And when they were quite concealed in the ashes of the fireplace I returned to the light and examined the wallet carefully. It seemed to me that there simply must be more to the matter than appeared. In any of those books which had so deep an influence upon my early thinking the discoverer of such a wallet would have surprised a jewel of value, secret documents popularly referred to as 'the papers,' or a marriage certificate which cleared the honor of the hero's mother, or something equally vital. And I must confess that I, in opening my find, rather anticipated some such discovery, but my expectations were doomed to disappointment, for it was in very truth what Peaches had suggested—a mileage ticket of some sort made out in Sandro's name!
- 43. I will saythat this end to my exciting evening was a trifle flat, but as my dear father used to say, our chief pleasure lies in anticipation and no disappointment in the event can cheat us of that. So I simply decided to put the thing carefully away in the bottom of my reticule in case it was ever needed. What with the war and all, one never can tell who is going to turn up a hero; and just think what souvenirs of Rupert Brooke, for example, are worth to-day, not to mention Napoleon and General Grant, and so forth, whose hero- value has, of course, been augmented with age. Well, at any rate, that was all there was to it at the time. I slept the sleep of duty well done, because I was determined to take care of Peaches in spite of herself, and the next morning rose refreshed, to make the early train for San Francisco, where we were to join Mr. Pegg and turn our faces eastward. The house which Sebastian Markheim had remodeled for his bride- to-be was already a sumptuous structure worthy of the famous collection of art treasures which it housed, and his efforts in altering it had been bent rather in the direction of improving its livableness and making it a cheerier spot to which to bring a young wife. The object of our visit was that Peaches be given the opportunity of making it completely to her liking in advance of her possession of it, and incidentally to make the acquaintance of her future neighbors, and of Mr. Markheim's set generally. He had planned a large house party as the means of introducing his fiancée to his social world, and she intended to procure her trousseau in New York during the intervals of gayety. Mr. Pegg was enchanted at the prospect thus opened up before him, and I was myself much elated at the thought of experiencing some real social life once more, for Abby's hospitality in dear old Italy, so lavish and yet in such excellent good form, had given me a taste for the gaieties my restricted youth had lacked. Even Peaches was gay, though not as of yore, but rather with a mature, stately gayety, and her manner toward me had become positively motherly.
- 44. There now, Free!she soothed me one day when I had expressed a mild concern about her state of mind. There now, Free, don't you worry about me! We all have to grow up sometime, don't we? Can't stay young plants forever—especially we women. Comes a time when we got to be grafted on to old stock and get ready for bearing —eh? Well, that's me, old thing! I was shocked at her indelicacy and did not hesitate to say so. If that is how you regard your forthcoming nuptials, I said stiffly, you ought to dissolve your betrothal. One should marry only for love—for love alone! Oh, should they? said Peaches. That's all you know about it. I'm very fond of Mr. Mark—of Sebastian, and he is the typical good husband. But you don't love him! I protested firmly. I love him as much as I am likely to love anyone, responded Peaches—like a young Portia, so stately and serious. And even if he is half a head shorter than I am he has a kind heart and he's a gentleman. And not over sixty years old! I retorted. Oh, Peaches, do you really want to do it? Suddenly she was serious. The defensively bantering light went out of her changeful eyes. Don't, Free! she pleaded. Yes, I do want to. I want to be a reasonable being—to make the best life I can for myself since I must go on living. I don't want to be a coward. I am still young and I haven't seen much of the world. Riches, art treasures, cultured people, and things—social position—there must be joy in these things or folks would not struggle for them so! And since they must be filling up the emptiness in a whole lot of lives I'm going to have a try at them too. Don't be afraid for me. I know just what I am doing. I know that I shall never care again. But I can like. And I can live,
- 45. and I'm goingto use my old beau to help me get the most out of life that I can when—when—well, you know, only don't say it, please! She was wonderful. So big and beautiful and full of health and common sense. I could not but admire her, though, of course, a few maidenly tears and vows of lifelong fidelity to the heroic dead would have been more suitable. But things had already gone too far for that. At the time the above-recorded conversation took place we were standing upon the steps of the Ritz in New York, waiting for the car which was to convey us up the river. Mr. Markheim had not expected us for another week and so hadn't been at the hotel to meet us, but was sending his chauffeur. And in a way Peaches' words reassured me. After all one must eventually resign oneself to fate, and if one had the good sense to take fate by the horns and as Peaches would say beat him to it— why, so much the better. We could all settle down to watch her live happily enough ever after if her program worked out. But would it? Despite her assurance I felt a faint misgiving. My dear father used always to say: Never you girls marry until Mister Right comes along. And we were brought up to honor and obey our parents—with the result that at the respective ages of fifty and sixty we girls were still single. However, I digress. In my youth, following the precepts of my father and seeking knowledge of the world through the medium of literature, I came upon the works of a lady of rank whose writings had for me the greatest fascination. As to what her actual name was I have to this day remained in ignorance, and her title, The Duchess, is all that I identify her by. But this estimable lady, while somewhat given to the recounting of scandalous episodes and the misfortunes peculiar to innocent maidens, had a wealth of descriptive power when she undertook the description of rich and aristocratic mansions or the interiors of castles of the less modest variety. But nothing ever recorded by her, not set forth for public inspection in the Boston Museum, could compare with the sumptuousness of Mr. Markheim's establishment.
- 46. I had beenprepared for something very fine, but this gorgeous replica of a famous Italian villa built upon terraces, its lovely low white façades rising in a symmetrical group one above the other, the whole nestling into the budding verdure of the hillside, its formal gardens descending step by step almost to the broad sweep of the Hudson below, was a veritable dream-palace. And the interior! Words almost fail me when I seek to describe it. Perhaps the most fitting thing I can say of it is that it was a home good enough for Peaches. Her great height, her gold-and-marble beauty, here found at last a fitting habitat. And then when I saw that little, comparatively speaking, Markheim man trotting about in front of her and giving her the place with a gesture as he displayed each treasure in turn, I felt sick and faint in my mind. And yet he was most kind and had never given me the least cause to criticize him, and certainly the house was enough to tempt any girl. I sighed, however, to think of the day when she would be married and living there. Mr. Markheim—Sebastian, I mean, I said—Mr. Pegg and I followed in the wake of the happy couple as they made the tour of the house —Sebastian, this place looks as if you had dug up the rich heart of Italy and transplanted it to America! Sebastian laughed. You have the right idea, Miss Freedom! The right idea—yes! he exclaimed with pride. More than half my collection is Italian—and if I do so say myself, it has taken a lot of patience and trouble to gather it—not to speak of the cost in money. They have a strict law against taking objects of art out of their country, you know, and it's been nip and tuck getting hold of a lot of this stuff—smuggled of course. Oh, don't look so shocked! If it's genuine it's smuggled—at the Italian end. But one doesn't call attention to the fact except in the privacy of one's own family! It sure is swell! said Mr. Pegg.
- 47. Sebastian laughed again—asound which never got him favor with me—and opened the door into the newest addition to the house— the library wing, which he had remodeled for the especial purpose of housing the Madonna of the Lamp. When I entered I could not refrain from an exclamation of delight, nor can I forbear to describe the place in some detail. To begin with it was almost round and very large, the ceiling being domed and the books being carried in long narrow stacks sunk into the paneling between the French windows as high as the carved molding. Above this an exquisite tone of blue with a few cleverly distributed stars gave a sense of infinite space, and despite the cumbersome old Florentine furniture the room was neither heavy nor dull. There was just enough gold to furnish flashes of light, and the warm old amber brocade on the chairs seemed to catch and hold the sunlight which poured through the long narrow windows at the west, all of which opened directly upon the first terrace of the rose garden. But the real triumph in lighting was the rose window of plain leaded glass on the north side of the room—the wall of which had been reconstructed to accommodate it in order that the Madonna might be properly illuminated by day. We gasped our admiration of its perfect lacery, and then turned about and faced the picture itself in reverent silence. Of course it is ridiculous to suppose there is anyone to whom the Madonna of the Lamp is not perfectly familiar, being, as she is, one of those paintings which are impressed upon the popular mind in spite of itself through endless repetition upon postal and Christmas cards, engravers' windows, magazine covers and Sunday-school prizes, to say nothing of Little Collections of Great Masters, gift photographs, furnishings for college rooms and appeals for public charities. Nevertheless, I will describe it, because as my dear father used to say, the collective mind of the public is not the public mind of the collector. It has to be told, in other words, when it can't be shown;
- 48. whereas, of course,you can tell a collector nothing—and get him to admit it. Well, at any rate, in case you do not recall it, the Madonna of the Lamp is a round canvas, not more than two and a half feet in diameter, and represents the Virgin with the Child curled up in a robe of sapphire blue which falls from her head in thick sweeping folds and crosses her knee in such a way as to give the appearance of being blown from behind by a wind and aiding in the circular effect. She is seated and bending over the Infant, protecting both him and the flickering lamp from the wind. Above her head is a single star visible through a patch of leaded window. Now you recall it, I am sure. It was painted in Florence by Raphael about the year 1506 and is one of the most famous monuments to his genius. And Markheim had provided a most wonderful setting for this jewel. The great window was of a design made from that behind the Virgin's head, and the carved panel upon which the painting hung was a skillful variation of the beautiful old carved frame about the canvas—the original frame, it was believed to be, and the motif of the design was carried out in a molding which diminished into a faint bas-relief at the outer edges of the large wall space above the mantel where it hung. Nor was the picture hung too high. Even I could have touched the bottom of the carvings; and the mantelpiece had no other ornament except two gigantic polychrome candlesticks of the same period. Truly it was a wonderfully successful arrangement and reflected great credit on the owner who had conceived it. Do you like it? was all he said, looking not at the Madonna but at Alicia. Do you like it, eh? Mr. Pegg took the question to himself. And you paid five hundred thousand dollars for that little picture? he asked incredulously. Why, from the price I expected something as big as a barn door!
- 49. Pa—don't be aboob—it's a diamond without a flaw, said Peaches, going closer, her face alight with pleasure. It's a real mother and child, she added. How big would you want them to be? They are immortal—isn't that big enough? Through the crudity of her rebuke I got one of those rare glimpses of her golden heart. Her crude parent, however, was unimpressed. Of course it's real pretty, he said. Which is more than can be said for most antiques. But five hundred thousand! My Lord, look at the profit? There can't be over ten dollars' worth of paint in it! Where is this feller, Raphael? Where the profit is doing him precious little good, chuckled Sebastian. Must be hell! commented Pinto. Very possibly, in spite of his choice of subjects! replied Markheim. Whereat he and I exchanged our first glance of thoroughly sympathetic understanding. I, of course, at once lowered my eyes, a burning sense of shame at my implied disloyalty struggling with my desire to spare Mr. Pegg the mortification of instruction. I had not forgotten and shall never forget how gently he led me to see the error of my ways when I first hit the ranch—as, for example, when I unknowingly made culls of his best tree of home fruit and he urged me to make marmalade of them and never told me until afterward that the way I had picked them by pulling them off the tree instead of clipping the stem made it impossible to use them for anything else. So now in my own realm I wished to lead him gradually into the paths of erudition and allow him to learn by inference whenever possible. Well, the rest of the house was beautiful as could be, and after we had finished inspecting it we had tea in a wonderful glass room filled with gay cretonnes and flowering plants, wicker chairs and caged canaries. Two menservants served the refection. Mr. Sebastian
- 50. Markheim had aconsiderable household, that was plain, and I began to regret that I had steadfastly stood with Peaches on refusing her father's suggestion of a personal maid. There's something too public about it, had been her objection, which I had sustained. But here amid all these servitors I felt differently. Not that I felt any indignity attached to our maidless condition, being, as I was, a self- supporting female well able to afford one if I desired such a thing. I could now live as I chose instead of as I aught, if you understand me. But I knew that Peaches would have to get a female attendant after she was married. Markheim was not the man to allow his wife to live in comfort when he could provide her with luxury. And at this juncture of my thought I stopped halfway through the sugared tea biscuit, a terrible realization overwhelming me for the first time. When Peaches was married she would no longer need me. Who then would need me? Nobody? Not Euphemia, who never answered my letters, though she always mutely cashed the inclosed checks. And would there be any checks to send her? Where would they come from? It was a chilling thought, as will readily be admitted. Why I had not thought of it sooner I cannot say. It must have been evident from the moment of Peaches' engagement that when the affair reached its consummation I would be, to put it vulgarly, out of a job. Of course I did not so greatly care for myself, but there was Euphemia, the dependent, to consider, whose tradition of useless gentility must not be disturbed in her declining years. True, I had saved a very considerable portion of my salary and had almost twenty thousand dollars distributed among six savings banks. That might conceivably tide us over for the remainder of our lives. But I had acquired the habit of remunerative occupation and close companionship with dear friends; also a taste for French heels and facial massage whenever practical. And the thought of the Chestnut Street house was, the more shame upon me for saying it of my father's home, almost intolerable. And Mr. Pegg—dear Pinto, how I should miss him! in a purely friendly way of course.
- 51. Fully realizing forthe first time the bitterness of my situation I refused a second sugared bun and rising remarked that as Sebastian expected dinner guests we had best retire and obtain a little rest before it was time to dress. Of course my intention was in part to leave the lovers together for a properly brief interval, but somewhat to my surprise Peaches rose also and said she would accompany me. My heart was heavy, and for once I would have preferred to be alone. But she slipped her arm about my neck, and we started for our rooms, chatting amiably while the men settled down for a cigar. Now one of the peculiarities of the Markheim palace was that it gave no appearance of modernity. Though it was in point of fact less than ten years built, it was so cunningly designed, so convincingly arranged, with such perfection of detail that it possessed an air of old mystery difficult to define, and under ordinary circumstances most fascinating—a real achievement on the part of architect and decorator alike. The ancient furniture stood so easily in the background provided for it that one could have sworn the walls had been made before it; the modern lighting was so well handled as to be absolutely unobtrusive. Slowly, affectionately, we crossed the main hall, pausing to look at the chased armor on the two silent figures at the foot of the beautiful winding stairs. A Gobelin tapestry fluttered faintly on the wall above us, stirred by the gentle sunset wind from the spring- scented river below, and the lingering twilight filled the great hall with mysterious shadows. There was not another soul in sight and not a sound to be heard except the distant murmur of the men's talk and the voice of a pleasure boat distantly upon the water. I accompanied Alicia up the stairs, feeling as if I were in some enchanted palace of medieval days, and above, the long dim corridor in which the lamps had not yet been lit was ghostly in the pale glimmer from its high mullioned windows. Isn't it spooky? said Peaches in a low tone.
- 52. Yes! I replied,whispering involuntarily. One might almost expect to see a ghost! And scarcely had I spoken the words when Peaches, the supernormal, who was a trifle ahead of me by now, uttered a shriek and leaned trembling against the stone wall of the passageway. But for a moment I could not come to her aid. My limbs seemed frozen, paralyzed. For there suddenly and soundlessly a form was towering vaguely before us, its white face luminous in a shaft of uncanny light. It was the Duke di Monteventi!
- 53. XI After one horribleendless moment the figure moved slightly and the corridor was flooded with the soft mellow light from half a dozen electric sconces. With a half-choked cry of Sandy! upon her lips Peaches moved toward him, only to stop short, her face going completely blank. The man was a servant, a valet presumably, carrying a folded suit of clothing carefully over one arm and wearing soft felt shoes, which had been the secret of his noiseless approach. His hair was thickly gray and his face was lined and scarred. He looked perhaps ten years older than Sandro—and yet the likeness was there— unmistakable, though in the full light not by any means so perfect. I beg pardon, ladies, he said in a measured voice, withdrawing another step. The lights should have been on. Then with a little bow he passed noiselessly down the corridor and entered one of the bedrooms, presumably that occupied by Markheim himself. Peaches made a little involuntary gesture as if to follow him, stretching out her hands toward his unconscious back, and then, as the door closed upon him, turned to me, her amber eyes afire. She seized me by the wrist in a manner positively painful and dragged me into her room, where she caused me to sit down abruptly and without personal selection upon a sort of hassock, the while she towered over me, fairly glowing with animation—far, far, more like her old self than she had been for almost six years. Free! she said. Was it? Was it? Oh, Free—say something! It couldn't have been! I replied shakily. And yet the resemblance —it was extraordinary!
- 54. It was amiracle! said Peaches. No two people could look so much alike. He had a brother, I began doubtfully, who was merely supposed to be dead. Sandro would have known you at once. But didn't he? she questioned, striding up and down the room with her long, clean gesture of body. Why didn't he speak at once? He was too much amazed! Nonsense! I exclaimed. How could he be amazed, when as a servant in this house—in all probability Sebastian's valet—he must have known in advance all about your coming here! That's so, said Peaches. And, of course there are differences—the grayness, the lines in his face. But something may have happened to him. Very likely! I replied dryly. Considering we have heard from Cousin Abby that he was killed in action. But it may have been a mistake, she whispered. Stranger things have happened. And a servant! No—even if he had gone quite mad and forgotten everything that would hardly be possible. Servant or not, if it is he, why on earth shouldn't he recognize you? I demanded. That's the sort of encounter which is supposed to bring people to their senses, you know. But didn't he recognize me? she replied with a doubt willfully sustained. Just for an instant, I was so sure! Well! What are we going to do about it? I said. If by chance it really is Sandro it's a nice situation, I'm sure! With your wedding only a few weeks off and, and—why, good gracious! It's simply terrible! But Peaches didn't look as if she thought it was simply terrible—not in the least. She was terrifically excited, but more beautiful than ever. Free! she cried. I know it is he! Do you suppose I could feel as I did—as I do, at the encounter unless it is Sandy? Lots of times
- 55. people know thingswithout evidence. And this is one of those times. I feel it is he. I don't care how differently he looked when the lights went up. But how on earth are you going to find out? I urged. Surely, Peaches, he cannot have forgotten you! Forgotten! she exclaimed, stopping short in her pacing of the floor. Forgotten! Good heavens, Free, you don't suppose that is it, do you? Of course I don't! I snapped, even though I was not entirely sure but that a young man who was capable of taking French leave in the way that Sandro had six years previously, was not capable of anything, including having an affaire de cœur with Peaches and then failing to recollect the incident. Some men are that way; I have it on the authority of The Duchess. This man is older! I went on. And we don't know for certain what his position in the household is. The best thing for you to do is to question Sebastian about him. Won't he think it strange if I let him on to the fact that I'm stuck on his valet? Peaches considered in her disconcertingly frank way. Good gracious, you must do nothing of the kind! I interposed. Besides, you don't know that you are, as you vulgarly put it, stuck on him. You only think it may be Sandy. Kindly keep that in mind, my dear! I think there is something damn funny about the whole shooting match! said Peaches vigorously. And I'm going to the bottom of it mighty pronto! With which she flung from the room to don one of her majestic evening gowns, leaving me in great distress of mind for fear of what she would do next. To array myself for the evening's festivities and to descend to them in a becomingly dignified manner was no easy task, but by the greatest effort at self-control I accomplished both the arrangement of my toilet and the adjustment of my manner
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