Calculus Volume 1

This companion book to Thinkwell's Calculus course is an optional resource to complement the video lessons in Thinkwell Calculus.

Volume 1

Calculus Volume 1

SENIOR CONTRIBUTING AUTHORS EDWIN "JED" HERMAN, UNIVERSITY OF WISCONSIN-STEVENS POINT GILBERT STRANG, MASSACHUSETTS INSTITUTE OF TECHNOLOGY

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Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 1: Functions and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1 Review of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Basic Classes of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.3 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.4 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 1.5 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Chapter2: Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.1 A Preview of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2.2 The Limit of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 2.3 The Limit Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 2.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 2.5 The Precise Definition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Chapter3: Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 3.1 Defining the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 3.2 The Derivative as a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 3.3 Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 3.4 Derivatives as Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 3.5 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 3.6 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 3.7 Derivatives of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 3.8 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 3.9 Derivatives of Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 319 Chapter 4: Applications of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 4.1 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 4.2 Linear Approximations and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 4.3 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 4.4 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 4.5 DerivativesandtheShapeofaGraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 4.6 Limits at Infinity and Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 4.7 Applied Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 4.8 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 4.9 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 4.10 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Chapter5: Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 5.1 Approximating Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 5.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 5.3 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 5.4 Integration Formulas and the Net Change Theorem . . . . . . . . . . . . . . . . . . . . . . 566 5.5 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 5.6 Integrals Involving Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . 595 5.7 Integrals Resulting in Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 608 Chapter 6: Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 6.1 Areas between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 6.2 Determining Volumes by Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 6.3 Volumes of Revolution: Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 6.4 Arc Length of a Curve and Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 6.5 Physical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 6.6 Moments and Centers of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 6.7 Integrals, Exponential Functions, and Logarithms . . . . . . . . . . . . . . . . . . . . . . . 721 6.8 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 6.9 Calculus of the Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Appendix A: Table of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 Appendix B: Table of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769 Appendix C: Review of Pre-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865

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Preface

PREFACE

Welcome to Calculus Volume 1 , an OpenStax resource. This textbook was written to increase student access to high-quality learning materials, maintaining highest standards of academic rigor at little to no cost. About OpenStax OpenStax is a nonprofit based at Rice University, and it’s our mission to improve student access to education. Our first openly licensed college textbook was published in 2012, and our library has since scaled to over 25 books for college and AP ® courses used by hundreds of thousands of students. OpenStax Tutor, our low-cost personalized learning tool, is being used in college courses throughout the country. Through our partnerships with philanthropic foundations and our alliance with other educational resource organizations, OpenStax is breaking down the most common barriers to learning Calculus Volume 1 is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC- BY-NC-SA) license, which means that you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its content contributors, do not use the content for commercial purposes, and distribute the content under the same CC-BY-NC-SA license. Because our books are openly licensed, you are free to use the entire book or pick and choose the sections that are most relevant to the needs of your course. Feel free to remix the content by assigning your students certain chapters and sections in your syllabus, in the order that you prefer. You can even provide a direct link in your syllabus to the sections in the web view of your book. Instructors also have the option of creating a customized version of their OpenStax book. The custom version can be made available to students in low-cost print or digital form through their campus bookstore. Visit your book page on OpenStax.org for more information. Errata All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors sometimes occur. Since our books are web based, we can make updates periodically when deemed pedagogically necessary. If you have a correction to suggest, submit it through the link on your book page on OpenStax.org. Subject matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you will also find a list of past errata changes on your book page on OpenStax.org. Format You can access this textbook for free in web view or PDF through OpenStax.org, and for a low cost in print. About Calculus Volume 1 and empowering students and instructors to succeed. About OpenStax's resources Customization Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 1 covers functions, limits, derivatives, and integration. Coverage and scope Our Calculus Volume 1 textbook adheres to the scope and sequence of most general calculus courses nationwide. We have worked to make calculus interesting and accessible to students while maintaining the mathematical rigor inherent in the subject. With this objective in mind, the content of the three volumes of Calculus have been developed and arranged to provide a logical progression from fundamental to more advanced concepts, building upon what students have already learned and emphasizing connections between topics and between theory and applications. The goal of each section is to enable students not just to recognize concepts, but work with them in ways that will be useful in later courses and future careers. The organization and pedagogical features were developed and vetted with feedback from mathematics educators dedicated to the project.

Preface

Volume 1

Chapter 1: Functions and Graphs Chapter 2: Limits Chapter 3: Derivatives Chapter 4: Applications of Derivatives Chapter 5: Integration Chapter 6: Applications of Integration

Volume 2

Chapter 1: Integration Chapter 2: Applications of Integration Chapter 3: Techniques of Integration Chapter 4: Introduction to Differential Equations Chapter 5: Sequences and Series Chapter 6: Power Series Chapter 7: Parametric Equations and Polar Coordinates Chapter 1: Parametric Equations and Polar Coordinates Chapter 2: Vectors in Space Chapter 3: Vector-Valued Functions Chapter 4: Differentiation of Functions of Several Variables Chapter 5: Multiple Integration Chapter 6: Vector Calculus Chapter 7: Second-Order Differential Equations

Volume 3

Pedagogical foundation Throughout Calculus Volume 1 you will find examples and exercises that present classical ideas and techniques as well as modern applications and methods. Derivations and explanations are based on years of classroom experience on the part of long-time calculus professors, striving for a balance of clarity and rigor that has proven successful with their students. Motivational applications cover important topics in probability, biology, ecology, business, and economics, as well as areas of physics, chemistry, engineering, and computer science. Student Projects in each chapter give students opportunities to explore interesting sidelights in pure and applied mathematics, from determining a safe distance between the grandstand and the track at a Formula One racetrack, to calculating the center of mass of the Grand Canyon Skywalk or the terminal speed of a skydiver. Chapter Opening Applications pose problems that are solved later in the chapter, using the ideas covered in that chapter. Problems include the hydraulic force against the Hoover Dam, and the comparison of relative intensity of two earthquakes. Definitions, Rules, and Theorems are highlighted throughout the text, including over 60 Proofs of theorems. Assessments that reinforce key concepts In-chapter Examples walk students through problems by posing a question, stepping out a solution, and then asking students to practice the skill with a “Checkpoint” question. The book also includes assessments at the end of each chapter so students can apply what they’ve learned through practice problems. Many exercises are marked with a [T] to indicate they are suitable for solution by technology, including calculators or Computer Algebra Systems (CAS). Answers for selected exercises are available in the AnswerKey at the back of the book. The book also includes assessments at the end of each chapter so students can apply what they’ve learned through practice problems. Early or late transcendentals Calculus Volume 1 is designed to accommodate both Early and Late Transcendental approaches to calculus. Exponential and logarithmic functions are introduced informally in Chapter 1 and presented in more rigorous terms in Chapter 6. Differentiation and integration of these functions is covered in Chapters 3–5 for instructors who want to include them with other types of functions. These discussions, however, are in separate sections that can be skipped for instructors who prefer to wait until the integral definitions are given before teaching the calculus derivations of exponentials and logarithms.

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Preface

Comprehensive art program Our art program is designed to enhance students’ understanding of concepts through clear and effective illustrations, diagrams, and photographs.

Additional resources Student and instructor resources

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Preface

About the authors Senior contributing authors Gilbert Strang, Massachusetts Institute of Technology

Dr. Strang received his PhD from UCLA in 1959 and has been teaching mathematics at MIT ever since. His Calculus online textbook is one of eleven that he has published and is the basis from which our final product has been derived and updated for today’s student. Strang is a decorated mathematician and past Rhodes Scholar at Oxford University. Edwin “Jed” Herman, University of Wisconsin-Stevens Point Dr. Herman earned a BS in Mathematics from Harvey Mudd College in 1985, an MA in Mathematics from UCLA in 1987, and a PhD in Mathematics from the University of Oregon in 1997. He is currently a Professor at the University of Wisconsin-Stevens Point. He has more than 20 years of experience teaching college mathematics, is a student research mentor, is experienced in course development/design, and is also an avid board game designer and player. Contributing authors Catherine Abbott, Keuka College Nicoleta Virginia Bila, Fayetteville State University Sheri J. Boyd, Rollins College

Joyati Debnath, Winona State University Valeree Falduto, Palm Beach State College Joseph Lakey, New Mexico State University Julie Levandosky, Framingham State University David McCune, William Jewell College Michelle Merriweather, Bronxville High School

Kirsten R. Messer, Colorado State University - Pueblo Alfred K. Mulzet, Florida State College at Jacksonville William Radulovich (retired), Florida State College at Jacksonville Erica M. Rutter, Arizona State University David Smith, University of the Virgin Islands Elaine A. Terry, Saint Joseph’s University David Torain, Hampton University Reviewers Marwan A. Abu-Sawwa, Florida State College at Jacksonville Kenneth J. Bernard, Virginia State University John Beyers, University of Maryland Charles Buehrle, Franklin & Marshall College William DeSalazar, Broward County School System Murray Eisenberg, University of Massachusetts Amherst Kristyanna Erickson, Cecil College Tiernan Fogarty, Oregon Institute of Technology David French, Tidewater Community College Marilyn Gloyer, Virginia Commonwealth University Shawna Haider, Salt Lake Community College Lance Hemlow, Raritan Valley Community College Matthew Cathey, Wofford College Michael Cohen, Hofstra University M.R. Khadivi, Jackson State University Robert J. Krueger, Concordia University Tor A. Kwembe, Jackson State University Jean-Marie Magnier, Springfield Technical Community College Cheryl Chute Miller, SUNY Potsdam Bagisa Mukherjee, Penn State University, Worthington Scranton Campus Jerry Jared, The Blue Ridge School Peter Jipsen, Chapman University David Johnson, Lehigh University

Kasso Okoudjou, University of Maryland College Park Peter Olszewski, Penn State Erie, The Behrend College Steven Purtee, Valencia College Alice Ramos, Bethel College

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Preface

Doug Shaw, University of Northern Iowa Hussain Elalaoui-Talibi, Tuskegee University Jeffrey Taub, Maine Maritime Academy William Thistleton, SUNY Polytechnic Institute A. David Trubatch, Montclair State University Carmen Wright, Jackson State University Zhenbu Zhang, Jackson State University

Preface

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Chapter 1 | Functions and Graphs

1| FUNCTIONS AND GRAPHS

Figure 1.1 A portion of the San Andreas Fault in California. Major faults like this are the sites of most of the strongest earthquakes ever recorded. (credit: modification of work by Robb Hannawacker, NPS)

Chapter Outline

1.1 Review of Functions 1.2 Basic Classes of Functions 1.3 Trigonometric Functions 1.4 Inverse Functions 1.5 Exponential and Logarithmic Functions Introduction

In the past few years, major earthquakes have occurred in several countries around the world. In January 2010, an earthquake of magnitude 7.3 hit Haiti. A magnitude 9 earthquake shook northeastern Japan in March 2011. In April 2014, an 8.2-magnitude earthquake struck off the coast of northern Chile. What do these numbers mean? In particular, how does a magnitude 9 earthquake compare with an earthquake of magnitude 8.2? Or 7.3? Later in this chapter, we show how logarithmic functions are used to compare the relative intensity of two earthquakes based on the magnitude of each earthquake (see Example 1.39 ). Calculus is the mathematics that describes changes in functions. In this chapter, we review all the functions necessary to study calculus. We define polynomial, rational, trigonometric, exponential, and logarithmic functions. We review how to evaluate these functions, and we show the properties of their graphs. We provide examples of equations with terms involving these functions and illustrate the algebraic techniques necessary to solve them. In short, this chapter provides the foundation for the material to come. It is essential to be familiar and comfortable with these ideas before proceeding to the formal introduction of calculus in the next chapter.

Chapter 1 | Functions and Graphs

1.1 | Review of Functions

Learning Objectives

1.1.1 Use functional notation to evaluate a function. 1.1.2 Determine the domain and range of a function. 1.1.3 Draw the graph of a function. 1.1.4 Find the zeros of a function. 1.1.5 Recognize a function from a table of values. 1.1.6 Make new functions from two or more given functions. 1.1.7 Describe the symmetry properties of a function.

In this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and terms related to functions. We also define composition of functions and symmetry properties. Most of this material will be a review for you, but it serves as a handy reference to remind you of some of the algebraic techniques useful for working with functions. Functions Given two sets A and B , a set with elements that are ordered pairs ( x , y ), where x is an element of A and y is an element of B , is a relation from A to B . A relation from A to B defines a relationship between those two sets. A function is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the input ; the element of the second set is called the output . Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input. For example, the area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them. Definition A function f consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The set of inputs is called the domain of the function. The set of outputs is called the range of the function. For example, consider the function f , where the domain is the set of all real numbers and the rule is to square the input. Then, the input x =3 is assigned to the output 3 2 =9. Since every nonnegative real number has a real-value square root, every nonnegative number is an element of the range of this function. Since there is no real number with a square that is negative, the negative real numbers are not elements of the range. We conclude that the range is the set of nonnegative real numbers. For a general function f with domain D , we often use x to denote the input and y to denote the output associated with x . When doing so, we refer to x as the independent variable and y as the dependent variable , because it depends on x . Using function notation, we write y = f ( x ), and we read this equation as “ y equals f of x .” For the squaring function described earlier, we write f ( x ) = x 2 . The concept of a function can be visualized using Figure 1.2 , Figure 1.3 , and Figure 1.4 .

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Chapter 1 | Functions and Graphs

Figure 1.2 A function can be visualized as an input/output device.

Figure 1.3 A function maps every element in the domain to exactly one element in the range. Although each input can be sent to only one output, two different inputs can be sent to the same output.

Figure 1.4 In this case, a graph of a function f has a domain of {1, 2, 3} and a range of {1, 2}. The independent variable is x and the dependent variable is y .

Visit this applet link (http://www.openstax.org/l/grapherrors) to see more about graphs of functions.

We can also visualize a function by plotting points ( x , y ) in the coordinate plane where y = f ( x ). The graph of a function is the set of all these points. For example, consider the function f , where the domain is the set D ={1, 2, 3} and the rule is f ( x ) =3− x . In Figure 1.5 , we plot a graph of this function.

Chapter 1 | Functions and Graphs

Figure 1.5 Here we see a graph of the function f with domain {1, 2, 3} and rule f ( x ) =3− x . The graph consists of the points ( x , f ( x )) for all x in the domain.

Every function has a domain. However, sometimes a function is described by an equation, as in f ( x ) = x 2 , with no specific domain given. In this case, the domain is taken to be the set of all real numbers x forwhich f ( x ) is a real number. For example, since any real number can be squared, if no other domain is specified, we consider the domain of f ( x ) = x 2 to be the set of all real numbers. On the other hand, the square root function f ( x ) = x only gives a real output if x is nonnegative. Therefore, the domain of the function f ( x ) = x is the set of nonnegative real numbers, sometimes called the natural domain . For the functions f ( x ) = x 2 and f ( x ) = x , the domains are sets with an infinite number of elements. Clearly we cannot list all these elements. When describing a set with an infinite number of elements, it is often helpful to use set-builder or interval notation. When using set-builder notation to describe a subset of all real numbers, denoted ℝ , we write

⎧ ⎩ ⎨ x | x has some property

⎫ ⎭ ⎬ .

We read this as the set of real numbers x such that x has some property. For example, if we were interested in the set of real numbers that are greater than one but less than five, we could denote this set using set-builder notation by writing { x | 1< x <5}. A set such as this, which contains all numbers greater than a and less than b , can also be denoted using the interval notation ( a , b ). Therefore, (1, 5) = ⎧ ⎩ ⎨ x | 1< x <5 ⎫ ⎭ ⎬ . The numbers 1 and 5 are called the endpoints of this set. If we want to consider the set that includes the endpoints, we would denote this set by writing [1, 5] ={ x | 1≤ x ≤5}. We can use similar notation if we want to include one of the endpoints, but not the other. To denote the set of nonnegative real numbers, we would use the set-builder notation { x | 0≤ x }. The smallest number in this set is zero, but this set does not have a largest number. Using interval notation, we would use the symbol ∞, which refers to positive infinity, and we would write the set as [0, ∞) ={ x | 0≤ x }. It is important to note that ∞ is not a real number. It is used symbolically here to indicate that this set includes all real numbers greater than or equal to zero. Similarly, if we wanted to describe the set of all nonpositive numbers, we could write (−∞, 0] ={ x | x ≤0}.

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Here, the notation −∞ refers to negative infinity, and it indicates that we are including all numbers less than or equal to zero, no matter how small. The set (−∞, ∞) = ⎧ ⎩ ⎨ x | x is any real number ⎫ ⎭ ⎬ refers to the set of all real numbers. Some functions are defined using different equations for different parts of their domain. These types of functions are known as piecewise-defined functions . For example, suppose we want to define a function f with a domain that is the set of all real numbers such that f ( x ) =3 x +1 for x ≥2 and f ( x ) = x 2 for x <2. We denote this function by writing

⎧ ⎩ ⎨ 3 x +1 x ≥2 x 2 x <2 .

f ( x ) =

When evaluating this function for an input x , the equation to use depends on whether x ≥2 or x <2. For example, since 5>2, we use the fact that f ( x ) =3 x +1 for x ≥2 and see that f (5) = 3(5) + 1 = 16. On the other hand, for x =−1, we use the fact that f ( x ) = x 2 for x <2 and see that f (−1) =1.

Example 1.1 Evaluating Functions

For the function f ( x ) =3 x 2 +2 x −1, evaluate a. f (−2) b. f ( 2) c. f ( a + h )

Solution Substitute the given value for x in the formula for f ( x ). a. f (−2) = 3(−2) 2 +2(−2)−1=12−4−1=7 b. f ( 2) =3( 2) 2 +2 2−1=6+2 2−1=5+2 2

f ( a + h ) =3( a + h ) 2 +2( a + h )−1 =3 ⎛

⎝ a 2 +2 ah + h 2 ⎞ ⎠ +2 a +2 h −1 =3 a 2 +6 ah +3 h 2 +2 a +2 h −1

For f ( x ) = x 2 −3 x +5, evaluate f (1) and f ( a + h ).

1.1

Example 1.2 Finding Domain and Range

For each of the following functions, determine the i. domain and ii. range.

Chapter 1 | Functions and Graphs

a. f ( x ) = ( x −4) 2 +5 b. f ( x ) = 3 x +2−1 c. f ( x ) = 3 x −2

Solution

a. Consider f ( x ) = ( x −4) 2 +5. i. Since f ( x ) = ( x −4) 2 +5 is a real number for any real number x , the domain of f is the interval (−∞, ∞). ii. Since ( x −4) 2 ≥0, we know f ( x ) = ( x −4) 2 +5≥5. Therefore, the range must be a subset of ⎧ ⎩ ⎨ y | y ≥5 ⎫ ⎭ ⎬ . To show that every element in this set is in the range, we need to show that for a given y in that set, there is a real number x such that f ( x ) = ( x −4) 2 +5= y . Solving this equation for x , we see that we need x such that ( x −4) 2 = y −5.

This equation is satisfied as long as there exists a real number x such that x −4= ± y −5.

Since y ≥5, the square root is well-defined. We conclude that for x =4± y −5, f ( x ) = y , and therefore the range is ⎧ ⎩ ⎨ y | y ≥5 ⎫ ⎭ ⎬ . b. Consider f ( x ) = 3 x +2−1. i. To find the domain of f , we need the expression 3 x +2≥0. Solving this inequality, we conclude that the domain is { x | x ≥−2/3}. ii. To find the range of f , we note that since 3 x +2≥0, f ( x ) = 3 x +2−1≥−1. Therefore, the range of f must be a subset of the set ⎧ ⎩ ⎨ y | y ≥−1 ⎫ ⎭ ⎬ . To show that every element in this set is in the range of f , we need to show that for all y in this set, there exists a real number x in the domain such that f ( x ) = y . Let y ≥−1. Then, f ( x ) = y if and only if 3 x +2−1= y .

Solving this equation for x , we see that x must solve the equation 3 x +2= y +1.

Since y ≥−1, such an x could exist. Squaring both sides of this equation, we have 3 x +2= ( y +1) 2 . Therefore, we need 3 x = ( y +1) 2 −2,

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Chapter 1 | Functions and Graphs

which implies

⎠ 2 − 2

x = 1 3 ⎛

⎝ y +1 ⎞

3 . We just need to verify that x is in the domain of f . Since the domain of f consists of all real numbers greater than or equal to −2/3, and

1 3 2 3 , there does exist an x in the domain of f . We conclude that the range of f is ⎧ ⎩ ⎛ ⎝ y +1 ⎞ ⎠ 2 − 2 3 ≥ −

⎨ y | y ≥−1 ⎫ ⎭ ⎬ .

c. Consider f ( x ) =3/( x −2). i. Since 3/( x −2) is defined when the denominator is nonzero, the domain is { x | x ≠2}. ii. To find the range of f , we need to find the values of y such that there exists a real number x in the domain with the property that 3 x −2 = y . Solving this equation for x , we find that x = 3 y +2. Therefore, as long as y ≠0, there exists a real number x in the domain such that f ( x ) = y . Thus, the range is ⎧ ⎩ ⎨ y | y ≠0 ⎫ ⎭ ⎬ .

1.2

Find the domain and range for f ( x ) = 4−2 x +5.

Representing Functions Typically, a function is represented using one or more of the following tools:

• A table • A graph • A formula We can identify a function in each form, but we can also use them together. For instance, we can plot on a graph the values from a table or create a table from a formula. Tables Functions described using a table of values arise frequently in real-world applications. Consider the following simple example. We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable x be the time after midnight, measured in hours, and the output variable y be the temperature x hours after midnight, measured in degrees Fahrenheit. We record our data in Table 1.1 .

Chapter 1 | Functions and Graphs

Temperature (°F)

Hours after Midnight

Table 1.1 Temperature as a Function of Time of Day

We can see from the table that temperature is a function of time, and the temperature decreases, then increases, and then decreases again. However, we cannot get a clear picture of the behavior of the function without graphing it. Graphs Given a function f described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in Table1.1 can give us a better idea of their fluctuation throughout the day. Figure1.6 shows the plot of the temperature function.

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Chapter 1 | Functions and Graphs

Figure 1.6 The graph of the data from Table 1.1 shows temperature as a function of time.

From the points plotted on the graph in Figure 1.6 , we can visualize the general shape of the graph. It is often useful to connect the dots in the graph, which represent the data from the table. In this example, although we cannot make any definitive conclusion regarding what the temperature was at any time for which the temperature was not recorded, given the number of data points collected and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern, as we can see in Figure 1.7 .

Figure 1.7 Connecting the dots in Figure 1.6 shows the general pattern of the data.

Algebraic Formulas Sometimes we are not given the values of a function in table form, rather we are given the values in an explicit formula. Formulas arise in many applications. For example, the area of a circle of radius r is given by the formula A ( r ) = πr 2 . When an object is thrown upward from the ground with an initial velocity v 0 ft/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula s ( t ) =−16 t 2 + v 0 t . When P dollars are invested in an account at an annual interest rate r compounded continuously, the amount of money after t years is given by the formula A ( t ) = Pe rt . Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form.

Chapter 1 | Functions and Graphs

Given an algebraic formula for a function f , the graph of f is the set of points ⎛ ⎝ x , f ( x ) ⎞ ⎠ , where x is in the domain of f and f ( x ) is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of f consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin. When creating a table of inputs and outputs, we typically check to determine whether zero is an output. Those values of x where f ( x ) =0 are called the zeros of a function . For example, the zeros of f ( x ) = x 2 −4 are x = ±2. The zeros determine where the graph of f intersects the x -axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the x -axis, or it may intersect multiple (or even infinitely many) times. Another point of interest is the y -intercept, if it exists. The y -intercept is given by ⎛ ⎝ 0, f (0) ⎞ ⎠ . Since a function has exactly one output for each input, the graph of a function can have, at most, one y -intercept. If x =0 is in the domain of a function f , then f has exactly one y -intercept. If x =0 is not in the domain of f , then f has no y -intercept. Similarly, for any real number c , if c is in the domain of f , there is exactly one output f ( c ), and the line x = c intersects the graph of f exactly once. On the other hand, if c is not in the domain of f , f ( c ) is not defined and the line x = c does not intersect the graph of f . This property is summarized in the vertical line test . Rule: Vertical Line Test Given a function f , every vertical line that may be drawn intersects the graph of f no more than once. If any vertical line intersects a set of points more than once, the set of points does not represent a function.

We can use this test to determine whether a set of plotted points represents the graph of a function ( Figure 1.8 ).

Figure 1.8 (a) The set of plotted points represents the graph of a function because every vertical line intersects the set of points, at most, once. (b) The set of plotted points does not represent the graph of a function because some vertical lines intersect the set of points more than once.

Example 1.3

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Chapter 1 | Functions and Graphs

Finding Zeros and y -Intercepts of a Function

Consider the function f ( x ) =−4 x +2. a. Find all zeros of f . b. Find the y -intercept (if any). c. Sketch a graph of f .

Solution a. To find the zeros, solve f ( x ) =−4 x +2=0. We discover that f has one zero at x =1/2. b. The y -intercept is given by ⎛ ⎝ 0, f (0) ⎞ ⎠ = (0, 2). c. Given that f is a linear function of the form f ( x ) = mx + b that passes through the points (1/2, 0) and (0, 2), we can sketch the graph of f ( Figure 1.9 ).

Figure 1.9 The function f ( x ) =−4 x +2 is a line with x -intercept (1/2, 0) and y -intercept (0, 2).

Example 1.4 Using Zeros and y -Intercepts to Sketch a Graph

Consider the function f ( x ) = x +3+1. a. Find all zeros of f . b. Find the y -intercept (if any). c. Sketch a graph of f .

Solution a. To find the zeros, solve x +3+1=0. This equation implies x +3=−1. Since x +3≥0 for all

Chapter 1 | Functions and Graphs

x , this equation has no solutions, and therefore f has no zeros. b. The y -intercept is given by ⎛ ⎝ 0, f (0) ⎞ ⎠ = (0, 3+1). c. To graph this function, we make a table of values. Since we need x +3≥0, we need to choose values of x ≥−3. We choose values that make the square-root function easy to evaluate.

−3 −2 1

f ( x )

Table 1.2 Making use of the table and knowing that, since the function is a square root, the graph of f should be similar to the graph of y = x , we sketch the graph ( Figure 1.10 ).

Figure 1.10 The graph of f ( x ) = x +3+1 has a y -intercept but no x -intercepts.

Find the zeros of f ( x ) = x 3 −5 x 2 +6 x .

1.3

Example 1.5 Finding the Height of a Free-Falling Object

If a ball is dropped from a height of 100 ft, its height s at time t is given by the function s ( t ) =−16 t 2 +100, where s is measured in feet and t is measured in seconds. The domain is restricted to the interval [0, c ], where t =0 is the time when the ball is dropped and t = c is the time when the ball hits the ground. a. Create a table showing the height s ( t ) when t = 0, 0.5, 1, 1.5, 2, and 2.5. Using the data from the table, determine the domain for this function. That is, find the time c when the ball hits the ground. b. Sketch a graph of s .

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Chapter 1 | Functions and Graphs

Solution a.

0.5

1.5

2.5

s ( t )

100

Table 1.3 Height s as a Function of Time t Since the ball hits the ground when t =2.5, the domain of this function is the interval [0, 2.5]. b.

Note that for this function and the function f ( x ) =−4 x +2 graphed in Figure 1.9 , the values of f ( x ) are getting smaller as x is getting larger. A function with this property is said to be decreasing. On the other hand, for the function f ( x ) = x +3+1 graphed in Figure 1.10 , the values of f ( x ) are getting larger as the values of x are getting larger. A function with this property is said to be increasing. It is important to note, however, that a function can be increasing on some interval or intervals and decreasing over a different interval or intervals. For example, using our temperature function in Figure 1.6 , we can see that the function is decreasing on the interval (0, 4), increasing on the interval (4, 14), and then decreasing on the interval (14, 23). We make the idea of a function increasing or decreasing over a particular interval more precise in the next definition.

Definition We say that a function f is increasing on the interval I if for all x 1 , x 2 ∈ I , f ( x 1 ) ≤ f ( x 2 )when x 1 < x 2 . We say f is strictly increasing on the interval I if for all x 1 , x 2 ∈ I ,

Chapter 1 | Functions and Graphs

f ( x 1 ) < f ( x 2 )when x 1 < x 2 . We say that a function f is decreasing on the interval I if for all x 1 , x 2 ∈ I , f ( x 1 ) ≥ f ( x 2 ) if x 1 < x 2 . We say that a function f is strictly decreasing on the interval I if for all x 1 , x 2 ∈ I , f ( x 1 ) > f ( x 2 ) if x 1 < x 2 .

For example, the function f ( x ) =3 x is increasing on the interval (−∞, ∞) because 3 x 1 <3 x 2 whenever x 1 < x 2 . On the other hand, the function f ( x ) =− x 3 is decreasing on the interval (−∞, ∞) because − x 1 3 > − x 2 3 whenever x 1 < x 2 ( Figure 1.11 ).

Figure 1.11 (a) The function f ( x ) =3 x is increasing on the interval (−∞, ∞). (b) The function f ( x ) =− x 3 is decreasing on the interval (−∞, ∞).

Combining Functions Now that we have reviewed the basic characteristics of functions, we can see what happens to these properties when we combine functions in different ways, using basic mathematical operations to create new functions. For example, if the cost for a company to manufacture x items is described by the function C ( x ) and the revenue created by the sale of x items is described by the function R ( x ), then the profit on the manufacture and sale of x items is defined as P ( x ) = R ( x )− C ( x ). Using the difference between two functions, we created a new function. Alternatively, we can create a new function by composing two functions. For example, given the functions f ( x ) = x 2 and g ( x ) =3 x +1, the composite function f ∘ g is defined such that ⎛ ⎝ f ∘ g ⎞ ⎠ ( x ) = f ⎛ ⎝ g ( x ) ⎞ ⎠ = ⎛ ⎝ g ( x ) ⎞ ⎠ 2 = (3 x +1) 2 .

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Chapter 1 | Functions and Graphs

The composite function g ∘ f is defined such that ⎛ ⎝ g ∘ f ⎞ ⎠ ( x ) = g ⎛ ⎝ f ( x ) ⎞

⎠ =3 f ( x )+1=3 x 2 +1.

Note that these two new functions are different from each other. Combining Functions with Mathematical Operators To combine functions using mathematical operators, we simply write the functions with the operator and simplify. Given two functions f and g , we can define four new functions: ⎛ ⎝ f + g ⎞ ⎠ ( x ) = f ( x )+ g ( x ) Sum ⎛ ⎝ f − g ⎞ ⎠ ( x ) = f ( x )− g ( x ) Difference ⎛ ⎝ f · g ⎞ ⎠ ( x ) = f ( x ) g ( x ) Product ⎛ ⎝ f g ⎞ ⎠ ( x ) = f ( x ) g ( x ) for g ( x ) ≠0 Quotient

Example 1.6 Combining Functions Using Mathematical Operations

Given the functions f ( x ) =2 x −3 and g ( x ) = x 2 −1, find each of the following functions and state its domain.

a. ( f + g )( x ) b. ( f − g )( x ) c. ( f · g )( x ) d. ⎛ ⎝ f g ⎞ ⎠ ( x )

Solution a. ⎛

⎝ f + g ⎞ ⎠ ( x ) = (2 x −3)+( x 2 −1) = x 2 +2 x −4. The domain of this function is the interval (−∞, ∞). b. ⎛ ⎝ f − g ⎞ ⎠ ( x ) = (2 x −3)−( x 2 −1) =− x 2 +2 x −2. The domain of this function is the interval (−∞, ∞). c. ⎛ ⎝ f · g ⎞ ⎠ ( x ) = (2 x −3)( x 2 −1) =2 x 3 −3 x 2 −2 x +3. The domain of this function is the interval (−∞, ∞). d. ⎛ ⎝ f g ⎞ ⎠ ( x ) = 2 x −3 x 2 −1 . The domain of this function is { x | x ≠±1}.

For f ( x ) = x 2 +3 and g ( x ) =2 x −5, find ⎛ ⎝ f / g ⎞

1.4

⎠ ( x ) and state its domain.

Function Composition When we compose functions, we take a function of a function. For example, suppose the temperature T on a given day is described as a function of time t (measured in hours after midnight) as in Table 1.1 . Suppose the cost C , to heat or cool a building for 1 hour, can be described as a function of the temperature T . Combining these two functions, we can describe

Chapter 1 | Functions and Graphs

the cost of heating or cooling a building as a function of time by evaluating C ⎛ ⎝ T ( t ) ⎞ ⎠ . We have defined a new function, denoted C ∘ T , which is defined such that ( C ∘ T )( t ) = C ( T ( t )) for all t in the domain of T . This new function is called a composite function. We note that since cost is a function of temperature and temperature is a function of time, it makes sense to define this new function ( C ∘ T )( t ). It does not make sense to consider ( T ∘ C )( t ), because temperature is not a function of cost. Definition Consider the function f with domain A and range B , and the function g with domain D and range E . If B is a subset of D , then the composite function ( g ∘ f )( x ) is the function with domain A such that (1.1) ⎛ ⎝ g ∘ f ⎞ ⎠ ( x ) = g ⎛ ⎝ f ( x ) ⎞ ⎠ . A composite function g ∘ f can be viewed in two steps. First, the function f maps each input x in the domain of f to its output f ( x ) in the range of f . Second, since the range of f is a subset of the domain of g , the output f ( x ) is an element in the domain of g , and therefore it is mapped to an output g ⎛ ⎝ f ( x ) ⎞ ⎠ in the range of g . In Figure 1.12 , we see a visual image of a composite function.

Figure 1.12 For the composite function g ∘ f , we have ⎛ ⎝ g ∘ f ⎞ ⎠ (1) =4, ⎛ ⎝ g ∘ f ⎞ ⎠ (2) =5, and ⎛ ⎝ g ∘ f ⎞ ⎠ (3) =4.

Example 1.7 Compositions of Functions Defined by Formulas

Consider the functions f ( x ) = x 2 +1 and g ( x ) =1/ x . a. Find ( g ∘ f )( x ) and state its domain and range. b. Evaluate ( g ∘ f )(4), ( g ∘ f )(−1/2). c. Find ( f ∘ g )( x ) and state its domain and range. d. Evaluate ( f ∘ g )(4), ( f ∘ g )(−1/2).

Solution a. We can find the formula for ( g ∘ f )( x ) in two different ways. We could write ( g ∘ f )( x ) = g ( f ( x )) = g ( x 2 +1) = 1 x 2 +1 .

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