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Calculus Volume 1 SENIOR CONTRIBUTING AUTHORS GILBERT STRANG, MASSACHUSETTS INSTITUTE OF TECHNOLOGY EDWIN “JED” HERMAN, UNIVERSITY OF WISCONSIN-STEVENS POINT
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Contents Preface 1 Functions and Graphs 71 Introduction 7 1.1 Review of Functions 8 1.2 Basic Classes of Functions 30 1.3 Trigonometric Functions 53 1.4 Inverse Functions 65 1.5 Exponential and Logarithmic Functions 80 Chapter Review 99 Limits 1052 Introduction 105 2.1 A Preview of Calculus 106 2.2 The Limit of a Function 116 2.3 The Limit Laws 140 2.4 Continuity 155 2.5 The Precise Definition of a Limit 169 Chapter Review 182 Derivatives 1873 Introduction 187 3.1 Defining the Derivative 188 3.2 The Derivative as a Function 203 3.3 Differentiation Rules 216 3.4 Derivatives as Rates of Change 230 3.5 Derivatives of Trigonometric Functions 241 3.6 The Chain Rule 250 3.7 Derivatives of Inverse Functions 259 3.8 Implicit Differentiation 268 3.9 Derivatives of Exponential and Logarithmic Functions 275 Chapter Review 288 Applications of Derivatives 2954 Introduction 295 4.1 Related Rates 296 4.2 Linear Approximations and Differentials 308 4.3 Maxima and Minima 317 4.4 The Mean Value Theorem 329 4.5 Derivatives and the Shape of a Graph 339 4.6 Limits at Infinity and Asymptotes 354 4.7 Applied Optimization Problems 381 4.8 L’Hôpital’s Rule 394 4.9 Newton’s Method 409
4.10 Antiderivatives 419 Chapter Review 431 Integration 4375 Introduction 437 5.1 Approximating Areas 438 5.2 The Definite Integral 456 5.3 The Fundamental Theorem of Calculus 472 5.4 Integration Formulas and the Net Change Theorem 488 5.5 Substitution 506 5.6 Integrals Involving Exponential and Logarithmic Functions 516 5.7 Integrals Resulting in Inverse Trigonometric Functions 526 Chapter Review 533 Applications of Integration 5396 Introduction 539 6.1 Areas between Curves 540 6.2 Determining Volumes by Slicing 551 6.3 Volumes of Revolution: Cylindrical Shells 569 6.4 Arc Length of a Curve and Surface Area 582 6.5 Physical Applications 594 6.6 Moments and Centers of Mass 610 6.7 Integrals, Exponential Functions, and Logarithms 626 6.8 Exponential Growth and Decay 636 6.9 Calculus of the Hyperbolic Functions 646 Chapter Review 657 Table of Integrals 665A Table of Derivatives 671B Review of Pre-Calculus 673C Answer Key 677 Index 759 Access for free at openstax.org
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 and empowering students and instructors to succeed. About OpenStax's resources CustomizationCalculus 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 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. Volume 1 • Chapter 1: Functions and Graphs • Chapter 2: Limits • Chapter 3: Derivatives • Chapter 4: Applications of Derivatives Preface 1
• 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 Volume 3 • 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 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 Answer Key at the back of the book. Early or late transcendentalsCalculus 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. Comprehensive art program Our art program is designed to enhance students’ understanding of concepts through clear and effective illustrations, diagrams, and photographs. 2 Preface Access for free at openstax.org
Answers to Questions in the Book Answers to Examples are provided just below the question in the book. All Checkpoint answers are provided in the Answer Key. Odd-numbered Exercises and Review Exercises questions are provided to students in the Answer Key as well as the Student Solution Guide on the Student Resources page. Even-numbered answers are provided only to instructors in the Instructor Answer Guide via the Instructor Resources page. Additional resources Student and instructor resources We’ve compiled additional resources for both students and instructors, including Getting Started Guides, an instructor solution manual, and PowerPoint slides. Instructor resources require a verified instructor account, which can be requested on your OpenStax.org log-in. Take advantage of these resources to supplement your OpenStax book. Community Hubs OpenStax partners with the Institute for the Study of Knowledge Management in Education (ISKME) to offer Community Hubs on OER Commons – a platform for instructors to share community-created resources that support OpenStax books, free of charge. Through our Community Hubs, instructors can upload their own materials or download resources to use in their own courses, including additional ancillaries, teaching material, multimedia, and relevant course content. We encourage instructors to join the hubs for the subjects most relevant to your teaching and research as an opportunity both to enrich your courses and to engage with other faculty. To reach the Community Hubs, visit www.oercommons.org/hubs/OpenStax. Partner resources OpenStax Partners are our allies in the mission to make high-quality learning materials affordable and accessible to students and instructors everywhere. Their tools integrate seamlessly with our OpenStax titles at a low cost. To access the partner resources for your text, visit your book page on OpenStax.org. 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 Preface 3
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 Matthew Cathey, Wofford College Michael Cohen, Hofstra University 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 Jerry Jared, The Blue Ridge School Peter Jipsen, Chapman University David Johnson, Lehigh 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 Kasso Okoudjou, University of Maryland College Park Peter Olszewski, Penn State Erie, The Behrend College Steven Purtee, Valencia College Alice Ramos, Bethel College Doug Shaw, University of Northern Iowa Hussain Elalaoui-Talibi, Tuskegee University Jeffrey Taub, Maine Maritime Academy William Thistleton, SUNY Polytechnic Institute 4 Preface Access for free at openstax.org
A. David Trubatch, Montclair State University Carmen Wright, Jackson State University Zhenbu Zhang, Jackson State University Preface 5
6 Preface Access for free at openstax.org
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. 1 FUNCTIONS AND GRAPHS 1 • Introduction 7
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 and a set with elements that are ordered pairs where is an element of and is an element of is a relation from to A relation from to 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 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 where the domain is the set of all real numbers and the rule is to square the input. Then, the input is assigned to the output 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 with domain we often use to denote the input and to denote the output associated with When doing so, we refer to as the independent variable and as the dependent variable, because it depends on Using function notation, we write and we read this equation as equals of For the squaring function described earlier, we write The concept of a function can be visualized using Figure 1.2, Figure 1.3, and Figure 1.4. Figure 1.2 A function can be visualized as an input/output device. 8 1 • Functions and Graphs Access for free at openstax.org
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 has a domain of and a range of The independent variable is and the dependent variable is MEDIA 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 in the coordinate plane where The graph of a function is the set of all these points. For example, consider the function where the domain is the set and the rule is In Figure 1.5, we plot a graph of this function. Figure 1.5 Here we see a graph of the function with domain and rule The graph consists of the points for all in the domain. Every function has a domain. However, sometimes a function is described by an equation, as in with no specific domain given. In this case, the domain is taken to be the set of all real numbers, , for which is a real number. For example, since any real number can be squared, if no other domain is specified, we consider the domain of to be the set of all real numbers. On the other hand, the square root function gives a real output only if is nonnegative. Therefore, the domain of the function is the set of nonnegative real numbers, sometimes called the natural domain. For the functions and 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 1.1 • Review of Functions 9
We read this as the set of real numbers such that 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 A set such as this, which contains all numbers greater than and less than can also be denoted using the interval notation Therefore, The numbers and 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 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 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 ∞ 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 ∞ 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 ∞ ∞ 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 with a domain that is the set of all real numbers such that for and for We denote this function by writing When evaluating this function for an input the equation to use depends on whether or For example, since we use the fact that for and see that On the other hand, for we use the fact that for and see that EXAMPLE 1.1 Evaluating Functions For the function evaluate a. b. c. Solution Substitute the given value for x in the formula for a. 10 1 • Functions and Graphs Access for free at openstax.org
b. c. 1.1 For evaluate and EXAMPLE 1.2 Finding Domain and Range For each of the following functions, determine the i. domain and ii. range. a. b. c. Solution a. Consider i. Since is a real number for any real number the domain of is the interval ∞ ∞ ii. Since we know Therefore, the range must be a subset of To show that every element in this set is in the range, we need to show that for a given in that set, there is a real number such that Solving this equation for we see that we need such that This equation is satisfied as long as there exists a real number such that Since the square root is well-defined. We conclude that for and therefore the range is b. Consider i. To find the domain of we need the expression Solving this inequality, we conclude that the domain is ii. To find the range of we note that since Therefore, the range of must be a subset of the set To show that every element in this set is in the range of we need to show that for all in this set, there exists a real number in the domain such that Let Then, if and only if Solving this equation for we see that must solve the equation Since such an could exist. Squaring both sides of this equation, we have Therefore, we need which implies We just need to verify that is in the domain of Since the domain of consists of all real numbers greater 1.1 • Review of Functions 11
than or equal to and there does exist an in the domain of We conclude that the range of is c. Consider i. Since is defined when the denominator is nonzero, the domain is ii. To find the range of we need to find the values of such that there exists a real number in the domain with the property that Solving this equation for we find that Therefore, as long as there exists a real number in the domain such that Thus, the range is 1.2 Find the domain and range for 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 be the time after midnight, measured in hours, and the output variable be the temperature hours after midnight, measured in degrees Fahrenheit. We record our data in Table 1.1. Hours after Midnight Temperature Hours after Midnight Temperature 0 58 12 84 1 54 13 85 2 53 14 85 3 52 15 83 4 52 16 82 5 55 17 80 Table 1.1 Temperature as a Function of Time of Day 12 1 • Functions and Graphs Access for free at openstax.org
Hours after Midnight Temperature Hours after Midnight Temperature 6 60 18 77 7 64 19 74 8 72 20 69 9 75 21 65 10 78 22 60 11 80 23 58 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 described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in Table 1.1 can give us a better idea of their fluctuation throughout the day. Figure 1.6 shows the plot of the temperature function. 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. 1.1 • Review of Functions 13
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 is given by the formula When an object is thrown upward from the ground with an initial velocity ft/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula When dollars are invested in an account at an annual interest rate compounded continuously, the amount of money after years is given by the formula Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form. Given an algebraic formula for a function the graph of is the set of points where is in the domain of and 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 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 where are called the zeros of a function. For example, the zeros of are The zeros determine where the graph of intersects the -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 -intercept, if it exists. The -intercept is given by Since a function has exactly one output for each input, the graph of a function can have, at most, one -intercept. If is in the domain of a function then has exactly one -intercept. If is not in the domain of then has no -intercept. Similarly, for any real number if is in the domain of there is exactly one output and the line intersects the graph of exactly once. On the other hand, if is not in the domain of is not defined and the line does not intersect the graph of This property is summarized in the vertical line test. Rule: Vertical Line Test Given a function every vertical line that may be drawn intersects the graph of 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). 14 1 • Functions and Graphs Access for free at openstax.org
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 Finding Zeros and -Intercepts of a Function Consider the function a. Find all zeros of b. Find the -intercept (if any). c. Sketch a graph of Solution a. To find the zeros, solve We discover that has one zero at b. The -intercept is given by c. Given that is a linear function of the form that passes through the points and we can sketch the graph of (Figure 1.9). Figure 1.9 The function is a line with -intercept and -intercept EXAMPLE 1.4 Using Zeros and -Intercepts to Sketch a Graph Consider the function 1.1 • Review of Functions 15
a. Find all zeros of b. Find the -intercept (if any). c. Sketch a graph of Solution a. To find the zeros, solve This equation implies Since for all this equation has no solutions, and therefore has no zeros. b. The -intercept is given by c. To graph this function, we make a table of values. Since we need we need to choose values of We choose values that make the square-root function easy to evaluate. Table 1.2 Making use of the table and knowing that, since the function is a square root, the graph of should be similar to the graph of we sketch the graph (Figure 1.10). Figure 1.10 The graph of has a -intercept but no -intercepts. 1.3 Find the zeros of EXAMPLE 1.5 Finding the Height of a Free-Falling Object If a ball is dropped from a height of ft, its height at time is given by the function where is measured in feet and is measured in seconds. The domain is restricted to the interval where is the time when the ball is dropped and is the time when the ball hits the ground. a. Create a table showing the height when Using the data from the table, determine the domain for this function. That is, find the time when the ball hits the ground. b. Sketch a graph of Solution a. Table 1.3 Height as a Function of Time 16 1 • Functions and Graphs Access for free at openstax.org
Since the ball hits the ground when the domain of this function is the interval b. Note that for this function and the function graphed in Figure 1.9, the values of are getting smaller as is getting larger. A function with this property is said to be decreasing. On the other hand, for the function graphed in Figure 1.10, the values of are getting larger as the values of 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 increasing on the interval and then decreasing on the interval 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 is increasing on the interval if for all We say is strictly increasing on the interval if for all We say that a function is decreasing on the interval if for all We say that a function is strictly decreasing on the interval if for all For example, the function is increasing on the interval ∞ ∞ because whenever On the other hand, the function is strictly decreasing on the interval ∞ ∞ because whenever (Figure 1.11). 1.1 • Review of Functions 17
Figure 1.11 (a) The function is increasing on the interval ∞ ∞ (b) The function 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 items is described by the function and the revenue created by the sale of items is described by the function then the profit on the manufacture and sale of items is defined as 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 and the composite function is defined such that The composite function is defined such that 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 and we can define four new functions: 18 1 • Functions and Graphs Access for free at openstax.org
EXAMPLE 1.6 Combining Functions Using Mathematical Operations Given the functions and find each of the following functions and state its domain. a. b. c. d. Solution a. The domain of this function is the interval ∞ ∞ b. The domain of this function is the interval ∞ ∞ c. The domain of this function is the interval ∞ ∞ d. The domain of this function is 1.4 For and find and state its domain. Function Composition When we compose functions, we take a function of a function. For example, suppose the temperature on a given day is described as a function of time (measured in hours after midnight) as in Table 1.1. Suppose the cost to heat or cool a building for 1 hour, can be described as a function of the temperature Combining these two functions, we can describe the cost of heating or cooling a building as a function of time by evaluating We have defined a new function, denoted which is defined such that for all in the domain of 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 It does not make sense to consider because temperature is not a function of cost. Definition Consider the function with domain and range and the function with domain and range If is a subset of then the composite function is the function with domain such that (1.1) A composite function can be viewed in two steps. First, the function maps each input in the domain of to its output in the range of Second, since the range of is a subset of the domain of the output is an element in the domain of and therefore it is mapped to an output in the range of In Figure 1.12, we see a visual image of a composite function. Figure 1.12 For the composite function we have and 1.1 • Review of Functions 19
EXAMPLE 1.7 Compositions of Functions Defined by Formulas Consider the functions and a. Find and state its domain and range. b. Evaluate c. Find and state its domain and range. d. Evaluate Solution a. We can find the formula for in two different ways. We could write Alternatively, we could write Since for all real numbers the domain of is the set of all real numbers. Since the range is, at most, the interval To show that the range is this entire interval, we let and solve this equation for to show that for all in the interval there exists a real number such that Solving this equation for we see that which implies that If is in the interval the expression under the radical is nonnegative, and therefore there exists a real number such that We conclude that the range of is the interval b. c. We can find a formula for in two ways. First, we could write Alternatively, we could write The domain of is the set of all real numbers such that To find the range of we need to find all values for which there exists a real number such that Solving this equation for we see that we need to satisfy which simplifies to 20 1 • Functions and Graphs Access for free at openstax.org
Finally, we obtain Since is a real number if and only if the range of is the set d. In Example 1.7, we can see that This tells us, in general terms, that the order in which we compose functions matters. 1.5 Let Let Find EXAMPLE 1.8 Composition of Functions Defined by Tables Consider the functions and described by Table 1.4 and Table 1.5. 0 1 2 3 4 0 4 2 4 0 4 Table 1.4 0 2 4 1 0 3 0 5 Table 1.5 a. Evaluate b. State the domain and range of c. Evaluate d. State the domain and range of Solution a. b. The domain of is the set Since the range of is the set the range of is the set c. d. The domain of is the set Since the range of is the set the range of is the set EXAMPLE 1.9 Application Involving a Composite Function A store is advertising a sale of off all merchandise. Caroline has a coupon that entitles her to an additional off any item, including sale merchandise. If Caroline decides to purchase an item with an original price of dollars, how much will she end up paying if she applies her coupon to the sale price? Solve this problem by using a composite 1.1 • Review of Functions 21
function. Solution Since the sale price is off the original price, if an item is dollars, its sale price is given by Since the coupon entitles an individual to off the price of any item, if an item is dollars, the price, after applying the coupon, is given by Therefore, if the price is originally dollars, its sale price will be and then its final price after the coupon will be 1.6 If items are on sale for off their original price, and a customer has a coupon for an additional off, what will be the final price for an item that is originally dollars, after applying the coupon to the sale price? Symmetry of Functions The graphs of certain functions have symmetry properties that help us understand the function and the shape of its graph. For example, consider the function shown in Figure 1.13(a). If we take the part of the curve that lies to the right of the y-axis and flip it over the y-axis, it lays exactly on top of the curve to the left of the y-axis. In this case, we say the function has symmetry about the y-axis. On the other hand, consider the function shown in Figure 1.13(b). If we take the graph and rotate it about the origin, the new graph will look exactly the same. In this case, we say the function has symmetry about the origin. Figure 1.13 (a) A graph that is symmetric about the -axis. (b) A graph that is symmetric about the origin. If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function has symmetry? Looking at Figure 1.13 again, we see that since is symmetric about the -axis, if the point is on the graph, the point is on the graph. In other words, If a function has this property, we say is an even function, which has symmetry about the y-axis. For example, is even because In contrast, looking at Figure 1.13 again, if a function is symmetric about the origin, then whenever the point is on the graph, the point is also on the graph. In other words, If has this property, we say is an odd function, which has symmetry about the origin. For example, is odd because Definition If for all in the domain of then is an even function. An even function is symmetric about the 22 1 • Functions and Graphs Access for free at openstax.org
y-axis. If for all in the domain of then is an odd function. An odd function is symmetric about the origin. EXAMPLE 1.10 Even and Odd Functions Determine whether each of the following functions is even, odd, or neither. a. b. c. Solution To determine whether a function is even or odd, we evaluate and compare it to f( x) and a. Therefore, is even. b. Now, Furthermore, noting that we see that Therefore, is neither even nor odd. c. Therefore, is odd. 1.7 Determine whether is even, odd, or neither. One symmetric function that arises frequently is the absolute value function, written as The absolute value function is defined as (1.2) Some students describe this function by stating that it “makes everything positive.” By the definition of the absolute value function, we see that if then and if then However, for Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if the output We conclude that the range of the absolute value function is In Figure 1.14, we see that the absolute value function is symmetric about the y-axis and is therefore an even function. Figure 1.14 The graph of is symmetric about the -axis. 1.1 • Review of Functions 23
EXAMPLE 1.11 Working with the Absolute Value Function Find the domain and range of the function Solution Since the absolute value function is defined for all real numbers, the domain of this function is ∞ ∞ Since for all the function Therefore, the range is, at most, the set To see that the range is, in fact, this whole set, we need to show that for there exists a real number such that A real number satisfies this equation as long as Since we know and thus the right-hand side of the equation is nonnegative, so it is possible that there is a solution. Furthermore, Therefore, we see there are two solutions: The range of this function is 1.8 For the function find the domain and range. SECTION 1.1 EXERCISESFor the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. 1 . −3 9 1 1 −2 4 2 4 −1 1 3 9 0 0 2 . −3 −2 1 1 −2 −8 2 8 −1 −1 3 −2 0 0 3 . 1 −3 1 1 2 −2 2 2 3 −1 3 3 0 0 24 1 • Functions and Graphs Access for free at openstax.org
4 . 1 1 5 1 2 1 6 1 3 1 7 1 4 1 5 . 3 3 15 1 5 2 21 2 8 1 33 3 10 0 6 . −7 11 1 −2 −2 5 3 4 −2 1 6 11 0 −1 For the following exercises, find the values for each function, if they exist, then simplify. a. b. c. d. e. f. 7 . 8 . 9 . 10. 11 . 12 . 13. For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. 14. 15 . 16 . 17. 18 . 19 . 20. 21 . For the following exercises, sketch the graph with the aid of the tables given: 22. −3 10 1 2 −2 5 2 5 −1 2 3 10 0 1 23 . −3 −15 1 −3 −2 −12 2 0 −1 −9 3 3 0 −6 24 . −3 1 −2 0 2 2 −1 3 0 1 1.1 • Review of Functions 25
25. −3 6 1 2 −2 4 2 4 −1 2 3 6 0 0 26 . −3 −9 1 −1 −2 −4 2 −4 −1 −1 3 −9 0 0 27 . −3 −27 1 1 −2 −8 2 8 −1 −1 3 27 0 0 For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph: a. Domain and range b. -intercept, if any (estimate where necessary) c. -intercept, if any (estimate where necessary) d. The intervals for which the function is increasing e. The intervals for which the function is decreasing f. The intervals for which the function is constant g. Symmetry about any axis and/or the origin h. Whether the function is even, odd, or neither 28. 29 . 30 . 31. 32 . 33 . 26 1 • Functions and Graphs Access for free at openstax.org
34. 35 . For the following exercises, for each pair of functions, find a. b. c. d. Determine the domain of each of these new functions. 36. 37 . 38 . 39. 40 . 41 . For the following exercises, for each pair of functions, find a. and b. Simplify the results. Find the domain of each of the results. 42. 43 . 44 . 45. 46. 47 . 1.1 • Review of Functions 27
48. 49 . The table below lists the NBA championship winners for the years 2001 to 2012. Year Winner 2001 LA Lakers 2002 LA Lakers 2003 San Antonio Spurs 2004 Detroit Pistons 2005 San Antonio Spurs 2006 Miami Heat 2007 San Antonio Spurs 2008 Boston Celtics 2009 LA Lakers 2010 LA Lakers 2011 Dallas Mavericks 2012 Miami Heat a. Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why not. b. Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why not. 50 . [T] The area of a square depends on the length of the side a. Write a function for the area of a square. b. Find and interpret c. Find the exact and the two-significant-digit approximation to the length of the sides of a square with area 56 square units. 28 1 • Functions and Graphs Access for free at openstax.org
51. [T] The volume of a cube depends on the length of the sides a. Write a function for the volume of a cube. b. Find and interpret 52 . [T] A rental car company rents cars for a flat fee of $20 and an hourly charge of $10.25. Therefore, the total cost to rent a car is a function of the hours the car is rented plus the flat fee. a. Write the formula for the function that models this situation. b. Find the total cost to rent a car for 2 days and 7 hours. c. Determine how long the car was rented if the bill is $430. 53 . [T] A vehicle has a 20-gal tank and gets 15 mpg. The number of miles N that can be driven depends on the amount of gas x in the tank. a. Write a formula that models this situation. b. Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) 3/4 of a tank of gas. c. Determine the domain and range of the function. d. Determine how many times the driver had to stop for gas if she has driven a total of 578 mi. 54. [T] The volume V of a sphere depends on the length of its radius as Because Earth is not a perfect sphere, we can use the mean radius when measuring from the center to its surface. The mean radius is the average distance from the physical center to the surface, based on a large number of samples. Find the volume of Earth with mean radius m. 55 . [T] A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by where t is time measured in hours since a circle of a 1-cm radius of the bacterium was put into the culture. a. Express the area of the bacteria as a function of time. b. Find the exact and approximate area of the bacterial culture in 3 hours. c. Express the circumference of the bacteria as a function of time. d. Find the exact and approximate circumference of the bacteria in 3 hours. 56 . [T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function where x is the number of U.S. dollars and is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is where x is the number of Euros and is the equivalent number of U.S. dollars. a. Find the composite function that converts directly from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process? b. Use (a) to determine how many U.S. dollars the tourist would get back at the end of her trip if she converted an extra $200 when she arrived in Paris. 1.1 • Review of Functions 29
57. [T] The manager at a skateboard shop pays his workers a monthly salary S of $750 plus a commission of $8.50 for each skateboard they sell. a. Write a function that models a worker’s monthly salary based on the number of skateboards x he or she sells. b. Find the monthly salary when a worker sells 25, 40, or 55 skateboards. c. Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of $1400. (Hint: Find the intersection of the function and the line 58 . [T] Use a graphing calculator to graph the half-circle Then, use the INTERCEPT feature to find the value of both the - and -intercepts. 1.2 Basic Classes of Functions Learning Objectives 1.2.1 Calculate the slope of a linear function and interpret its meaning. 1.2.2 Recognize the degree of a polynomial. 1.2.3 Find the roots of a quadratic polynomial. 1.2.4 Describe the graphs of basic odd and even polynomial functions. 1.2.5 Identify a rational function. 1.2.6 Describe the graphs of power and root functions. 1.2.7 Explain the difference between algebraic and transcendental functions. 1.2.8 Graph a piecewise-defined function. 1.2.9 Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position. We have studied the general characteristics of functions, so now let’s examine some specific classes of functions. We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with examples of piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form. Linear Functions and Slope The easiest type of function to consider is a linear function. Linear functions have the form where and are constants. In Figure 1.15, we see examples of linear functions when is positive, negative, and zero. Note that if the graph of the line rises as increases. In other words, is increasing on If the graph of the line falls as increases. In this case, is decreasing on If the line is horizontal. 30 1 • Functions and Graphs Access for free at openstax.org
Figure 1.15 These linear functions are increasing or decreasing on and one function is a horizontal line. As suggested by Figure 1.15, the graph of any linear function is a line. One of the distinguishing features of a line is its slope. The slope is the change in for each unit change in The slope measures both the steepness and the direction of a line. If the slope is positive, the line points upward when moving from left to right. If the slope is negative, the line points downward when moving from left to right. If the slope is zero, the line is horizontal. To calculate the slope of a line, we need to determine the ratio of the change in versus the change in To do so, we choose any two points and on the line and calculate In Figure 1.16, we see this ratio is independent of the points chosen. Figure 1.16 For any linear function, the slope is independent of the choice of points and on the line. Definition Consider line passing through points and Let and denote the changes in and respectively. The slope of the line is (1.3) We now examine the relationship between slope and the formula for a linear function. Consider the linear function given by the formula As discussed earlier, we know the graph of a linear function is given by a line. We can use our definition of slope to calculate the slope of this line. As shown, we can determine the slope by calculating for any points and on the line. Evaluating the function at we see that 1.2 • Basic Classes of Functions 31
is a point on this line. Evaluating this function at we see that is also a point on this line. Therefore, the slope of this line is We have shown that the coefficient is the slope of the line. We can conclude that the formula describes a line with slope Furthermore, because this line intersects the -axis at the point we see that the -intercept for this linear function is We conclude that the formula tells us the slope, and the -intercept, for this line. Since we often use the symbol to denote the slope of a line, we can write to denote the slope-intercept form of a linear function. Sometimes it is convenient to express a linear function in different ways. For example, suppose the graph of a linear function passes through the point and the slope of the line is Since any other point on the graph of must satisfy the equation this linear function can be expressed by writing We call this equation the point-slope equation for that linear function. Since every nonvertical line is the graph of a linear function, the points on a nonvertical line can be described using the slope-intercept or point-slope equations. However, a vertical line does not represent the graph of a function and cannot be expressed in either of these forms. Instead, a vertical line is described by the equation for some constant Since neither the slope-intercept form nor the point-slope form allows for vertical lines, we use the notation where are both not zero, to denote the standard form of a line. Definition Consider a line passing through the point with slope The equation (1.4) is the point-slope equation for that line. Consider a line with slope and -intercept The equation (1.5) is an equation for that line in slope-intercept form. The standard form of a line is given by the equation (1.6) where and are not both zero. This form is more general because it allows for a vertical line, EXAMPLE 1.12 Finding the Slope and Equations of Lines Consider the line passing through the points and as shown in Figure 1.17. 32 1 • Functions and Graphs Access for free at openstax.org
Figure 1.17 Finding the equation of a linear function with a graph that is a line between two given points. a. Find the slope of the line. b. Find an equation for this linear function in point-slope form. c. Find an equation for this linear function in slope-intercept form. Solution a. The slope of the line is b. To find an equation for the linear function in point-slope form, use the slope and choose any point on the line. If we choose the point we get the equation c. To find an equation for the linear function in slope-intercept form, solve the equation in part b. for When we do this, we get the equation 1.9 Consider the line passing through points and Find the slope of the line. Find an equation of that line in point-slope form. Find an equation of that line in slope-intercept form. EXAMPLE 1.13 A Linear Distance Function Jessica leaves her house at 5:50 a.m. and goes for a 9-mile run. She returns to her house at 7:08 a.m. Answer the following questions, assuming Jessica runs at a constant pace. a. Describe the distance (in miles) Jessica runs as a linear function of her run time (in minutes). b. Sketch a graph of c. Interpret the meaning of the slope. Solution a. At time Jessica is at her house, so At time minutes, Jessica has finished running mi, so The slope of the linear function is 1.2 • Basic Classes of Functions 33
The -intercept is so the equation for this linear function is b. To graph use the fact that the graph passes through the origin and has slope c. The slope describes the distance (in miles) Jessica runs per minute, or her average velocity. Polynomials A linear function is a special type of a more general class of functions: polynomials. A polynomial function is any function that can be written in the form (1.7) for some integer and constants where In the case when we allow for if the function is called the zero function. The value is called the degree of the polynomial; the constant is called the leading coefficient. A linear function of the form is a polynomial of degree 1 if and degree 0 if A polynomial of degree 0 is also called a constant function. A polynomial function of degree 2 is called a quadratic function. In particular, a quadratic function has the form where A polynomial function of degree is called a cubic function. Power Functions Some polynomial functions are power functions. A power function is any function of the form where and are any real numbers. The exponent in a power function can be any real number, but here we consider the case when the exponent is a positive integer. (We consider other cases later.) If the exponent is a positive integer, then is a polynomial. If is even, then is an even function because if is even. If is odd, then is an odd function because if is odd (Figure 1.18). 34 1 • Functions and Graphs Access for free at openstax.org
Figure 1.18 (a) For any even integer is an even function. (b) For any odd integer is an odd function. Behavior at Infinity To determine the behavior of a function as the inputs approach infinity, we look at the values as the inputs, become larger. For some functions, the values of approach a finite number. For example, for the function the values become closer and closer to zero for all values of as they get larger and larger. For this function, we say approaches two as goes to infinity,” and we write as ∞ The line is a horizontal asymptote for the function because the graph of the function gets closer to the line as gets larger. For other functions, the values may not approach a finite number but instead may become larger for all values of as they get larger. In that case, we say approaches infinity as approaches infinity,” and we write ∞ as∞ For example, for the function the outputs become larger as the inputs get larger. We can conclude that the function approaches infinity as approaches infinity, and we write ∞ as ∞ The behavior as ∞ and the meaning of ∞ as ∞ or ∞ can be defined similarly. We can describe what happens to the values of as ∞ and as ∞ as the end behavior of the function. To understand the end behavior for polynomial functions, we can focus on quadratic and cubic functions. The behavior for higher-degree polynomials can be analyzed similarly. Consider a quadratic function If the values ∞ as ∞ If the values as ∞ Since the graph of a quadratic function is a parabola, the parabola opens upward if the parabola opens downward if (See Figure 1.19(a).) Now consider a cubic function If then ∞ as ∞ and as If then as ∞ and ∞ as As we can see from both of these graphs, the leading term of the polynomial determines the end behavior. (See Figure 1.19(b).) 1.2 • Basic Classes of Functions 35
Figure 1.19 (a) For a quadratic function, if the leading coefficient the parabola opens upward. If the parabola opens downward. (b) For a cubic function if the leading coefficient the values ∞ as ∞ and the values as If the leading coefficient the opposite is true. Zeros of Polynomial Functions Another characteristic of the graph of a polynomial function is where it intersects the -axis. To determine where a function intersects the -axis, we need to solve the equation for x. In the case of the linear function the -intercept is given by solving the equation In this case, we see that the -intercept is given by In the case of a quadratic function, finding the -intercept(s) requires finding the zeros of a quadratic equation: In some cases, it is easy to factor the polynomial to find the zeros. If not, we make use of the quadratic formula. Rule: The Quadratic Formula Consider the quadratic equation where The solutions of this equation are given by the quadratic formula (1.8) If the discriminant this formula tells us there are two real numbers that satisfy the quadratic equation. If this formula tells us there is only one solution, and it is a real number. If no real numbers satisfy the quadratic equation. In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the -axis. In some instances, it is possible to find the -intercepts by factoring the polynomial to find its zeros. In other cases, it is impossible to calculate the exact values of the -intercepts. However, as we see later in the text, in cases such as this, we can use analytical tools to approximate (to a very high degree) where the -intercepts are located. Here we focus on the graphs of polynomials for which we can calculate their zeros explicitly. EXAMPLE 1.14 Graphing Polynomial Functions For the following functions a. and b., i. describe the behavior of as ∞ ii. find all zeros of and iii. sketch a 36 1 • Functions and Graphs Access for free at openstax.org
graph of a. b. Solution a. The function is a quadratic function. i. Because ∞ ii. To find the zeros of use the quadratic formula. The zeros are iii. To sketch the graph of use the information from your previous answers and combine it with the fact that the graph is a parabola opening downward. b. The function is a cubic function. i. Because ∞ ∞ As ii. To find the zeros of we need to factor the polynomial. First, when we factor out of all the terms, we find Then, when we factor the quadratic function we find Therefore, the zeros of are iii. Combining the results from parts i. and ii., draw a rough sketch of 1.2 • Basic Classes of Functions 37
1.10 Consider the quadratic function Find the zeros of Does the parabola open upward or downward? Mathematical Models A large variety of real-world situations can be described using mathematical models. A mathematical model is a method of simulating real-life situations with mathematical equations. Physicists, engineers, economists, and other researchers develop models by combining observation with quantitative data to develop equations, functions, graphs, and other mathematical tools to describe the behavior of various systems accurately. Models are useful because they help predict future outcomes. Examples of mathematical models include the study of population dynamics, investigations of weather patterns, and predictions of product sales. As an example, let’s consider a mathematical model that a company could use to describe its revenue for the sale of a particular item. The amount of revenue a company receives for the sale of items sold at a price of dollars per item is described by the equation The company is interested in how the sales change as the price of the item changes. Suppose the data in Table 1.6 show the number of units a company sells as a function of the price per item. Table 1.6 Number of Units Sold (in Thousands) as a Function of Price per Unit (in Dollars) In Figure 1.20, we see the graph the number of units sold (in thousands) as a function of price (in dollars). We note from the shape of the graph that the number of units sold is likely a linear function of price per item, and the data can be closely approximated by the linear function for where predicts the number of units sold 38 1 • Functions and Graphs Access for free at openstax.org
in thousands. Using this linear function, the revenue (in thousands of dollars) can be estimated by the quadratic function for In Example 1.15, we use this quadratic function to predict the amount of revenue the company receives depending on the price the company charges per item. Note that we cannot conclude definitively the actual number of units sold for values of for which no data are collected. However, given the other data values and the graph shown, it seems reasonable that the number of units sold (in thousands) if the price charged is dollars may be close to the values predicted by the linear function Figure 1.20 The data collected for the number of items sold as a function of price is roughly linear. We use the linear function to estimate this function. EXAMPLE 1.15 Maximizing Revenue A company is interested in predicting the amount of revenue it will receive depending on the price it charges for a particular item. Using the data from Table 1.6, the company arrives at the following quadratic function to model revenue (in thousands of dollars) as a function of price per item for a. Predict the revenue if the company sells the item at a price of and b. Find the zeros of this function and interpret the meaning of the zeros. c. Sketch a graph of d. Use the graph to determine the value of that maximizes revenue. Find the maximum revenue. Solution a. Evaluating the revenue function at and we can conclude that 1.2 • Basic Classes of Functions 39
b. The zeros of this function can be found by solving the equation When we factor the quadratic expression, we get The solutions to this equation are given by For these values of the revenue is zero. When the revenue is zero because the company is giving away its merchandise for free. When the revenue is zero because the price is too high, and no one will buy any items. c. Knowing the fact that the function is quadratic, we also know the graph is a parabola. Since the leading coefficient is negative, the parabola opens downward. One property of parabolas is that they are symmetric about the axis, so since the zeros are at and the parabola must be symmetric about the line halfway between them, or d. The function is a parabola with zeros at and and it is symmetric about the line so the maximum revenue occurs at a price of per item. At that price, the revenue is Algebraic Functions By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions. Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a rational function is any function of the form where and are polynomials. For example, are rational functions. A root function is a power function of the form where is a positive integer greater than one. For example, is the square-root function and is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, is an algebraic function. 40 1 • Functions and Graphs Access for free at openstax.org
EXAMPLE 1.16 Finding Domain and Range for Algebraic Functions For each of the following functions, find the domain and range. a. b. Solution a. It is not possible to divide by zero, so the domain is the set of real numbers such that To find the range, we need to find the values for which there exists a real number such that When we multiply both sides of this equation by we see that must satisfy the equation From this equation, we can see that must satisfy If this equation has no solution. On the other hand, as long as satisfies this equation. We can conclude that the range of is b. To find the domain of , we need to identify values of with . Note that when or . Solving is equivalent to solving . If this inequality is true, then both factors are positive or both are negative. Case 1: Both factors are positive. So, the expression is positive for all values in the interval . Case 2: Both factors are negative. There is no value of that can satisfy both inequalities. The domain of the function is . To identify the range, note that if , then . Therefore, . The range of is . 1.11 Find the domain and range for the function The root functions have defining characteristics depending on whether is odd or even. For all even integers the domain of is the interval ∞ For all odd integers the domain of is the set of all real numbers. Since for odd integers is an odd function if is odd. See the graphs of root functions for different values of in Figure 1.21. 1.2 • Basic Classes of Functions 41
Figure 1.21 (a) If is even, the domain of is ∞ (b) If is odd, the domain of is ∞ and the function is an odd function. EXAMPLE 1.17 Finding Domains for Algebraic Functions For each of the following functions, determine the domain of the function. a. b. c. d. Solution a. You cannot divide by zero, so the domain is the set of values such that Therefore, the domain is b. You need to determine the values of for which the denominator is zero. Since for all real numbers the denominator is never zero. Therefore, the domain is ∞ c. Since the square root of a negative number is not a real number, the domain is the set of values for which Therefore, the domain is d. The cube root is defined for all real numbers, so the domain is the interval 1.12 Find the domain for each of the following functions: and Transcendental Functions Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. Atrigonometric function relates the ratios of two sides of a right triangle. They are (We discuss trigonometric functions later in the chapter.) An exponential function is a function of the form where the base A logarithmic function is a function of the form for some constant where if and only if (We also discuss exponential and logarithmic functions later in the chapter.) EXAMPLE 1.18 Classifying Algebraic and Transcendental Functions Classify each of the following functions, a. through c., as algebraic or transcendental. 42 1 • Functions and Graphs Access for free at openstax.org
a. b. c. Solution a. Since this function involves basic algebraic operations only, it is an algebraic function. b. This function cannot be written as a formula that involves only basic algebraic operations, so it is transcendental. (Note that algebraic functions can only have powers that are rational numbers.) c. As in part b., this function cannot be written using a formula involving basic algebraic operations only; therefore, this function is transcendental. 1.13 Is an algebraic or a transcendental function? Piecewise-Defined Functions Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function. The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of Other piecewise-defined functions may be represented by completely different formulas, depending on the part of the domain in which a point falls. To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for and we need to pay special attention to what happens at when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at We examine this in the next example. EXAMPLE 1.19 Graphing a Piecewise-Defined Function Sketch a graph of the following piecewise-defined function: Solution Graph the linear function on the interval and graph the quadratic function on the interval∞ Since the value of the function at is given by the formula we see that To indicate this on the graph, we draw a closed circle at the point The value of the function is given by for all but not at To indicate this on the graph, we draw an open circle at 1.2 • Basic Classes of Functions 43
Figure 1.22 This piecewise-defined function is linear for and quadratic for 1.14 Sketch a graph of the function EXAMPLE 1.20 Parking Fees Described by a Piecewise-Defined Function In a big city, drivers are charged variable rates for parking in a parking garage. They are charged $10 for the first hour or any part of the first hour and an additional $2 for each hour or part thereof up to a maximum of $30 for the day. The parking garage is open from 6 a.m. to 12 midnight. a. Write a piecewise-defined function that describes the cost to park in the parking garage as a function of hours parked b. Sketch a graph of this function Solution a. Since the parking garage is open 18 hours each day, the domain for this function is The cost to park a car at this parking garage can be described piecewise by the function b. The graph of the function consists of several horizontal line segments. 44 1 • Functions and Graphs Access for free at openstax.org
1.15 The cost of mailing a letter is a function of the weight of the letter. Suppose the cost of mailing a letter is for the first ounce and for each additional ounce. Write a piecewise-defined function describing the cost as a function of the weight for where is measured in cents and is measured in ounces. Transformations of Functions We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. In the previous example, for instance, we subtracted 2 from the argument of the function to get the function This subtraction represents a shift of the function two units to the right. A shift, horizontally or vertically, is a type of transformation of a function. Other transformations include horizontal and vertical scalings, and reflections about the axes. A vertical shift of a function occurs if we add or subtract the same constant to each output For the graph of is a shift of the graph of up units, whereas the graph of is a shift of the graph of down units. For example, the graph of the function is the graph of shifted up units; the graph of the function is the graph of shifted down units (Figure 1.23). 1.2 • Basic Classes of Functions 45
Figure 1.23 (a) For the graph of is a vertical shift up units of the graph of (b) For the graph of is a vertical shift down units of the graph of A horizontal shift of a function occurs if we add or subtract the same constant to each input For the graph of is a shift of the graph of to the left units; the graph of is a shift of the graph of to the right units. Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let’s look at an example. Consider the function and evaluate this function at Since and the graph of is the graph of shifted left 3 units. Similarly, the graph of is the graph of shifted right units (Figure 1.24). Figure 1.24 (a) For the graph of is a horizontal shift left units of the graph of (b) For the graph of is a horizontal shift right units of the graph of A vertical scaling of a graph occurs if we multiply all outputs of a function by the same positive constant. For the graph of the function is the graph of scaled vertically by a factor of If the values of the outputs for the function are larger than the values of the outputs for the function therefore, the graph has been stretched vertically. If then the outputs of the function are smaller, so the graph has been compressed. For example, the graph of the function is the graph of stretched vertically by a factor of 3, whereas the graph of is the graph of compressed vertically by a factor of (Figure 1.25). 46 1 • Functions and Graphs Access for free at openstax.org
Figure 1.25 (a) If the graph of is a vertical stretch of the graph of (b) If the graph of is a vertical compression of the graph of The horizontal scaling of a function occurs if we multiply the inputs by the same positive constant. For the graph of the function is the graph of scaled horizontally by a factor of If the graph of is the graph of compressed horizontally. If the graph of is the graph of stretched horizontally. For example, consider the function and evaluate at Since the graph of is the graph of compressed horizontally. The graph of is a horizontal stretch of the graph of (Figure 1.26). Figure 1.26 (a) If the graph of is a horizontal compression of the graph of (b) If the graph of is a horizontal stretch of the graph of We have explored what happens to the graph of a function when we multiply by a constant to get a new function We have also discussed what happens to the graph of a function when we multiply the independent variable by to get a new function However, we have not addressed what happens to the graph of the function if the constant is negative. If we have a constant we can write c as a positive number multiplied by but, what kind of transformation do we get when we multiply the function or its argument by When we multiply all the outputs by we get a reflection about the -axis. When we multiply all inputs by we get a reflection about the -axis. For example, the graph of is the graph of reflected about the -axis. The graph of is the graph of reflected about the -axis (Figure 1.27). 1.2 • Basic Classes of Functions 47
Figure 1.27 (a) The graph of is the graph of reflected about the -axis. (b) The graph of is the graph of reflected about the -axis. If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function the graph of the related function can be obtained from the graph of by performing the transformations in the following order. 1. Horizontal shift of the graph of If shift left. If shift right. 2. Horizontal scaling of the graph of by a factor of If reflect the graph about the -axis. 3. Vertical scaling of the graph of by a factor of If reflect the graph about the -axis. 4. Vertical shift of the graph of If shift up. If shift down. We can summarize the different transformations and their related effects on the graph of a function in the following table. Transformation of Effect on the graph of Vertical shift up units Vertical shift down units Shift left by units Shift right by units Vertical stretch if vertical compression if Horizontal stretch if horizontal compression if Reflection about the -axis Reflection about the -axis Table 1.7 Transformations of Functions 48 1 • Functions and Graphs Access for free at openstax.org
EXAMPLE 1.21 Transforming a Function For each of the following functions, a. and b., sketch a graph by using a sequence of transformations of a well-known function. a. b. Solution a. Starting with the graph of shift units to the left, reflect about the -axis, and then shift down 3 units. Figure 1.28 The function can be viewed as a sequence of three transformations of the function b. Starting with the graph of reflect about the -axis, stretch the graph vertically by a factor of 3, and move up 1 unit. Figure 1.29 The function can be viewed as a sequence of three transformations of the function 1.2 • Basic Classes of Functions 49
1.16 Describe how the function can be graphed using the graph of and a sequence of transformations. SECTION 1.2 EXERCISESFor the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. 59. and 60 . and 61 . and 62. and 63 . and 64 . and 65. and 66 . and For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. 67. Slope passes through 68 . Slope passes through 69 . Slope passes through 70. Slope -intercept 71 . Passing through and 72 . Passing through and 73. -intercept and -intercept 74 . -intercept and -intercept For the following exercises, for each linear equation, a. give the slope and -intercept b, if any, and b. graph the line. 75. 76 . 77 . 78. 79 . 80 . 81. 82 . For the following exercises, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the -intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. 83. 84 . 85 . 86. 87 . For the following exercises, use the graph of to graph each transformed function 88. 89 . For the following exercises, use the graph of to graph each transformed function 90. 91 . 50 1 • Functions and Graphs Access for free at openstax.org
For the following exercises, use the graph of to graph each transformed function 92 . 93. For the following exercises, for each of the piecewise-defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph. 94. 95. 96. 97. For the following exercises, determine whether the statement is true or false. Explain why. 98. is a transcendental function. 99 . is an odd root function 100 . A logarithmic function is an algebraic function. 101. A function of the form where is a real valued constant, is an exponential function. 102 . The domain of an even root function is all real numbers. 103 . [T] A company purchases some computer equipment for $20,500. At the end of a 3-year period, the value of the equipment has decreased linearly to $12,300. a. Find a function that determines the value V of the equipment at the end of t years. b. Find and interpret the meaning of the - and -intercepts for this situation. c. What is the value of the equipment at the end of 5 years? d. When will the value of the equipment be $3000? 1.2 • Basic Classes of Functions 51
104. [T] Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion. a. Find a linear function S that estimates the total online holiday sales in the year t. b. Interpret the slope of the graph of S. c. Use part a. to predict the year when online shopping during Christmas will reach $60 billion. 105 . [T] A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $125 to set up a cupcake stand. The owner estimates that it costs $0.75 to make each cupcake. The owner is interested in determining the total cost as a function of number of cupcakes made. a. Find a linear function that relates cost C to x, the number of cupcakes made. b. Find the cost to bake 160 cupcakes. c. If the owner sells the cupcakes for $1.50 apiece, how many cupcakes does she need to sell to start making profit? (Hint: Use the INTERSECTION function on a calculator to find this number.) 106 . [T] A house purchased for $250,000 is expected to be worth twice its purchase price in 18 years. a. Find a linear function that models the price P of the house versus the number of years t since the original purchase. b. Interpret the slope of the graph of P. c. Find the price of the house 15 years from when it was originally purchased. 107. [T] A car was purchased for $26,000. The value of the car depreciates by $1500 per year. a. Find a linear function that models the value V of the car after t years. b. Find and interpret 108 . [T] A condominium in an upscale part of the city was purchased for $432,000. In 35 years it is worth $60,500. Find the rate of depreciation. 109 . [T] The total cost C to produce a certain item is modeled by the function where x is the number of items produced. Determine the cost to produce 175 items. 52 1 • Functions and Graphs Access for free at openstax.org
110. [T] A professor asks her class to report the amount of time t they spent writing two assignments. Most students report that it takes them about 45 minutes to type a four- page assignment and about 1.5 hours to type a nine-page assignment. a. Find the linear function that models this situation, where is the number of pages typed and t is the time in minutes. b. Use part a. to determine how many pages can be typed in 2 hours. c. Use part a. to determine how long it takes to type a 20-page assignment. 111 . [T] The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function where t is time in years and corresponds to the beginning of 2000. Use the model to predict the approximate percentage output at the beginning of 2015. 112 . [T] The admissions office at a public university estimates that 65% of the students offered admission to the class of 2019 will actually enroll. a. Find the linear function where is the number of students that actually enroll and is the number of all students offered admission to the class of 2019. b. If the university wants the 2019 freshman class size to be 1350, determine how many students should be offered admission. 1.3 Trigonometric Functions Learning Objectives 1.3.1 Convert angle measures between degrees and radians. 1.3.2 Recognize the triangular and circular definitions of the basic trigonometric functions. 1.3.3 Write the basic trigonometric identities. 1.3.4 Identify the graphs and periods of the trigonometric functions. 1.3.5 Describe the shift of a sine or cosine graph from the equation of the function. Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums. In fact, almost any repetitive, or cyclical, motion can be modeled by some combination of trigonometric functions. In this section, we define the six basic trigonometric functions and look at some of the main identities involving these functions. Radian Measure To use trigonometric functions, we first must understand how to measure the angles. Although we can use both radians and degrees, radians are a more natural measurement because they are related directly to the unit circle, a circle with radius 1. The radian measure of an angle is defined as follows. Given an angle let be the length of the corresponding arc on the unit circle (Figure 1.30). We say the angle corresponding to the arc of length 1 has radian measure 1. Figure 1.30 The radian measure of an angle is the arc length of the associated arc on the unit circle. 1.3 • Trigonometric Functions 53
Since an angle of corresponds to the circumference of a circle, or an arc of length we conclude that an angle with a degree measure of has a radian measure of Similarly, we see that is equivalent to radians. Table 1.8 shows the relationship between common degree and radian values. Degrees Radians Degrees Radians 0 0 120 30 135 45 150 60 180 90 Table 1.8 Common Angles Expressed in Degrees and Radians EXAMPLE 1.22 Converting between Radians and Degrees a. Express using radians. b. Express rad using degrees. Solution Use the fact that is equivalent to radians as a conversion factor: a. rad b. rad = 1.17 Express using radians. Express rad using degrees. The Six Basic Trigonometric Functions Trigonometric functions allow us to use angle measures, in radians or degrees, to find the coordinates of a point on any circle—not only on a unit circle—or to find an angle given a point on a circle. They also define the relationship among the sides and angles of a triangle. To define the trigonometric functions, first consider the unit circle centered at the origin and a point on the unit circle. Let be an angle with an initial side that lies along the positive -axis and with a terminal side that is the line segment An angle in this position is said to be in standard position (Figure 1.31). We can then define the values of the six trigonometric functions for in terms of the coordinates and 54 1 • Functions and Graphs Access for free at openstax.org
Figure 1.31 The angle is in standard position. The values of the trigonometric functions for are defined in terms of the coordinates and Definition Let be a point on the unit circle centered at the origin Let be an angle with an initial side along the positive -axis and a terminal side given by the line segment The trigonometric functions are then defined as (1.9) If and are undefined. If then and are undefined. We can see that for a point on a circle of radius with a corresponding angle the coordinates and satisfy The values of the other trigonometric functions can be expressed in terms of and (Figure 1.32). Figure 1.32 For a point on a circle of radius the coordinates and satisfy and Table 1.9 shows the values of sine and cosine at the major angles in the first quadrant. From this table, we can determine the values of sine and cosine at the corresponding angles in the other quadrants. The values of the other trigonometric functions are calculated easily from the values of and 1.3 • Trigonometric Functions 55
Table 1.9 Values of and at Major Angles in the First Quadrant EXAMPLE 1.23 Evaluating Trigonometric Functions Evaluate each of the following expressions. a. b. c. Solution a. On the unit circle, the angle corresponds to the point Therefore, b. An angle corresponds to a revolution in the negative direction, as shown. Therefore, 56 1 • Functions and Graphs Access for free at openstax.org
c. An angle Therefore, this angle corresponds to more than one revolution, as shown. Knowing the fact that an angle of corresponds to the point we can conclude that 1.18 Evaluate and As mentioned earlier, the ratios of the side lengths of a right triangle can be expressed in terms of the trigonometric functions evaluated at either of the acute angles of the triangle. Let be one of the acute angles. Let be the length of the adjacent leg, be the length of the opposite leg, and be the length of the hypotenuse. By inscribing the triangle into a circle of radius as shown in Figure 1.33, we see that and satisfy the following relationships with Figure 1.33 By inscribing a right triangle in a circle, we can express the ratios of the side lengths in terms of the trigonometric functions evaluated at 1.3 • Trigonometric Functions 57
EXAMPLE 1.24 Constructing a Wooden Ramp A wooden ramp is to be built with one end on the ground and the other end at the top of a short staircase. If the top of the staircase is ft from the ground and the angle between the ground and the ramp is to be how long does the ramp need to be? Solution Let denote the length of the ramp. In the following image, we see that needs to satisfy the equation Solving this equation for we see that ft. 1.19 A house painter wants to lean a -ft ladder against a house. If the angle between the base of the ladder and the ground is to be how far from the house should she place the base of the ladder? Trigonometric Identities A trigonometric identity is an equation involving trigonometric functions that is true for all angles for which the functions are defined. We can use the identities to help us solve or simplify equations. The main trigonometric identities are listed next. Rule: Trigonometric Identities Reciprocal identities Pythagorean identities Addition and subtraction formulas Double-angle formulas EXAMPLE 1.25 Solving Trigonometric Equations For each of the following equations, use a trigonometric identity to find all solutions. a. b. Solution a. Using the double-angle formula for we see that is a solution of if and only if 58 1 • Functions and Graphs Access for free at openstax.org
which is true if and only if To solve this equation, it is important to note that we need to factor the left-hand side and not divide both sides of the equation by The problem with dividing by is that it is possible that is zero. In fact, if we did divide both sides of the equation by we would miss some of the solutions of the original equation. Factoring the left-hand side of the equation, we see that is a solution of this equation if and only if Since when and when we conclude that the set of solutions to this equation is b. Using the double-angle formula for and the reciprocal identity for the equation can be written as To solve this equation, we multiply both sides by to eliminate the denominator, and say that if satisfies this equation, then satisfies the equation However, we need to be a little careful here. Even if satisfies this new equation, it may not satisfy the original equation because, to satisfy the original equation, we would need to be able to divide both sides of the equation by However, if we cannot divide both sides of the equation by Therefore, it is possible that we may arrive at extraneous solutions. So, at the end, it is important to check for extraneous solutions. Returning to the equation, it is important that we factor out of both terms on the left-hand side instead of dividing both sides of the equation by Factoring the left-hand side of the equation, we can rewrite this equation as Therefore, the solutions are given by the angles such that or The solutions of the first equation are The solutions of the second equation are After checking for extraneous solutions, the set of solutions to the equation is 1.20 Find all solutions to the equation EXAMPLE 1.26 Proving a Trigonometric Identity Prove the trigonometric identity 1.3 • Trigonometric Functions 59
Solution We start with the identity Dividing both sides of this equation by we obtain Since and we conclude that 1.21 Prove the trigonometric identity Graphs and Periods of the Trigonometric Functions We have seen that as we travel around the unit circle, the values of the trigonometric functions repeat. We can see this pattern in the graphs of the functions. Let be a point on the unit circle and let be the corresponding angle Since the angle and correspond to the same point the values of the trigonometric functions at and at are the same. Consequently, the trigonometric functions are periodic functions. The period of a function is defined to be the smallest positive value such that for all values in the domain of The sine, cosine, secant, and cosecant functions have a period of Since the tangent and cotangent functions repeat on an interval of length their period is (Figure 1.34). Figure 1.34 The six trigonometric functions are periodic. Just as with algebraic functions, we can apply transformations to trigonometric functions. In particular, consider the following function: 60 1 • Functions and Graphs Access for free at openstax.org
(1.10) In Figure 1.35, the constant causes a horizontal or phase shift. The factor changes the period. This transformed sine function will have a period The factor results in a vertical stretch by a factor of We say is the “amplitude of ” The constant causes a vertical shift. Figure 1.35 A graph of a general cosine function. Notice in Figure 1.34 that the graph of is the graph of shifted to the left units. Therefore, we can write Similarly, we can view the graph of as the graph of shifted right units, and state that A shifted sine curve arises naturally when graphing the number of hours of daylight in a given location as a function of the day of the year. For example, suppose a city reports that June 21 is the longest day of the year with hours and December 21 is the shortest day of the year with hours. It can be shown that the function is a model for the number of hours of daylight as a function of day of the year (Figure 1.36). Figure 1.36 The hours of daylight as a function of day of the year can be modeled by a shifted sine curve. EXAMPLE 1.27 Sketching the Graph of a Transformed Sine Curve Sketch a graph of 1.3 • Trigonometric Functions 61
Solution This graph is a horizontal compression by a factor of 2, a phase shift to the right by π/4 units, followed by a vertical stretch by a factor of 3, and then a vertical shift by 1 unit. The period of is 1.22 Describe the relationship between the graph of and the graph of SECTION 1.3 EXERCISESFor the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of 113. 114 . 115 . 116. 117 . For the following exercises, convert each angle in radians to degrees. 118. 119 . 120 . 121. 122 . Evaluate the following functional values. 123. 124 . 125 . 126. 127 . 128 . For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, simplify to a fraction or round to three decimal places. 129 . 130 . 131. 132 . 133 . 62 1 • Functions and Graphs Access for free at openstax.org
134. For the following exercises, is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle with a terminal side that passes through point Rationalize denominators. 135. 136 . 137 . 138. For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. 139. 140 . 141 . 142. 143 . 144 . 145. 146 . For the following exercises, verify that each equation is an identity. 147. 148 . 149 . 150. 151 . 152 . 153. 154 . For the following exercises, solve the trigonometric equations on the interval 155. 156 . 157 . 158. 159 . 160 . 161. 162 . For the following exercises, each graph is of the form or where Write the equation of the graph. 163. 164 . 1.3 • Trigonometric Functions 63
165. 166 . For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function. 167. 168 . 169 . 170. 171 . 172 . 173. [T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of how many inches does it move? Approximate to the nearest whole inch. 174 . [T] Find the length of the arc intercepted by central angle in a circle of radius r. Round to the nearest hundredth. a. cm, rad b. cm, rad c. cm, d. cm, 175 . [T] As a point P moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, and is given by where is in radians and t is time. Find the angular speed for the given data. Round to the nearest thousandth. a. sec b. sec c. min d. min 176. [T] A total of 250,000 m2 of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular. a. Find the radius of the circular land area. b. If the land area is to form a sector of a circle instead of a whole circle, find the length of the curved side. 177 . [T] The area of an isosceles triangle with equal sides of length x is where is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle rad. 178 . [T] A particle travels in a circular path at a constant angular speed The angular speed is modeled by the function Determine the angular speed at sec. 64 1 • Functions and Graphs Access for free at openstax.org
179. [T] An alternating current for outlets in a home has voltage given by the function where V is the voltage in volts at time t in seconds. a. Find the period of the function and interpret its meaning. b. Determine the number of periods that occur when 1 sec has passed. 180 . [T] The number of hours of daylight in a northeast city is modeled by the function where t is the number of days after January 1. a. Find the amplitude and period. b. Determine the number of hours of daylight on the longest day of the year. c. Determine the number of hours of daylight on the shortest day of the year. d. Determine the number of hours of daylight 90 days after January 1. e. Sketch the graph of the function for one period starting on January 1. 181 . [T] Suppose that is a mathematical model of the temperature (in degrees Fahrenheit) at t hours after midnight on a certain day of the week. a. Determine the amplitude and period. b. Find the temperature 7 hours after midnight. c. At what time does d. Sketch the graph of over 182. [T] The function models the height H (in feet) of the tide t hours after midnight. Assume that is midnight. a. Find the amplitude and period. b. Graph the function over one period. c. What is the height of the tide at 4:30 a.m.? 1.4 Inverse Functions Learning Objectives 1.4.1 Determine the conditions for when a function has an inverse. 1.4.2 Use the horizontal line test to recognize when a function is one-to-one. 1.4.3 Find the inverse of a given function. 1.4.4 Draw the graph of an inverse function. 1.4.5 Evaluate inverse trigonometric functions. An inverse function reverses the operation done by a particular function. In other words, whatever a function does, the inverse function undoes it. In this section, we define an inverse function formally and state the necessary conditions for an inverse function to exist. We examine how to find an inverse function and study the relationship between the graph of a function and the graph of its inverse. Then we apply these ideas to define and discuss properties of the inverse trigonometric functions. Existence of an Inverse Function We begin with an example. Given a function and an output we are often interested in finding what value or values were mapped to by For example, consider the function Since any output we can solve this equation for to find that the input is This equation defines as a function of Denoting this function as and writing we see that for any in the domain of Thus, this new function, “undid” what the original function did. A function with this property is called the inverse function of the original function. 1.4 • Inverse Functions 65
Definition Given a function with domain and range its inverse function (if it exists) is the function with domain and range such that if In other words, for a function and its inverse (1.11) Note that is read as “f inverse.” Here, the is not used as an exponent and Figure 1.37 shows the relationship between the domain and range of f and the domain and range of Figure 1.37 Given a function and its inverse if and only if The range of becomes the domain of and the domain of becomes the range of Recall that a function has exactly one output for each input. Therefore, to define an inverse function, we need to map each input to exactly one output. For example, let’s try to find the inverse function for Solving the equation for we arrive at the equation This equation does not describe as a function of because there are two solutions to this equation for every The problem with trying to find an inverse function for is that two inputs are sent to the same output for each output The function discussed earlier did not have this problem. For that function, each input was sent to a different output. A function that sends each input to a different output is called a one-to-one function. Definition We say a is a one-to-one function if when One way to determine whether a function is one-to-one is by looking at its graph. If a function is one-to-one, then no two inputs can be sent to the same output. Therefore, if we draw a horizontal line anywhere in the -plane, according to the horizontal line test, it cannot intersect the graph more than once. We note that the horizontal line test is different from the vertical line test. The vertical line test determines whether a graph is the graph of a function. The horizontal line test determines whether a function is one-to-one (Figure 1.38). Rule: Horizontal Line Test A function is one-to-one if and only if every horizontal line intersects the graph of no more than once. 66 1 • Functions and Graphs Access for free at openstax.org
Figure 1.38 (a) The function is not one-to-one because it fails the horizontal line test. (b) The function is one-to-one because it passes the horizontal line test. EXAMPLE 1.28 Determining Whether a Function Is One-to-One For each of the following functions, use the horizontal line test to determine whether it is one-to-one. a. b. 1.4 • Inverse Functions 67
Solution a. Since the horizontal line for any integer intersects the graph more than once, this function is not one-to- one. b. Since every horizontal line intersects the graph once (at most), this function is one-to-one. 1.23 Is the function graphed in the following image one-to-one? 68 1 • Functions and Graphs Access for free at openstax.org
Finding a Function’s Inverse We can now consider one-to-one functions and show how to find their inverses. Recall that a function maps elements in the domain of to elements in the range of The inverse function maps each element from the range of back to its corresponding element from the domain of Therefore, to find the inverse function of a one-to-one function given any in the range of we need to determine which in the domain of satisfies Since is one-to-one, there is exactly one such value We can find that value by solving the equation for Doing so, we are able to write as a function of where the domain of this function is the range of and the range of this new function is the domain of Consequently, this function is the inverse of and we write Since we typically use the variable to denote the independent variable and to denote the dependent variable, we often interchange the roles of and and write Representing the inverse function in this way is also helpful later when we graph a function and its inverse on the same axes. Problem-Solving Strategy Finding an Inverse Function 1. Solve the equation for 2. Interchange the variables and and write EXAMPLE 1.29 Finding an Inverse Function Find the inverse for the function State the domain and range of the inverse function. Verify that Solution Follow the steps outlined in the strategy. Step 1. If then and Step 2. Rewrite as and let Therefore, Since the domain of is ∞ ∞ the range of is ∞ ∞ Since the range of is ∞ ∞ the domain of is∞ ∞ You can verify that by writing Note that for to be the inverse of both and for all x in the domain of the inside function. 1.24 Find the inverse of the function State the domain and range of the inverse function. Graphing Inverse Functions Let’s consider the relationship between the graph of a function and the graph of its inverse. Consider the graph of shown in Figure 1.39 and a point on the graph. Since then Therefore, when we graph the point is on the graph. As a result, the graph of is a reflection of the graph of about the line 1.4 • Inverse Functions 69
Figure 1.39 (a) The graph of this function shows point on the graph of (b) Since is on the graph of the point is on the graph of The graph of is a reflection of the graph of about the line EXAMPLE 1.30 Sketching Graphs of Inverse Functions For the graph of in the following image, sketch a graph of by sketching the line and using symmetry. Identify the domain and range of Solution Reflect the graph about the line The domain of is ∞ The range of is ∞ By using the preceding strategy for finding inverse functions, we can verify that the inverse function is as shown in the graph. 70 1 • Functions and Graphs Access for free at openstax.org
1.25 Sketch the graph of and the graph of its inverse using the symmetry property of inverse functions. Restricting Domains As we have seen, does not have an inverse function because it is not one-to-one. However, we can choose a subset of the domain of such that the function is one-to-one. This subset is called a restricted domain. By restricting the domain of we can define a new function such that the domain of is the restricted domain of and for all in the domain of Then we can define an inverse function for on that domain. For example, since is one-to-one on the interval ∞ we can define a new function such that the domain of is ∞ and for all in its domain. Since is a one-to-one function, it has an inverse function, given by the formula On the other hand, the function is also one-to-one on the domain ∞ Therefore, we could also define a new function such that the domain of is ∞ and for all in the domain of Then is a one-to-one function and must also have an inverse. Its inverse is given by the formula (Figure 1.40). Figure 1.40 (a) For restricted to ∞ (b) For restricted to∞ EXAMPLE 1.31 Restricting the Domain Consider the function a. Sketch the graph of and use the horizontal line test to show that is not one-to-one. b. Show that is one-to-one on the restricted domain ∞ Determine the domain and range for the inverse of on this restricted domain and find a formula for Solution a. The graph of is the graph of shifted left 1 unit. Since there exists a horizontal line intersecting the graph more than once, is not one-to-one. 1.4 • Inverse Functions 71
b. On the interval ∞ is one-to-one. The domain and range of are given by the range and domain of respectively. Therefore, the domain of is ∞ and the range of is ∞ To find a formula for solve the equation for If then Since we are restricting the domain to the interval where we need Therefore, Interchanging and we write and conclude that 1.26 Consider restricted to the domain ∞ Verify that is one-to-one on this domain. Determine the domain and range of the inverse of and find a formula for Inverse Trigonometric Functions The six basic trigonometric functions are periodic, and therefore they are not one-to-one. However, if we restrict the domain of a trigonometric function to an interval where it is one-to-one, we can define its inverse. Consider the sine function (Figure 1.34). The sine function is one-to-one on an infinite number of intervals, but the standard convention is to restrict the domain to the interval By doing so, we define the inverse sine function on the domain such that for any in the interval the inverse sine function tells us which angle in the interval satisfies Similarly, we can restrict the domains of the other trigonometric functions to define inverse trigonometric functions, which are functions that tell us which angle in a certain interval has a specified trigonometric value. 72 1 • Functions and Graphs Access for free at openstax.org
Definition The inverse sine function, denoted or arcsin, and the inverse cosine function, denoted or arccos, are defined on the domain as follows: (1.12) The inverse tangent function, denoted or arctan, and inverse cotangent function, denoted or arccot, are defined on the domain ∞ ∞ as follows: (1.13) The inverse cosecant function, denoted or arccsc, and inverse secant function, denoted or arcsec, are defined on the domain as follows: (1.14) To graph the inverse trigonometric functions, we use the graphs of the trigonometric functions restricted to the domains defined earlier and reflect the graphs about the line (Figure 1.41). Figure 1.41 The graph of each of the inverse trigonometric functions is a reflection about the line of the corresponding restricted trigonometric function. 1.4 • Inverse Functions 73
MEDIA Go to the following site (http://www.openstax.org/l/20_inversefun) for more comparisons of functions and their inverses. When evaluating an inverse trigonometric function, the output is an angle. For example, to evaluate we need to find an angle such that Clearly, many angles have this property. However, given the definition of we need the angle that not only solves this equation, but also lies in the interval We conclude that We now consider a composition of a trigonometric function and its inverse. For example, consider the two expressions and For the first one, we simplify as follows: For the second one, we have The inverse function is supposed to “undo” the original function, so why isn’t Recalling our definition of inverse functions, a function and its inverse satisfy the conditions for all in the domain of and for all in the domain of so what happened here? The issue is that the inverse sine function, is the inverse of the restricted sine function defined on the domain Therefore, for in the interval it is true that However, for values of outside this interval, the equation does not hold, even though is defined for all real numbers What about Does that have a similar issue? The answer is no. Since the domain of is the interval we conclude that if and the expression is not defined for other values of To summarize, and Similarly, for the cosine function, and Similar properties hold for the other trigonometric functions and their inverses. EXAMPLE 1.32 Evaluating Expressions Involving Inverse Trigonometric Functions Evaluate each of the following expressions. a. b. c. d. 74 1 • Functions and Graphs Access for free at openstax.org
Solution a. Evaluating is equivalent to finding the angle such that and The angle satisfies these two conditions. Therefore, b. First we use the fact that Then Therefore, c. To evaluate first use the fact that Then we need to find the angle such that and Since satisfies both these conditions, we have d. Since we need to evaluate That is, we need to find the angle such that and Since satisfies both these conditions, we can conclude that STUDENT PROJECT The Maximum Value of a Function In many areas of science, engineering, and mathematics, it is useful to know the maximum value a function can obtain, even if we don’t know its exact value at a given instant. For instance, if we have a function describing the strength of a roof beam, we would want to know the maximum weight the beam can support without breaking. If we have a function that describes the speed of a train, we would want to know its maximum speed before it jumps off the rails. Safe design often depends on knowing maximum values. This project describes a simple example of a function with a maximum value that depends on two equation coefficients. We will see that maximum values can depend on several factors other than the independent variable x. 1. Consider the graph in Figure 1.42 of the function Describe its overall shape. Is it periodic? How do you know? Figure 1.42 The graph of Using a graphing calculator or other graphing device, estimate the - and -values of the maximum point for the graph (the first such point where x > 0). It may be helpful to express the -value as a multiple of π. 2. Now consider other graphs of the form for various values of A and B. Sketch the graph when A = 2 and B = 1, and find the - and y-values for the maximum point. (Remember to express the x-value as a multiple of π, if possible.) Has it moved? 3. Repeat for A = 1, B = 2. Is there any relationship to what you found in part (2)? 4. Complete the following table, adding a few choices of your own for A and B: 1.4 • Inverse Functions 75
A B x y A B x y 0 1 1 1 0 1 1 1 12 5 1 2 5 12 2 1 2 2 3 4 4 3 5. Try to figure out the formula for the y-values. 6. The formula for the -values is a little harder. The most helpful points from the table are (Hint: Consider inverse trigonometric functions.) 7. If you found formulas for parts (5) and (6), show that they work together. That is, substitute the -value formula you found into and simplify it to arrive at the -value formula you found. SECTION 1.4 EXERCISESFor the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one. 183. 184 . 76 1 • Functions and Graphs Access for free at openstax.org
185. 186 . 187. 188 . For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function. 189. 190 . 191 . 192. 193 . 194 . For the following exercises, use the graph of to sketch the graph of its inverse function. 195. 196 . 1.4 • Inverse Functions 77
197. 198 . For the following exercises, use composition to determine which pairs of functions are inverses. 199. 200 . 201 . 202. 203. 204. 205. 206. For the following exercises, evaluate the functions. Give the exact value. 207. 208 . 209 . 210. 211 . 212 . 213. 214 . 215 . 216. The function converts degrees Fahrenheit to degrees Celsius. a. Find the inverse function b. What is the inverse function used for? 217 . [T] The velocity V (in centimeters per second) of blood in an artery at a distance x cm from the center of the artery can be modeled by the function for a. Find b. Interpret what the inverse function is used for. c. Find the distance from the center of an artery with a velocity of 15 cm/ sec, 10 cm/sec, and 5 cm/sec. 218. A function that converts dress sizes in the United States to those in Europe is given by a. Find the European dress sizes that correspond to sizes 6, 8, 10, and 12 in the United States. b. Find the function that converts European dress sizes to U.S. dress sizes. c. Use part b. to find the dress sizes in the United States that correspond to 46, 52, 62, and 70. 78 1 • Functions and Graphs Access for free at openstax.org
219. [T] The cost to remove a toxin from a lake is modeled by the function where is the cost (in thousands of dollars) and is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than 85 ppb. a. Find the cost to remove 25 ppb, 40 ppb, and 50 ppb of the toxin from the lake. b. Find the inverse function. c. Use part b. to determine how much of the toxin is removed for $50,000. 220 . [T] A race car is accelerating at a velocity given by where v is the velocity (in feet per second) at time t. a. Find the velocity of the car at 10 sec. b. Find the inverse function. c. Use part b. to determine how long it takes for the car to reach a speed of 150 ft/sec. 221 . [T] An airplane’s Mach number M is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by Find the Mach angle (to the nearest degree) for the following Mach numbers. a. b. c. 222. [T] Using find the Mach number M for the following angles. a. b. c. 223 . [T] The average temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function where is time in months and corresponds to January 1. Determine the day(s) (month and day) when the average temperature is Use the integer portion of your answer(s) as the month and calculate the day of the month from the decimal portion. 224 . [T] The depth (in feet) of water at a dock changes with the rise and fall of tides. It is modeled by the function where is the number of hours after midnight. Determine the first time after midnight when the depth is 11.75 ft. 225. [T] An object moving in simple harmonic motion is modeled by the function where is measured in inches and is measured in seconds. Determine the first time when the distance moved is 4.5 in. 226 . [T] A local art gallery has a portrait 3 ft in height that is hung 2.5 ft above the eye level of an average person. The viewing angle can be modeled by the function where is the distance (in feet) from the portrait. Find the viewing angle when a person is 4 ft from the portrait. 227 . [T] Use a calculator to evaluate and Explain the results of each. 228. [T] Use a calculator to evaluate and Explain the results of each. 1.4 • Inverse Functions 79
1.5 Exponential and Logarithmic Functions Learning Objectives 1.5.1 Identify the form of an exponential function. 1.5.2 Explain the difference between the graphs of and 1.5.3 Recognize the significance of the number 1.5.4 Identify the form of a logarithmic function. 1.5.5 Explain the relationship between exponential and logarithmic functions. 1.5.6 Describe how to calculate a logarithm to a different base. 1.5.7 Identify the hyperbolic functions, their graphs, and basic identities. In this section we examine exponential and logarithmic functions. We use the properties of these functions to solve equations involving exponential or logarithmic terms, and we study the meaning and importance of the number We also define hyperbolic and inverse hyperbolic functions, which involve combinations of exponential and logarithmic functions. (Note that we present alternative definitions of exponential and logarithmic functions in the chapter Applications of Integrations, and prove that the functions have the same properties with either definition.) Exponential Functions Exponential functions arise in many applications. One common example is population growth. For example, if a population starts with individuals and then grows at an annual rate of its population after 1 year is Its population after 2 years is In general, its population after years is which is an exponential function. More generally, any function of the form where is an exponential function with base and exponent x. Exponential functions have constant bases and variable exponents. Note that a function of the form for some constant is not an exponential function but a power function. To see the difference between an exponential function and a power function, we compare the functions and In Table 1.10, we see that both and approach infinity as ∞ Eventually, however, becomes larger than and grows more rapidly as ∞ In the opposite direction, as ∞ ∞ whereas The line is a horizontal asymptote for Table 1.10 Values of and In Figure 1.43, we graph both and to show how the graphs differ. 80 1 • Functions and Graphs Access for free at openstax.org
Figure 1.43 Both and approach infinity as ∞ but grows more rapidly than As ∞ ∞ whereas Evaluating Exponential Functions Recall the properties of exponents: If is a positive integer, then we define (with factors of If is a negative integer, then for some positive integer and we define Also, is defined to be If is a rational number, then where and are integers and For example, However, how is defined if is an irrational number? For example, what do we mean by This is too complex a question for us to answer fully right now; however, we can make an approximation. In Table 1.11, we list some rational numbers approaching and the values of for each rational number are presented as well. We claim that if we choose rational numbers getting closer and closer to the values of get closer and closer to some number We define that number to be Table 1.11 Values of for a List of Rational Numbers Approximating EXAMPLE 1.33 Bacterial Growth Suppose a particular population of bacteria is known to double in size every hours. If a culture starts with bacteria, the number of bacteria after hours is The number of bacteria after hours is In general, the number of bacteria after hours is Letting we see that the number of bacteria after hours is Find the number of bacteria after hours, hours, and hours. Solution The number of bacteria after 6 hours is given by bacteria. The number of bacteria after hours is given by bacteria. The number of bacteria after hours is given by bacteria. 1.27 Given the exponential function evaluate and 1.5 • Exponential and Logarithmic Functions 81
MEDIA Go to Population Balance (http://www.openstax.org/l/20_exponengrow) for another example of exponential population growth. Graphing Exponential Functions For any base the exponential function is defined for all real numbers and Therefore, the domain of is ∞ ∞ and the range is ∞ To graph we note that for is increasing on∞ ∞ and ∞ as ∞ whereas as ∞ On the other hand, if is decreasing on ∞ ∞ and as ∞ whereas ∞ as ∞ (Figure 1.44). Figure 1.44 If then is increasing on ∞ ∞ If then is decreasing on ∞ ∞ MEDIA Visit this site (http://www.openstax.org/l/20_inverse) for more exploration of the graphs of exponential functions. Note that exponential functions satisfy the general laws of exponents. To remind you of these laws, we state them as rules. Rule: Laws of Exponents For any constants and for all x and y, 1. 2. 3. 4. 5. EXAMPLE 1.34 Using the Laws of Exponents Use the laws of exponents to simplify each of the following expressions. a. b. Solution a. We can simplify as follows: 82 1 • Functions and Graphs Access for free at openstax.org
b. We can simplify as follows: 1.28 Use the laws of exponents to simplify The Number e A special type of exponential function appears frequently in real-world applications. To describe it, consider the following example of exponential growth, which arises from compounding interest in a savings account. Suppose a person invests dollars in a savings account with an annual interest rate compounded annually. The amount of money after 1 year is The amount of money after years is More generally, the amount after years is If the money is compounded 2 times per year, the amount of money after half a year is The amount of money after year is After years, the amount of money in the account is More generally, if the money is compounded times per year, the amount of money in the account after years is given by the function What happens as ∞ To answer this question, we let and write and examine the behavior of as ∞ using a table of values (Table 1.12). Table 1.12 Values of as ∞ Looking at this table, it appears that is approaching a number between and as ∞ In fact, 1.5 • Exponential and Logarithmic Functions 83
does approach some number as ∞ We call this number . To six decimal places of accuracy, The letter was first used to represent this number by the Swiss mathematician Leonhard Euler during the 1720s. Although Euler did not discover the number, he showed many important connections between and logarithmic functions. We still use the notation today to honor Euler’s work because it appears in many areas of mathematics and because we can use it in many practical applications. Returning to our savings account example, we can conclude that if a person puts dollars in an account at an annual interest rate compounded continuously, then This function may be familiar. Since functions involving base arise often in applications, we call the function the natural exponential function. Not only is this function interesting because of the definition of the number but also, as discussed next, its graph has an important property. Since we know is increasing on ∞ ∞ In Figure 1.45, we show a graph of along with a tangent line to the graph of at We give a precise definition of tangent line in the next chapter; but, informally, we say a tangent line to a graph of at is a line that passes through the point and has the same “slope” as at that point The function is the only exponential function with tangent line at that has a slope of 1. As we see later in the text, having this property makes the natural exponential function the most simple exponential function to use in many instances. Figure 1.45 The graph of has a tangent line with slope at EXAMPLE 1.35 Compounding Interest Suppose is invested in an account at an annual interest rate of compounded continuously. a. Let denote the number of years after the initial investment and denote the amount of money in the account at time Find a formula for b. Find the amount of money in the account after years and after years. Solution a. If dollars are invested in an account at an annual interest rate compounded continuously, then Here and Therefore, b. After years, the amount of money in the account is After years, the amount of money in the account is 1.29 If is invested in an account at an annual interest rate of compounded continuously, find a formula for the amount of money in the account after years. Find the amount of money after years. Logarithmic Functions Using our understanding of exponential functions, we can discuss their inverses, which are the logarithmic functions. These come in handy when we need to consider any phenomenon that varies over a wide range of values, such as pH in chemistry or decibels in sound levels. 84 1 • Functions and Graphs Access for free at openstax.org
The exponential function is one-to-one, with domain ∞ ∞ and range ∞ Therefore, it has an inverse function, called the logarithmic function with base For any the logarithmic function with base b, denoted has domain ∞ and range ∞ ∞ and satisfies For example, Furthermore, since and are inverse functions, The most commonly used logarithmic function is the function Since this function uses natural as its base, it is called the natural logarithm. Here we use the notation or to mean For example, Since the functions and are inverses of each other, and their graphs are symmetric about the line (Figure 1.46). Figure 1.46 The functions and are inverses of each other, so their graphs are symmetric about the line MEDIA At this site (http://www.openstax.org/l/20_logscale) you can see an example of a base-10 logarithmic scale. In general, for any base the function is symmetric about the line with the function Using this fact and the graphs of the exponential functions, we graph functions for several values of (Figure 1.47). 1.5 • Exponential and Logarithmic Functions 85
Figure 1.47 Graphs of are depicted for Before solving some equations involving exponential and logarithmic functions, let’s review the basic properties of logarithms. Rule: Properties of Logarithms If and is any real number, then EXAMPLE 1.36 Solving Equations Involving Exponential Functions Solve each of the following equations for a. b. Solution a. Applying the natural logarithm function to both sides of the equation, we have Using the power property of logarithms, Therefore, b. Multiplying both sides of the equation by we arrive at the equation Rewriting this equation as we can then rewrite it as a quadratic equation in Now we can solve the quadratic equation. Factoring this equation, we obtain 86 1 • Functions and Graphs Access for free at openstax.org
Therefore, the solutions satisfy and Taking the natural logarithm of both sides gives us the solutions 1.30 Solve EXAMPLE 1.37 Solving Equations Involving Logarithmic Functions Solve each of the following equations for a. b. c. Solution a. By the definition of the natural logarithm function, Therefore, the solution is b. Using the product and power properties of logarithmic functions, rewrite the left-hand side of the equation as Therefore, the equation can be rewritten as The solution is c. Using the power property of logarithmic functions, we can rewrite the equation as Using the quotient property, this becomes Therefore, which implies We should then check for any extraneous solutions. 1.31 Solve When evaluating a logarithmic function with a calculator, you may have noticed that the only options are or log, called the common logarithm, or ln, which is the natural logarithm. However, exponential functions and logarithm functions can be expressed in terms of any desired base If you need to use a calculator to evaluate an expression with a different base, you can apply the change-of-base formulas first. Using this change of base, we typically write a given exponential or logarithmic function in terms of the natural exponential and natural logarithmic functions. Rule: Change-of-Base Formulas Let and 1. for any real number If this equation reduces to 1.5 • Exponential and Logarithmic Functions 87
2. for any real number If this equation reduces to Proof For the first change-of-base formula, we begin by making use of the power property of logarithmic functions. We know that for any base Therefore, In addition, we know that and are inverse functions. Therefore, Combining these last two equalities, we conclude that To prove the second property, we show that Let and We will show that By the definition of logarithmic functions, we know that and From the previous equations, we see that Therefore, Since exponential functions are one-to-one, we can conclude that □ EXAMPLE 1.38 Changing Bases Use a calculating utility to evaluate with the change-of-base formula presented earlier. Solution Use the second equation with and 1.32 Use the change-of-base formula and a calculating utility to evaluate EXAMPLE 1.39 Chapter Opener: The Richter Scale for Earthquakes Figure 1.48 (credit: modification of work by Robb Hannawacker, NPS) In 1935, Charles Richter developed a scale (now known as the Richter scale) to measure the magnitude of an earthquake. The scale is a base-10 logarithmic scale, and it can be described as follows: Consider one earthquake with magnitude on the Richter scale and a second earthquake with magnitude on the Richter scale. Suppose which means the earthquake of magnitude is stronger, but how much stronger is it than the other earthquake? A way of measuring the intensity of an earthquake is by using a seismograph to measure the amplitude of the earthquake waves. If is the amplitude measured for the first earthquake and is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation: 88 1 • Functions and Graphs Access for free at openstax.org
Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then, Therefore, which implies or Since is 10 times the size of we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation Therefore, That is, the first earthquake is 100 times more intense than the second earthquake. How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010? Solution To compare the Japan and Haiti earthquakes, we can use an equation presented earlier: Therefore, and we conclude that the earthquake in Japan was approximately times more intense than the earthquake in Haiti. 1.33 Compare the relative severity of a magnitude earthquake with a magnitude earthquake. Hyperbolic Functions The hyperbolic functions are defined in terms of certain combinations of and These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary (Figure 1.49). If we introduce a coordinate system so that the low point of the chain lies along the -axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions. 1.5 • Exponential and Logarithmic Functions 89
Figure 1.49 The shape of a strand of silk in a spider’s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight. (credit: “Mtpaley”, Wikimedia Commons) Definition Hyperbolic cosine Hyperbolic sine Hyperbolic tangent Hyperbolic cosecant Hyperbolic secant Hyperbolic cotangent The name cosh rhymes with “gosh,” whereas the name sinh is pronounced “cinch.” Tanh, sech, csch, and coth are pronounced “tanch,” “seech,” “coseech,” and “cotanch,” respectively. Using the definition of and principles of physics, it can be shown that the height of a hanging chain, such as the one in Figure 1.49, can be described by the function for certain constants and But why are these functions called hyperbolic functions? To answer this question, consider the quantity Using the definition of and we see that 90 1 • Functions and Graphs Access for free at openstax.org
This identity is the analog of the trigonometric identity Here, given a value the point lies on the unit hyperbola (Figure 1.50). Figure 1.50 The unit hyperbola Graphs of Hyperbolic Functions To graph and we make use of the fact that both functions approach as ∞ since as∞ As ∞ approaches whereas approaches Therefore, using the graphs of and as guides, we graph and To graph we use the fact that for all as ∞ and as ∞ The graphs of the other three hyperbolic functions can be sketched using the graphs of and (Figure 1.51). 1.5 • Exponential and Logarithmic Functions 91
Figure 1.51 The hyperbolic functions involve combinations of and Identities Involving Hyperbolic Functions The identity shown in Figure 1.50, is one of several identities involving the hyperbolic functions, some of which are listed next. The first four properties follow easily from the definitions of hyperbolic sine and hyperbolic cosine. Except for some differences in signs, most of these properties are analogous to identities for trigonometric functions. Rule: Identities Involving Hyperbolic Functions 1. 2. 3. 4. 5. 6. 7. 8. 9. 92 1 • Functions and Graphs Access for free at openstax.org
EXAMPLE 1.40 Evaluating Hyperbolic Functions a. Simplify b. If find the values of the remaining five hyperbolic functions. Solution a. Using the definition of the function, we write b. Using the identity we see that Since for all we must have Then, using the definitions for the other hyperbolic functions, we conclude that and 1.34 Simplify Inverse Hyperbolic Functions From the graphs of the hyperbolic functions, we see that all of them are one-to-one except and If we restrict the domains of these two functions to the interval ∞ then all the hyperbolic functions are one-to-one, and we can define the inverse hyperbolic functions. Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions. Definition Inverse Hyperbolic Functions Let’s look at how to derive the first equation. The others follow similarly. Suppose Then, and, by the definition of the hyperbolic sine function, Therefore, Multiplying this equation by we obtain This can be solved like a quadratic equation, with the solution Since the only solution is the one with the positive sign. Applying the natural logarithm to both sides of the equation, we conclude that 1.5 • Exponential and Logarithmic Functions 93
EXAMPLE 1.41 Evaluating Inverse Hyperbolic Functions Evaluate each of the following expressions. Solution 1.35 Evaluate SECTION 1.5 EXERCISESFor the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal. 229. a. b. c. 230 . a. b. c. 231 . a. b. c. 232. a. b. c. For the following exercises, match the exponential equation to the correct graph. a. b. c. d. e. f. 233. 234 . 94 1 • Functions and Graphs Access for free at openstax.org
235. 236 . 237. 238 . For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote. 239. 240. 241 . 242. 243. 244 . 245. For the following exercises, write the equation in equivalent exponential form. 246. 247 . 248 . 249. 250 . 251 . 252. 253 . For the following exercises, write the equation in equivalent logarithmic form. 254. 255 . 256 . 257. 258 . 259 . 260. 261 . 262 . 263. For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. 264. 265. 266 . 1.5 • Exponential and Logarithmic Functions 95
267. 268. 269 . For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. 270. 271 . 272 . 273. 274 . 275 . For the following exercises, solve the exponential equation exactly. 276. 277 . 278 . 279. 280 . 281 . 282. 283 . For the following exercises, solve the logarithmic equation exactly, if possible. 284. 285 . 286 . 287. 288 . 289 . 290. 291. For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places. 292. 293 . 294 . 295. 296 . 297 . 298. Rewrite the following expressions in terms of exponentials and simplify. a. b. c. d. 299 . [T] The number of bacteria N in a culture after t days can be modeled by the function Find the number of bacteria present after 15 days. 300 . [T] The demand D (in millions of barrels) for oil in an oil-rich country is given by the function where p is the price (in dollars) of a barrel of oil. Find the amount of oil demanded (to the nearest million barrels) when the price is between $15 and $20. 96 1 • Functions and Graphs Access for free at openstax.org
301. [T] The amount A of a $100,000 investment paying continuously and compounded for t years is given by Find the amount A accumulated in 5 years. 302 . [T] An investment is compounded monthly, quarterly, or yearly and is given by the function where is the value of the investment at time is the initial principle that was invested, is the annual interest rate, and is the number of time the interest is compounded per year. Given a yearly interest rate of 3.5% and an initial principle of $100,000, find the amount accumulated in 5 years for interest that is compounded a. daily, b., monthly, c. quarterly, and d. yearly. 303 . [T] The concentration of hydrogen ions in a substance is denoted by measured in moles per liter. The pH of a substance is defined by the logarithmic function This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base. a. Find the pH of the following substances. Round answers to one digit. b. Determine whether the substance is an acid or a base. i. Eggs: mol/L ii. Beer: mol/L iii. Tomato Juice: mol/L 304. [T] Iodine-131 is a radioactive substance that decays according to the function where is the initial quantity of a sample of the substance and t is in days. Determine how long it takes (to the nearest day) for 95% of a quantity to decay. 305 . [T] According to the World Bank, at the end of 2013 ( ) the U.S. population was 316 million and was increasing according to the following model: where P is measured in millions of people and t is measured in years after 2013. a. Based on this model, what will be the population of the United States in 2020? b. Determine when the U.S. population will be twice what it is in 2013. 306 . [T] The amount A accumulated after 1000 dollars is invested for t years at an interest rate of 4% is modeled by the function a. Find the amount accumulated after 5 years and 10 years. b. Determine how long it takes for the original investment to triple. 1.5 • Exponential and Logarithmic Functions 97
307. [T] A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present. a. Use the exponential function to determine the value which is the growth rate of the bacteria. Round to four decimal places. b. Determine approximately how long it takes for 200,000 bacteria to grow. 308 . [T] The rabbit population on a game reserve doubles every 6 months. Suppose there were 120 rabbits initially. a. Use the exponential function to determine the growth rate constant Round to four decimal places. b. Use the function in part a. to determine approximately how long it takes for the rabbit population to reach 3500. 309 . [T] The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time, in Japan, an earthquake with magnitude 4.9 caused only minor damage. Approximately how much more intense was the San Francisco earthquake than the Japanese earthquake? See the definition of Richter Scale in [link] in this section. 98 1 • Functions and Graphs Access for free at openstax.org
Chapter Review Key Terms absolute value function algebraic function a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable base the number in the exponential function and the logarithmic function composite function given two functions and a new function, denoted such that cubic function a polynomial of degree 3; that is, a function of the form where decreasing on the interval a function decreasing on the interval if, for all if degree for a polynomial function, the value of the largest exponent of any term dependent variable the output variable for a function domain the set of inputs for a function even function a function is even if for all in the domain of exponent the value in the expression function a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output graph of a function the set of points such that is in the domain of and horizontal line test a function is one-to-one if and only if every horizontal line intersects the graph of at most, once hyperbolic functions the functions denoted and which involve certain combinations of and increasing on the interval a function increasing on the interval if for all if independent variable the input variable for a function inverse function for a function the inverse function satisfies if inverse hyperbolic functions the inverses of the hyperbolic functions where and are restricted to the domain ∞ each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function inverse trigonometric functions the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions linear function a function that can be written in the form logarithmic function a function of the form for some base such that if and only if mathematical model A method of simulating real-life situations with mathematical equations natural exponential function the function natural logarithm the function number e as gets larger, the quantity gets closer to some real number; we define that real number to be the value of is approximately odd function a function is odd if for all in the domain of one-to-one function a function is one-to-one if if periodic function a function is periodic if it has a repeating pattern as the values of move from left to right piecewise-defined function a function that is defined differently on different parts of its domain point-slope equation equation of a linear function indicating its slope and a point on the graph of the function polynomial function a function of the form power function a function of the form for any positive integer quadratic function a polynomial of degree 2; that is, a function of the form where radians for a circular arc of length on a circle of radius 1, the radian measure of the associated angle is range the set of outputs for a function rational function a function of the form where and are polynomials restricted domain a subset of the domain of a function root function a function of the form for any integer slope the change in y for each unit change in x slope-intercept form equation of a linear function indicating its slope and y-intercept symmetry about the origin the graph of a function is symmetric about the origin if is on the graph of whenever is on the graph symmetry about the y-axis the graph of a function is symmetric about the -axis if is on the graph of whenever is on the graph 1 • Chapter Review 99
table of values a table containing a list of inputs and their corresponding outputs transcendental function a function that cannot be expressed by a combination of basic arithmetic operations transformation of a function a shift, scaling, or reflection of a function trigonometric functions functions of an angle defined as ratios of the lengths of the sides of a right triangle trigonometric identity an equation involving trigonometric functions that is true for all angles for which the functions in the equation are defined vertical line test given the graph of a function, every vertical line intersects the graph, at most, once zeros of a function when a real number is a zero of a function Key Equations Composition of two functions Absolute value function Point-slope equation of a line Slope-intercept form of a line Standard form of a line Polynomial function Generalized sine function Inverse functions Key Concepts 1.1 Review of Functions • A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input. • If no domain is stated for a function the domain is considered to be the set of all real numbers for which the function is defined. • When sketching the graph of a function each vertical line may intersect the graph, at most, once. • A function may have any number of zeros, but it has, at most, one y-intercept. • To define the composition the range of must be contained in the domain of • Even functions are symmetric about the -axis whereas odd functions are symmetric about the origin. 1.2 Basic Classes of Functions • The power function is an even function if is even and and it is an odd function if is odd. • The root function has the domain ∞ if is even and the domain ∞ if is odd. If is odd, then is an odd function. • The domain of the rational function where and are polynomial functions, is the set of such that • Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions. • A polynomial function with degree satisfies ∞ as ∞ The sign of the output as ∞ depends on the sign of the leading coefficient only and on whether is even or odd. • Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the - and -axes are examples of transformations of functions. 1.3 Trigonometric Functions • Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian 100 1 • Chapter Review Access for free at openstax.org
measure 1. An angle with a degree measure of has a radian measure of rad. • For acute angles the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is • For a general angle let be a point on a circle of radius corresponding to this angle The trigonometric functions can be written as ratios involving and • The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period The tangent and cotangent functions have period 1.4 Inverse Functions • For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test. • If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain. • For a function and its inverse for all in the domain of and for all in the domain of • Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions. • The graph of a function and its inverse are symmetric about the line 1.5 Exponential and Logarithmic Functions • The exponential function is increasing if and decreasing if Its domain is ∞ ∞ and its range is ∞ • The logarithmic function is the inverse of Its domain is ∞ and its range is ∞ ∞ • The natural exponential function is and the natural logarithmic function is • Given an exponential function or logarithmic function in base we can make a change of base to convert this function to any base We typically convert to base • The hyperbolic functions involve combinations of the exponential functions and As a result, the inverse hyperbolic functions involve the natural logarithm. Review ExercisesTrue or False? Justify your answer with a proof or a counterexample. 310. A function is always one- to-one. 311 . assuming f and g are functions. 312 . A relation that passes the horizontal and vertical line tests is a one-to-one function. 313. A relation passing the horizontal line test is a function. For the following problems, state the domain and range of the given functions: 314. h 315 . g 316 . 317. Find the degree, y-intercept, and zeros for the following polynomial functions. 318. 319 . Simplify the following trigonometric expressions. 320. 321 . 1 • Chapter Review 101
Solve the following trigonometric equations on the interval exactly. 322. 323 . Solve the following logarithmic equations. 324. 325 . Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse of the function. Justify your answer. 326. 327 . For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain. 328. 329 . 330 . A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn 55° to keep up with the car. How fast is the car traveling? For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts. 331. a. Find the equation that describes the total cost as a function of number of shirts and b. determine how many shirts he must sell to break even if he sells the shirts for $10 each. 332 . a. Find the inverse function and describe the meaning of this function. b. Determine how many shirts the owner can buy if he has $8000 to spend. For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season. 333. The population can be modeled by where is time in months represents January 1) and is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000? 334 . In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as where is time in months ( represents January 1) and is population (in thousands). When is the first time the population reaches 200,000? 102 1 • Chapter Review Access for free at openstax.org
For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation where is the ratio of radiocarbon still present in the material, is the number of years passed, and is the decay rate of radiocarbon. 335. If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present? 336 . Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton? 1 • Chapter Review 103
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Figure 2.1 The vision of human exploration by the National Aeronautics and Space Administration (NASA) to distant parts of the universe illustrates the idea of space travel at high speeds. But, is there a limit to how fast a spacecraft can go? (credit: NASA) Chapter Outline 2.1 A Preview of Calculus 2.2 The Limit of a Function 2.3 The Limit Laws 2.4 Continuity 2.5 The Precise Definition of a Limit Introduction Science fiction writers often imagine spaceships that can travel to far-off planets in distant galaxies. However, back in 1905, Albert Einstein showed that a limit exists to how fast any object can travel. The problem is that the faster an object moves, the more mass it attains (in the form of energy), according to the equation where m0 is the object’s mass at rest, v is its speed, and c is the speed of light. What is this speed limit? (We explore this problem further in Example 2.12.) The idea of a limit is central to all of calculus. We begin this chapter by examining why limits are so important. Then, we 2 LIMITS 2 • Introduction 105
go on to describe how to find the limit of a function at a given point. Not all functions have limits at all points, and we discuss what this means and how we can tell if a function does or does not have a limit at a particular value. This chapter has been created in an informal, intuitive fashion, but this is not always enough if we need to prove a mathematical statement involving limits. The last section of this chapter presents the more precise definition of a limit and shows how to prove whether a function has a limit. 2.1 A Preview of Calculus Learning Objectives 2.1.1 Describe the tangent problem and how it led to the idea of a derivative. 2.1.2 Explain how the idea of a limit is involved in solving the tangent problem. 2.1.3 Recognize a tangent to a curve at a point as the limit of secant lines. 2.1.4 Identify instantaneous velocity as the limit of average velocity over a small time interval. 2.1.5 Describe the area problem and how it was solved by the integral. 2.1.6 Explain how the idea of a limit is involved in solving the area problem. 2.1.7 Recognize how the ideas of limit, derivative, and integral led to the studies of infinite series and multivariable calculus. As we embark on our study of calculus, we shall see how its development arose from common solutions to practical problems in areas such as engineering physics—like the space travel problem posed in the chapter opener. Two key problems led to the initial formulation of calculus: (1) the tangent problem, or how to determine the slope of a line tangent to a curve at a point; and (2) the area problem, or how to determine the area under a curve. The Tangent Problem and Differential Calculus Rate of change is one of the most critical concepts in calculus. We begin our investigation of rates of change by looking at the graphs of the three lines and shown in Figure 2.2. Figure 2.2 The rate of change of a linear function is constant in each of these three graphs, with the constant determined by the slope. As we move from left to right along the graph of we see that the graph decreases at a constant rate. For every 1 unit we move to the right along the x-axis, the y-coordinate decreases by 2 units. This rate of change is determined by the slope (−2) of the line. Similarly, the slope of 1/2 in the function tells us that for every change in x of 1 unit there is a corresponding change in y of 1/2 unit. The function has a slope of zero, indicating that the values of the function remain constant. We see that the slope of each linear function indicates the rate of change of the function. Compare the graphs of these three functions with the graph of (Figure 2.3). The graph of starts from the left by decreasing rapidly, then begins to decrease more slowly and level off, and then finally begins to increase—slowly at first, followed by an increasing rate of increase as it moves toward the right. Unlike a linear function, no single number represents the rate of change for this function. We quite naturally ask: How do we measure the rate of change of a nonlinear function? 106 2 • Limits Access for free at openstax.org
Figure 2.3 The function does not have a constant rate of change. We can approximate the rate of change of a function at a point on its graph by taking another point on the graph of drawing a line through the two points, and calculating the slope of the resulting line. Such a line is called a secant line. Figure 2.4 shows a secant line to a function at a point Figure 2.4 The slope of a secant line through a point estimates the rate of change of the function at the point We formally define a secant line as follows: Definition The secant to the function through the points and is the line passing through these points. Its slope is given by (2.1) The accuracy of approximating the rate of change of the function with a secant line depends on how close x is to a. As we see in Figure 2.5, if x is closer to a, the slope of the secant line is a better measure of the rate of change of at a. 2.1 • A Preview of Calculus 107
Figure 2.5 As x gets closer to a, the slope of the secant line becomes a better approximation to the rate of change of the function at a. The secant lines themselves approach a line that is called the tangent to the function at a (Figure 2.6). The slope of the tangent line to the graph at a measures the rate of change of the function at a. This value also represents the derivative of the function at a, or the rate of change of the function at a. This derivative is denoted by Differential calculus is the field of calculus concerned with the study of derivatives and their applications. MEDIA For an interactive demonstration of the slope of a secant line that you can manipulate yourself, visit this applet (Note: this site requires a Java browser plugin): Math Insight (http://www.openstax.org/l/20_mathinsight). Figure 2.6 Solving the Tangent Problem: As x approaches a, the secant lines approach the tangent line. Example 2.1 illustrates how to find slopes of secant lines. These slopes estimate the slope of the tangent line or, equivalently, the rate of change of the function at the point at which the slopes are calculated. EXAMPLE 2.1 Finding Slopes of Secant Lines Estimate the slope of the tangent line (rate of change) to at by finding slopes of secant lines through and each of the following points on the graph of a. b. Solution Use the formula for the slope of a secant line from the definition. 108 2 • Limits Access for free at openstax.org
a. b. The point in part b. is closer to the point so the slope of 2.5 is closer to the slope of the tangent line. A good estimate for the slope of the tangent would be in the range of 2 to 2.5 (Figure 2.7). Figure 2.7 The secant lines to at through (a) and (b) provide successively closer approximations to the tangent line to at 2.1 Estimate the slope of the tangent line (rate of change) to at by finding the slope of the secant line through and the point on the graph of We continue our investigation by exploring a related question. Keeping in mind that velocity may be thought of as the rate of change of position, suppose that we have a function, that gives the position of an object along a coordinate axis at any given time t. Can we use these same ideas to create a reasonable definition of the instantaneous velocity at a given time We start by approximating the instantaneous velocity with an average velocity. First, recall that the speed of an object traveling at a constant rate is the ratio of the distance traveled to the length of time it has traveled. We define the average velocity of an object over a time period to be the change in its position divided by the length of the time period. Definition Let be the position of an object moving along a coordinate axis at time t. The average velocity of the object over a time interval where (or if is (2.2) As t is chosen closer to a, the average velocity becomes closer to the instantaneous velocity. Note that finding the average velocity of a position function over a time interval is essentially the same as finding the slope of a secant line to a function. Furthermore, to find the slope of a tangent line at a point a, we let the x-values approach a in the slope of the secant line. Similarly, to find the instantaneous velocity at time a, we let the t-values approach a in the average velocity. This process of letting x or t approach a in an expression is called taking a limit. Thus, we may define the instantaneous velocity as follows. 2.1 • A Preview of Calculus 109
Definition For a position function the instantaneous velocity at a time is the value that the average velocities approach on intervals of the form and as the values of t become closer to a, provided such a value exists. Example 2.2 illustrates this concept of limits and average velocity. EXAMPLE 2.2 Finding Average Velocity A rock is dropped from a height of 64 ft. It is determined that its height (in feet) above ground t seconds later (for is given by Find the average velocity of the rock over each of the given time intervals. Use this information to guess the instantaneous velocity of the rock at time a. b. Solution Substitute the data into the formula for the definition of average velocity. a. b. The instantaneous velocity is somewhere between −15.84 and −16.16 ft/sec. A good guess might be −16 ft/sec. 2.2 An object moves along a coordinate axis so that its position at time t is given by Estimate its instantaneous velocity at time by computing its average velocity over the time interval The Area Problem and Integral Calculus We now turn our attention to a classic question from calculus. Many quantities in physics—for example, quantities of work—may be interpreted as the area under a curve. This leads us to ask the question: How can we find the area between the graph of a function and the x-axis over an interval (Figure 2.8)? Figure 2.8 The Area Problem: How do we find the area of the shaded region? As in the answer to our previous questions on velocity, we first try to approximate the solution. We approximate the area by dividing up the interval into smaller intervals in the shape of rectangles. The approximation of the area comes from adding up the areas of these rectangles (Figure 2.9). 110 2 • Limits Access for free at openstax.org
Figure 2.9 The area of the region under the curve is approximated by summing the areas of thin rectangles. As the widths of the rectangles become smaller (approach zero), the sums of the areas of the rectangles approach the area between the graph of and the x-axis over the interval Once again, we find ourselves taking a limit. Limits of this type serve as a basis for the definition of the definite integral. Integral calculus is the study of integrals and their applications. EXAMPLE 2.3 Estimation Using Rectangles Estimate the area between the x-axis and the graph of over the interval by using the three rectangles shown in Figure 2.10. Figure 2.10 The area of the region under the curve of can be estimated using rectangles. Solution The areas of the three rectangles are 1 unit2, 2 unit2, and 5 unit2. Using these rectangles, our area estimate is 8 unit2. 2.3 Estimate the area between the x-axis and the graph of over the interval by using the three rectangles shown here: 2.1 • A Preview of Calculus 111
Other Aspects of Calculus So far, we have studied functions of one variable only. Such functions can be represented visually using graphs in two dimensions; however, there is no good reason to restrict our investigation to two dimensions. Suppose, for example, that instead of determining the velocity of an object moving along a coordinate axis, we want to determine the velocity of a rock fired from a catapult at a given time, or of an airplane moving in three dimensions. We might want to graph real- value functions of two variables or determine volumes of solids of the type shown in Figure 2.11. These are only a few of the types of questions that can be asked and answered using multivariable calculus. Informally, multivariable calculus can be characterized as the study of the calculus of functions of two or more variables. However, before exploring these and other ideas, we must first lay a foundation for the study of calculus in one variable by exploring the concept of a limit. Figure 2.11 We can use multivariable calculus to find the volume between a surface defined by a function of two variables and a plane. 112 2 • Limits Access for free at openstax.org
SECTION 2.1 EXERCISESFor the following exercises, points and are on the graph of the function 1 . [T] Complete the following table with the appropriate values: y-coordinate of Q, the point and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits. x y msec 1.1 a. e. i. 1.01 b. f. j. 1.001 c. g. k. 1.0001 d. h. l. 2 . Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to f at 3 . Use the value in the preceding exercise to find an equation of the tangent line at point P. Graph and the tangent line. For the following exercises, points and are on the graph of the function 4 . [T] Complete the following table with the appropriate values: y-coordinate of Q, the point and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits. x y msec 1.1 a. e. i. 1.01 b. f. j. 1.001 c. g. k. 1.0001 d. h. l. 5 . Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at 6 . Use the value in the preceding exercise to find an equation of the tangent line at point P. Graph and the tangent line. 2.1 • A Preview of Calculus 113
For the following exercises, points and are on the graph of the function 7 . [T] Complete the following table with the appropriate values: y-coordinate of Q, the point and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits. x y msec 4.1 a. e. i. 4.01 b. f. j. 4.001 c. g. k. 4.0001 d. h. l. 8 . Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at 9 . Use the value in the preceding exercise to find an equation of the tangent line at point P. For the following exercises, points and are on the graph of the function 10. [T] Complete the following table with the appropriate values: y-coordinate of Q, the point and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits. ϕ y msec 1.4 a. e. i. 1.49 b. f. j. 1.499 c. g. k. 1.4999 d. h. l. 11. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at 12. Use the value in the preceding exercise to find an equation of the tangent line at point P. 114 2 • Limits Access for free at openstax.org
For the following exercises, points and are on the graph of the function 13. [T] Complete the following table with the appropriate values: y-coordinate of Q, the point and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits. x y msec −1.05 a. e. i. −1.01 b. f. j. −1.005 c. g. k. −1.001 d. h. l. 14. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to f at 15 . Use the value in the preceding exercise to find an equation of the tangent line at point P. For the following exercises, the position function of a ball dropped from the top of a 200-meter tall building is given by where position s is measured in meters and time t is measured in seconds. Round your answer to eight significant digits. 16. [T] Compute the average velocity of the ball over the given time intervals. a. b. c. d. 17 . Use the preceding exercise to guess the instantaneous velocity of the ball at sec. For the following exercises, consider a stone tossed into the air from ground level with an initial velocity of 15 m/sec. Its height in meters at time t seconds is 18. [T] Compute the average velocity of the stone over the given time intervals. a. b. c. d. 19 . Use the preceding exercise to guess the instantaneous velocity of the stone at sec. For the following exercises, consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by where t is measured in seconds. 20. [T] Compute the average velocity of the rocket over the given time intervals. a. b. c. d. 21 . Use the preceding exercise to guess the instantaneous velocity of the rocket at sec. 2.1 • A Preview of Calculus 115
For the following exercises, consider an athlete running a 40-m dash. The position of the athlete is given by where d is the position in meters and t is the time elapsed, measured in seconds. 22. [T] Compute the average velocity of the runner over the given time intervals. a. b. c. d. 23 . Use the preceding exercise to guess the instantaneous velocity of the runner at sec. For the following exercises, consider the function 24. Sketch the graph of f over the interval and shade the region above the x-axis. 25 . Use the preceding exercise to find the aproximate value of the area between the x-axis and the graph of f over the interval using rectangles. For the rectangles, use the square units, and approximate both above and below the lines. Use geometry to find the exact answer. For the following exercises, consider the function (Hint: This is the upper half of a circle of radius 1 positioned at 26. Sketch the graph of f over the interval 27 . Use the preceding exercise to find the aproximate area between the x-axis and the graph of f over the interval using rectangles. For the rectangles, use squares 0.4 by 0.4 units, and approximate both above and below the lines. Use geometry to find the exact answer. For the following exercises, consider the function 28. Sketch the graph of f over the interval 29 . Approximate the area of the region between the x-axis and the graph of f over the interval 2.2 The Limit of a Function Learning Objectives 2.2.1 Using correct notation, describe the limit of a function. 2.2.2 Use a table of values to estimate the limit of a function or to identify when the limit does not exist. 2.2.3 Use a graph to estimate the limit of a function or to identify when the limit does not exist. 2.2.4 Define one-sided limits and provide examples. 2.2.5 Explain the relationship between one-sided and two-sided limits. 2.2.6 Using correct notation, describe an infinite limit. 2.2.7 Define a vertical asymptote. The concept of a limit or limiting process, essential to the understanding of calculus, has been around for thousands of years. In fact, early mathematicians used a limiting process to obtain better and better approximations of areas of 116 2 • Limits Access for free at openstax.org
circles. Yet, the formal definition of a limit—as we know and understand it today—did not appear until the late 19th century. We therefore begin our quest to understand limits, as our mathematical ancestors did, by using an intuitive approach. At the end of this chapter, armed with a conceptual understanding of limits, we examine the formal definition of a limit. We begin our exploration of limits by taking a look at the graphs of the functions which are shown in Figure 2.12. In particular, let’s focus our attention on the behavior of each graph at and around Figure 2.12 These graphs show the behavior of three different functions around Each of the three functions is undefined at but if we make this statement and no other, we give a very incomplete picture of how each function behaves in the vicinity of To express the behavior of each graph in the vicinity of 2 more completely, we need to introduce the concept of a limit. Intuitive Definition of a Limit Let’s first take a closer look at how the function behaves around in Figure 2.12. As the values of x approach 2 from either side of 2, the values of approach 4. Mathematically, we say that the limit of as x approaches 2 is 4. Symbolically, we express this limit as From this very brief informal look at one limit, let’s start to develop an intuitive definition of the limit. We can think of the limit of a function at a number a as being the one real number L that the functional values approach as the x-values approach a, provided such a real number L exists. Stated more carefully, we have the following definition: Definition Let be a function defined at all values in an open interval containing a, with the possible exception of a itself, and let L be a real number. If all values of the function approach the real number L as the values of approach the number a, then we say that the limit of as x approaches a is L. (More succinct, as x gets closer to a, gets closer and stays close to L.) Symbolically, we express this idea as (2.3) We can estimate limits by constructing tables of functional values and by looking at their graphs. This process is described in the following Problem-Solving Strategy. 2.2 • The Limit of a Function 117
Problem-Solving Strategy Evaluating a Limit Using a Table of Functional Values 1. To evaluate we begin by completing a table of functional values. We should choose two sets ofx-values—one set of values approaching a and less than a, and another set of values approaching a and greater than a. Table 2.1 demonstrates what your tables might look like. x x Use additional values as necessary. Use additional values as necessary. Table 2.1 Table of Functional Values for 2. Next, let’s look at the values in each of the columns and determine whether the values seem to be approaching a single value as we move down each column. In our columns, we look at the sequence and so on, and and so on. ( Note: Although we have chosen the x-values and so forth, and these values will probably work nearly every time, on very rare occasions we may need to modify our choices.) 3. If both columns approach a common y-value L, we state We can use the following strategy to confirm the result obtained from the table or as an alternative method for estimating a limit. 4. Using a graphing calculator or computer software that allows us to graph functions, we can plot the function making sure the functional values of for x-values near a are in our window. We can use the trace feature to move along the graph of the function and watch the y-value readout as the x-values approach a. If the y-values approach L as our x-values approach a from both directions, then We may need to zoom in on our graph and repeat this process several times. We apply this Problem-Solving Strategy to compute a limit in Example 2.4. EXAMPLE 2.4 Evaluating a Limit Using a Table of Functional Values 1 Evaluate using a table of functional values. Solution We have calculated the values of for the values of x listed in Table 2.2. 118 2 • Limits Access for free at openstax.org
x x −0.1 0.998334166468 0.1 0.998334166468 −0.01 0.999983333417 0.01 0.999983333417 −0.001 0.999999833333 0.001 0.999999833333 −0.0001 0.999999998333 0.0001 0.999999998333 Table 2.2 Table of Functional Values for Note: The values in this table were obtained using a calculator and using all the places given in the calculator output. As we read down each column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that A calculator or computer-generated graph of would be similar to that shown in Figure 2.13, and it confirms our estimate. Figure 2.13 The graph of confirms the estimate from Table 2.2. EXAMPLE 2.5 Evaluating a Limit Using a Table of Functional Values 2 Evaluate using a table of functional values. Solution As before, we use a table—in this case, Table 2.3—to list the values of the function for the given values of x. x x 3.9 0.251582341869 4.1 0.248456731317 3.99 0.25015644562 4.01 0.24984394501 3.999 0.250015627 4.001 0.249984377 Table 2.3 Table of Functional Values for 2.2 • The Limit of a Function 119
x x 3.9999 0.250001563 4.0001 0.249998438 3.99999 0.25000016 4.00001 0.24999984 Table 2.3 Table of Functional Values for After inspecting this table, we see that the functional values less than 4 appear to be decreasing toward 0.25 whereas the functional values greater than 4 appear to be increasing toward 0.25. We conclude that We confirm this estimate using the graph of shown in Figure 2.14. Figure 2.14 The graph of confirms the estimate from Table 2.3. 2.4 Estimate using a table of functional values. Use a graph to confirm your estimate. At this point, we see from Example 2.4 and Example 2.5 that it may be just as easy, if not easier, to estimate a limit of a function by inspecting its graph as it is to estimate the limit by using a table of functional values. In Example 2.6, we evaluate a limit exclusively by looking at a graph rather than by using a table of functional values. EXAMPLE 2.6 Evaluating a Limit Using a Graph For shown in Figure 2.15, evaluate 120 2 • Limits Access for free at openstax.org
Figure 2.15 The graph of includes one value not on a smooth curve. Solution Despite the fact that as the x-values approach −1 from either side, the values approach 3. Therefore, Note that we can determine this limit without even knowing the algebraic expression of the function. Based on Example 2.6, we make the following observation: It is possible for the limit of a function to exist at a point, and for the function to be defined at this point, but the limit of the function and the value of the function at the point may be different. 2.5 Use the graph of in Figure 2.16 to evaluate if possible. Figure 2.16 Looking at a table of functional values or looking at the graph of a function provides us with useful insight into the value of the limit of a function at a given point. However, these techniques rely too much on guesswork. We eventually need to develop alternative methods of evaluating limits. These new methods are more algebraic in nature and we explore them in the next section; however, at this point we introduce two special limits that are foundational to the techniques to come. 2.2 • The Limit of a Function 121
Theorem 2.1 Two Important Limits Let a be a real number and c be a constant. i. (2.4) ii. (2.5) We can make the following observations about these two limits. i. For the first limit, observe that as x approaches a, so does because Consequently, ii. For the second limit, consider Table 2.4. x x c c c c c c c c Table 2.4 Table of Functional Values for Observe that for all values of x (regardless of whether they are approaching a), the values remain constant at c. We have no choice but to conclude The Existence of a Limit As we consider the limit in the next example, keep in mind that for the limit of a function to exist at a point, the functional values must approach a single real-number value at that point. If the functional values do not approach a single value, then the limit does not exist. EXAMPLE 2.7 Evaluating a Limit That Fails to Exist Evaluate using a table of values. Solution Table 2.5 lists values for the function for the given values of x. x x −0.1 0.544021110889 0.1 −0.544021110889 −0.01 0.50636564111 0.01 −0.50636564111 −0.001 −0.8268795405312 0.001 0.826879540532 −0.0001 0.305614388888 0.0001 −0.305614388888 Table 2.5 Table of Functional Values for 122 2 • Limits Access for free at openstax.org
x x −0.00001 −0.035748797987 0.00001 0.035748797987 −0.000001 0.349993504187 0.000001 −0.349993504187 Table 2.5 Table of Functional Values for After examining the table of functional values, we can see that the y-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let’s take a more systematic approach. Take the following sequence of x-values approaching 0: The corresponding y-values are At this point we can indeed conclude that does not exist. (Mathematicians frequently abbreviate “does not exist” as DNE. Thus, we would write DNE.) The graph of is shown in Figure 2.17 and it gives a clearer picture of the behavior of as x approaches 0. You can see that oscillates ever more wildly between −1 and 1 as x approaches 0. Figure 2.17 The graph of oscillates rapidly between −1 and 1 as x approaches 0. 2.6 Use a table of functional values to evaluate if possible. One-Sided Limits Sometimes indicating that the limit of a function fails to exist at a point does not provide us with enough information about the behavior of the function at that particular point. To see this, we now revisit the function introduced at the beginning of the section (see Figure 2.12(b)). As we pick values of x close to 2, does not approach a single value, so the limit as x approaches 2 does not exist—that is, DNE. However, this statement alone does not give us a complete picture of the behavior of the function around the x-value 2. To provide a more accurate description, we introduce the idea of a one-sided limit. For all values to the left of 2 (or the negative side of 2), Thus, as x approaches 2 from the left, approaches −1. Mathematically, we say that the limit as x approaches 2 from the left is −1. Symbolically, we express this idea as 2.2 • The Limit of a Function 123
Similarly, as x approaches 2 from the right (or from the positive side), approaches 1. Symbolically, we express this idea as We can now present an informal definition of one-sided limits. Definition We define two types of one-sided limits.Limit from the left: Let be a function defined at all values in an open interval of the form ( c, a), and let L be a real number. If the values of the function approach the real number L as the values of x (where approach the number a, then we say that L is the limit of as x approaches a from the left. Symbolically, we express this idea as (2.6) Limit from the right: Let be a function defined at all values in an open interval of the form and let L be a real number. If the values of the function approach the real number L as the values of x (where approach the number a, then we say that L is the limit of as x approaches a from the right. Symbolically, we express this idea as (2.7) EXAMPLE 2.8 Evaluating One-Sided Limits For the function evaluate each of the following limits. a. b. Solution We can use tables of functional values again Table 2.6. Observe that for values of x less than 2, we use and for values of x greater than 2, we use x x 1.9 2.9 2.1 0.41 1.99 2.99 2.01 0.0401 1.999 2.999 2.001 0.004001 1.9999 2.9999 2.0001 0.00040001 1.99999 2.99999 2.00001 0.0000400001 Table 2.6 Table of Functional Values for Based on this table, we can conclude that a. and b. Therefore, the (two-sided) limit of 124 2 • Limits Access for free at openstax.org
does not exist at Figure 2.18 shows a graph of and reinforces our conclusion about these limits. Figure 2.18 The graph of has a break at 2.7 Use a table of functional values to estimate the following limits, if possible. a. b. Let us now consider the relationship between the limit of a function at a point and the limits from the right and left at that point. It seems clear that if the limit from the right and the limit from the left have a common value, then that common value is the limit of the function at that point. Similarly, if the limit from the left and the limit from the right take on different values, the limit of the function does not exist. These conclusions are summarized in Relating One-Sided and Two-Sided Limits. Theorem 2.2 Relating One-Sided and Two-Sided Limits Let be a function defined at all values in an open interval containing a, with the possible exception of a itself, and let L be a real number. Then, Infinite Limits Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits. We now turn our attention to the third and final function introduced at the beginning of this section (see Figure 2.12(c)). From its graph we see that as the values of x approach 2, the values of become larger and larger and, in fact, become infinite. Mathematically, we say that the limit of as x approaches 2 is positive infinity. Symbolically, we express this idea as 2.2 • The Limit of a Function 125
∞ More generally, we define infinite limits as follows: Definition We define three types of infinite limits.Infinite limits from the left: Let be a function defined at all values in an open interval of the form i. If the values of increase without bound as the values of x (where approach the number a, then we say that the limit as x approaches a from the left is positive infinity and we write ∞ (2.8) ii. If the values of decrease without bound as the values of x (where approach the number a, then we say that the limit as x approaches a from the left is negative infinity and we write ∞ (2.9) Infinite limits from the right: Let be a function defined at all values in an open interval of the form i. If the values of increase without bound as the values of x (where approach the number a, then we say that the limit as x approaches a from the right is positive infinity and we write ∞ (2.10) ii. If the values of decrease without bound as the values of x (where approach the number a, then we say that the limit as x approaches a from the right is negative infinity and we write ∞ (2.11) Two-sided infinite limit: Let be defined for all in an open interval containing a. i. If the values of increase without bound as the values of x (where approach the number a, then we say that the limit as x approaches a is positive infinity and we write ∞ (2.12) ii. If the values of decrease without bound as the values of x (where approach the number a, then we say that the limit as x approaches a is negative infinity and we write ∞ (2.13) It is important to understand that when we write statements such as ∞ or ∞ we are describing the behavior of the function, as we have just defined it. We are not asserting that a limit exists. For the limit of a function to exist at a, it must approach a real number L as x approaches a. That said, if, for example, ∞ we always write ∞ rather than DNE. EXAMPLE 2.9 Recognizing an Infinite Limit Evaluate each of the following limits, if possible. Use a table of functional values and graph to confirm your conclusion. a. b. c. 126 2 • Limits Access for free at openstax.org
Solution Begin by constructing a table of functional values. x x −0.1 −10 0.1 10 −0.01 −100 0.01 100 −0.001 −1000 0.001 1000 −0.0001 −10,000 0.0001 10,000 −0.00001 −100,000 0.00001 100,000 −0.000001 −1,000,000 0.000001 1,000,000 Table 2.7 Table of Functional Values for a. The values of decrease without bound as x approaches 0 from the left. We conclude that ∞ b. The values of increase without bound as x approaches 0 from the right. We conclude that ∞ c. Since ∞ and ∞ have different values, we conclude that The graph of in Figure 2.19 confirms these conclusions. Figure 2.19 The graph of confirms that the limit as x approaches 0 does not exist. 2.8 Evaluate each of the following limits, if possible. Use a table of functional values and graph to confirm your conclusion. 2.2 • The Limit of a Function 127
a. b. c. It is useful to point out that functions of the form where n is a positive integer, have infinite limits as x approaches a from either the left or right (Figure 2.20). These limits are summarized in Infinite Limits from Positive Integers. Figure 2.20 The function has infinite limits at a. Theorem 2.3 Infinite Limits from Positive Integers If n is a positive even integer, then ∞ If n is a positive odd integer, then ∞ and ∞ We should also point out that in the graphs of points on the graph having x-coordinates very near toa are very close to the vertical line That is, as x approaches a, the points on the graph of are closer to the line The line is called a vertical asymptote of the graph. We formally define a vertical asymptote as follows: Definition Let be a function. If any of the following conditions hold, then the line is a vertical asymptote of 128 2 • Limits Access for free at openstax.org
∞ ∞ ∞ EXAMPLE 2.10 Finding a Vertical Asymptote Evaluate each of the following limits using Infinite Limits from Positive Integers. Identify any vertical asymptotes of the function a. b. c. Solution We can use Infinite Limits from Positive Integers directly. a. ∞ b. ∞ c. ∞ The function has a vertical asymptote of 2.9 Evaluate each of the following limits. Identify any vertical asymptotes of the function a. b. c. In the next example we put our knowledge of various types of limits to use to analyze the behavior of a function at several different points. EXAMPLE 2.11 Behavior of a Function at Different Points Use the graph of in Figure 2.21 to determine each of the following values: a. b. c. d. 2.2 • The Limit of a Function 129
Figure 2.21 The graph shows Solution Using Infinite Limits from Positive Integers and the graph for reference, we arrive at the following values: a. b. is undefined c. DNE; d. ∞ ∞ ∞ is undefined 2.10 Evaluate for shown here: 130 2 • Limits Access for free at openstax.org
EXAMPLE 2.12 Chapter Opener: Einstein’s Equation Figure 2.22 (credit: NASA) In the chapter opener we mentioned briefly how Albert Einstein showed that a limit exists to how fast any object can travel. Given Einstein’s equation for the mass of a moving object, what is the value of this bound? Solution Our starting point is Einstein’s equation for the mass of a moving object, where is the object’s mass at rest, v is its speed, and c is the speed of light. To see how the mass changes at high speeds, we can graph the ratio of masses as a function of the ratio of speeds, (Figure 2.23). Figure 2.23 This graph shows the ratio of masses as a function of the ratio of speeds in Einstein’s equation for the mass of a moving object. We can see that as the ratio of speeds approaches 1—that is, as the speed of the object approaches the speed of light—the ratio of masses increases without bound. In other words, the function has a vertical asymptote at We can try a few values of this ratio to test this idea. 0.99 0.1411 7.089 0.999 0.0447 22.37 0.9999 0.0141 70.71 Table 2.8 Ratio of Masses and Speeds for a Moving Object 2.2 • The Limit of a Function 131
Thus, according to Table 2.8, if an object with mass 100 kg is traveling at 0.9999 c, its mass becomes 7071 kg. Since no object can have an infinite mass, we conclude that no object can travel at or more than the speed of light. SECTION 2.2 EXERCISESFor the following exercises, consider the function 30. [T] Complete the following table for the function. Round your solutions to four decimal places. x x 0.9 a. 1.1 e. 0.99 b. 1.01 f. 0.999 c. 1.001 g. 0.9999 d. 1.0001 h. 31 . What do your results in the preceding exercise indicate about the two-sided limit Explain your response. For the following exercises, consider the function 32. [T] Make a table showing the values of f for and for Round your solutions to five decimal places.x x −0.01 a. 0.01 e. −0.001 b. 0.001 f. −0.0001 c. 0.0001 g. −0.00001 d. 0.00001 h. 33 . What does the table of values in the preceding exercise indicate about the function 34 . To which mathematical constant does the limit in the preceding exercise appear to be getting closer? 132 2 • Limits Access for free at openstax.org
In the following exercises, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places. 35. [T] x x −0.1 a. 0.1 e. −0.01 b. 0.01 f. −0.001 c. 0.001 g. −0.0001 d. 0.0001 h. 36. [T] ±0.1, ±0.01, ±0.001, ±0.0001 X x −0.1 a. 0.1 e. −0.01 b. 0.01 f. −0.001 c. 0.001 g. −0.0001 d. 0.0001 h. 37. Use the preceding two exercises to conjecture (guess) the value of the following limit: for a, a positive real value. [T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits. 38. x x 1.9 a. 2.1 e. 1.99 b. 2.01 f. 1.999 c. 2.001 g. 1.9999 d. 2.0001 h. 39 . x x 0.9 a. 1.1 e. 0.99 b. 1.01 f. 0.999 c. 1.001 g. 0.9999 d. 1.0001 h. 40. x x −0.1 a. 0.1 e. −0.01 b. 0.01 f. −0.001 c. 0.001 g. −0.0001 d. 0.0001 h. 41 . z z −0.1 a. 0.1 e. −0.01 b. 0.01 f. −0.001 c. 0.001 g. −0.0001 d. 0.0001 h. 2.2 • The Limit of a Function 133
42. t 0.1 a. 0.01 b. 0.001 c. 0.0001 d. 43 . x x 1.9 a. 2.1 e. 1.99 b. 2.01 f. 1.999 c. 2.001 g. 1.9999 d. 2.0001 h. [T] In the following exercises, set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail? 44. θ θ −0.1 a. 0.1 e. −0.01 b. 0.01 f. −0.001 c. 0.001 g. −0.0001 d. 0.0001 h. 45 . 0.1 a. 0.01 b. 0.001 c. 0.0001 d. 134 2 • Limits Access for free at openstax.org
In the following exercises, consider the graph of the function shown here. Which of the statements about are true and which are false? Explain why a statement is false. 46. 47. 48 . 49 . In the following exercises, use the following graph of the function to find the values, if possible. Estimate when necessary. 50 . 51. 52 . 53 . 54. 2.2 • The Limit of a Function 135
In the following exercises, use the graph of the function shown here to find the values, if possible. Estimate when necessary. 55 . 56. 57 . 58 . In the following exercises, use the graph of the function shown here to find the values, if possible. Estimate when necessary. 59 . 60. 61 . 62 . 63. 64 . 136 2 • Limits Access for free at openstax.org
In the following exercises, use the graph of the function shown here to find the values, if possible. Estimate when necessary. 65 . 66. 67 . In the following exercises, use the graph of the function shown here to find the values, if possible. Estimate when necessary. 68 . 69. 70 . 2.2 • The Limit of a Function 137
In the following exercises, use the graph of the function shown here to find the values, if possible. Estimate when necessary. 71 . 72. 73 . 74 . 75. In the following exercises, sketch the graph of a function with the given properties. 76. is not defined. 77. ∞ ∞ ∞ ∞ ∞ 78. ∞ ∞ ∞ ∞ 79. ∞ ∞ ∞ 138 2 • Limits Access for free at openstax.org
80. ∞ ∞ ∞ ∞ ∞ ∞ 81. Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, x, is shown here. We are mainly interested in the location of the front of the shock, labeled in the diagram. a. Evaluate b. Evaluate c. Evaluate Explain the physical meanings behind your answers. 2.2 • The Limit of a Function 139
82. A track coach uses a camera with a fast shutter to estimate the position of a runner with respect to time. A table of the values of position of the athlete versus time is given here, where x is the position in meters of the runner and t is time in seconds. What is What does it mean physically? t (sec) x (m) 1.75 4.5 1.95 6.1 1.99 6.42 2.01 6.58 2.05 6.9 2.25 8.5 2.3 The Limit Laws Learning Objectives 2.3.1 Recognize the basic limit laws. 2.3.2 Use the limit laws to evaluate the limit of a function. 2.3.3 Evaluate the limit of a function by factoring. 2.3.4 Use the limit laws to evaluate the limit of a polynomial or rational function. 2.3.5 Evaluate the limit of a function by factoring or by using conjugates. 2.3.6 Evaluate the limit of a function by using the squeeze theorem. In the previous section, we evaluated limits by looking at graphs or by constructing a table of values. In this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. We begin by restating two useful limit results from the previous section. These two results, together with the limit laws, serve as a foundation for calculating many limits. Evaluating Limits with the Limit Laws The first two limit laws were stated in Two Important Limits and we repeat them here. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions. Theorem 2.4 Basic Limit Results For any real number a and any constant c, i. (2.14) ii. (2.15) 140 2 • Limits Access for free at openstax.org
EXAMPLE 2.13 Evaluating a Basic Limit Evaluate each of the following limits using Basic Limit Results. a. b. Solution a. The limit of x as x approaches a is a: b. The limit of a constant is that constant: We now take a look at the limit laws, the individual properties of limits. The proofs that these laws hold are omitted here. Theorem 2.5 Limit Laws Let and be defined for all over some open interval containing a. Assume that L and M are real numbers such that and Let c be a constant. Then, each of the following statements holds: Sum law for limits: Difference law for limits: Constant multiple law for limits: Product law for limits: Quotient law for limits: for Power law for limits: for every positive integer n. Root law for limits: for all L if n is odd and for if n is even and . We now practice applying these limit laws to evaluate a limit. EXAMPLE 2.14 Evaluating a Limit Using Limit Laws Use the limit laws to evaluate Solution Let’s apply the limit laws one step at a time to be sure we understand how they work. We need to keep in mind the requirement that, at each application of a limit law, the new limits must exist for the limit law to be applied. 2.3 • The Limit Laws 141
EXAMPLE 2.15 Using Limit Laws Repeatedly Use the limit laws to evaluate Solution To find this limit, we need to apply the limit laws several times. Again, we need to keep in mind that as we rewrite the limit in terms of other limits, each new limit must exist for the limit law to be applied. 2.11 Use the limit laws to evaluate In each step, indicate the limit law applied. Limits of Polynomial and Rational Functions By now you have probably noticed that, in each of the previous examples, it has been the case that This is not always true, but it does hold for all polynomials for any choice of a and for all rational functions at all values ofa for which the rational function is defined. Theorem 2.6 Limits of Polynomial and Rational Functions Let and be polynomial functions. Let a be a real number. Then, To see that this theorem holds, consider the polynomial By applying the sum, constant multiple, and power laws, we end up with It now follows from the quotient law that if and are polynomials for which then 142 2 • Limits Access for free at openstax.org
Example 2.16 applies this result. EXAMPLE 2.16 Evaluating a Limit of a Rational Function Evaluate the Solution Since 3 is in the domain of the rational function we can calculate the limit by substituting 3 for x into the function. Thus, 2.12 Evaluate Additional Limit Evaluation Techniques As we have seen, we may evaluate easily the limits of polynomials and limits of some (but not all) rational functions by direct substitution. However, as we saw in the introductory section on limits, it is certainly possible for to exist when is undefined. The following observation allows us to evaluate many limits of this type: If for all over some open interval containing a, then To understand this idea better, consider the limit The function and the function are identical for all values of The graphs of these two functions are shown in Figure 2.24. Figure 2.24 The graphs of and are identical for all Their limits at 1 are equal. We see that 2.3 • The Limit Laws 143
The limit has the form where and (In this case, we say that has the indeterminate form The following Problem-Solving Strategy provides a general outline for evaluating limits of this type. Problem-Solving Strategy Calculating a Limit When has the Indeterminate Form 0/0 1. First, we need to make sure that our function has the appropriate form and cannot be evaluated immediately using the limit laws. 2. We then need to find a function that is equal to for all over some interval containing a. To do this, we may need to try one or more of the following steps: a. If and are polynomials, we should factor each function and cancel out any common factors. b. If the numerator or denominator contains a difference involving a square root, we should try multiplying the numerator and denominator by the conjugate of the expression involving the square root. c. If is a complex fraction, we begin by simplifying it. 3. Last, we apply the limit laws. The next examples demonstrate the use of this Problem-Solving Strategy. Example 2.17 illustrates the factor-and-cancel technique; Example 2.18 shows multiplying by a conjugate. In Example 2.19, we look at simplifying a complex fraction. EXAMPLE 2.17 Evaluating a Limit by Factoring and Canceling Evaluate Solution Step 1. The function is undefined for In fact, if we substitute 3 into the function we get which is indeterminate. Factoring and canceling is a good strategy: Step 2. For all Therefore, Step 3. Evaluate using the limit laws: 2.13 Evaluate 144 2 • Limits Access for free at openstax.org
EXAMPLE 2.18 Evaluating a Limit by Multiplying by a Conjugate Evaluate Solution Step 1. has the form at −1. Let’s begin by multiplying by the conjugate of on the numerator and denominator: Step 2. We then multiply out the numerator. We don’t multiply out the denominator because we are hoping that the in the denominator cancels out in the end: Step 3. Then we cancel: Step 4. Last, we apply the limit laws: 2.14 Evaluate EXAMPLE 2.19 Evaluating a Limit by Simplifying a Complex Fraction Evaluate Solution Step 1. has the form at 1. We simplify the algebraic fraction by multiplying by Step 2. Next, we multiply through the numerators. Do not multiply the denominators because we want to be able to cancel the factor Step 3. Then, we simplify the numerator: Step 4. Now we factor out −1 from the numerator: 2.3 • The Limit Laws 145
Step 5. Then, we cancel the common factors of Step 6. Last, we evaluate using the limit laws: 2.15 Evaluate Example 2.20 does not fall neatly into any of the patterns established in the previous examples. However, with a little creativity, we can still use these same techniques. EXAMPLE 2.20 Evaluating a Limit When the Limit Laws Do Not Apply Evaluate Solution Both and fail to have a limit at zero. Since neither of the two functions has a limit at zero, we cannot apply the sum law for limits; we must use a different strategy. In this case, we find the limit by performing addition and then applying one of our previous strategies. Observe that Thus, 2.16 Evaluate Let’s now revisit one-sided limits. Simple modifications in the limit laws allow us to apply them to one-sided limits. For example, to apply the limit laws to a limit of the form we require the function to be defined over an open interval of the form for a limit of the form we require the function to be defined over an open interval of the form Example 2.21 illustrates this point. EXAMPLE 2.21 Evaluating a One-Sided Limit Using the Limit Laws Evaluate each of the following limits, if possible. a. b. 146 2 • Limits Access for free at openstax.org
Solution Figure 2.25 illustrates the function and aids in our understanding of these limits. Figure 2.25 The graph shows the function a. The function is defined over the interval ∞ Since this function is not defined to the left of 3, we cannot apply the limit laws to compute In fact, since is undefined to the left of 3, does not exist. b. Since is defined to the right of 3, the limit laws do apply to By applying these limit laws we obtain In Example 2.22 we look at one-sided limits of a piecewise-defined function and use these limits to draw a conclusion about a two-sided limit of the same function. EXAMPLE 2.22 Evaluating a Two-Sided Limit Using the Limit Laws For evaluate each of the following limits: a. b. c. Solution Figure 2.26 illustrates the function and aids in our understanding of these limits. Figure 2.26 This graph shows a function a. Since for all x in ∞ replace in the limit with and apply the limit laws: 2.3 • The Limit Laws 147
b. Since for all x in ∞ replace in the limit with and apply the limit laws: c. Since and we conclude that does not exist. 2.17 Graph and evaluate We now turn our attention to evaluating a limit of the form where where and That is, has the form at a. EXAMPLE 2.23 Evaluating a Limit of the Form Using the Limit Laws Evaluate Solution Step 1. After substituting in we see that this limit has the form That is, as x approaches 2 from the left, the numerator approaches −1; and the denominator approaches 0. Consequently, the magnitude of becomes infinite. To get a better idea of what the limit is, we need to factor the denominator: Step 2. Since is the only part of the denominator that is zero when 2 is substituted, we then separate from the rest of the function: Step 3. and ∞ Therefore, the product of and has a limit of ∞ 2.18 Evaluate The Squeeze Theorem The techniques we have developed thus far work very well for algebraic functions, but we are still unable to evaluate limits of very basic trigonometric functions. The next theorem, called the squeeze theorem, proves very useful for establishing basic trigonometric limits. This theorem allows us to calculate limits by “squeezing” a function, with a limit at a point a that is unknown, between two functions having a common known limit at a. Figure 2.27 illustrates this idea. 148 2 • Limits Access for free at openstax.org
Figure 2.27 The Squeeze Theorem applies when and Theorem 2.7 The Squeeze Theorem Let and be defined for all over an open interval containing a. If for all in an open interval containing a and where L is a real number, then EXAMPLE 2.24 Applying the Squeeze Theorem Apply the squeeze theorem to evaluate Solution Because for all x, we have . Since from the squeeze theorem, we obtain The graphs of and are shown in Figure 2.28. 2.3 • The Limit Laws 149
Figure 2.28 The graphs of and are shown around the point 2.19 Use the squeeze theorem to evaluate We now use the squeeze theorem to tackle several very important limits. Although this discussion is somewhat lengthy, these limits prove invaluable for the development of the material in both the next section and the next chapter. The first of these limits is Consider the unit circle shown in Figure 2.29. In the figure, we see that is the y-coordinate on the unit circle and it corresponds to the line segment shown in blue. The radian measure of angle θ is the length of the arc it subtends on the unit circle. Therefore, we see that for Figure 2.29 The sine function is shown as a line on the unit circle. Because and by using the squeeze theorem we conclude that To see that as well, observe that for and hence, Consequently, It follows that An application of the squeeze theorem produces the desired limit. Thus, since and (2.16) Next, using the identity for we see that 150 2 • Limits Access for free at openstax.org
(2.17) We now take a look at a limit that plays an important role in later chapters—namely, To evaluate this limit, we use the unit circle in Figure 2.30. Notice that this figure adds one additional triangle to Figure 2.30. We see that the length of the side opposite angle θ in this new triangle is Thus, we see that for Figure 2.30 The sine and tangent functions are shown as lines on the unit circle. By dividing by in all parts of the inequality, we obtain Equivalently, we have Since we conclude that By applying a manipulation similar to that used in demonstrating that we can show that Thus, (2.18) In Example 2.25 we use this limit to establish This limit also proves useful in later chapters. EXAMPLE 2.25 Evaluating an Important Trigonometric Limit Evaluate Solution In the first step, we multiply by the conjugate so that we can use a trigonometric identity to convert the cosine in the numerator to a sine: 2.3 • The Limit Laws 151
Therefore, (2.19) 2.20 Evaluate STUDENT PROJECT Deriving the Formula for the Area of a Circle Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. The Greek mathematician Archimedes (ca. 287−212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. He never came up with the idea of a limit, but we can use this idea to see what his geometric constructions could have predicted about the limit. We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think of the regular polygon as being made up of n triangles. By taking the limit as the vertex angle of these triangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps: 1. Express the height h and the base b of the isosceles triangle in Figure 2.31 in terms of and r. Figure 2.31 2. Using the expressions that you obtained in step 1, express the area of the isosceles triangle in terms of θ and r. (Substitute for in your expression.) 3. If an n-sided regular polygon is inscribed in a circle of radius r, find a relationship between θ and n. Solve this for n. Keep in mind there are 2π radians in a circle. (Use radians, not degrees.) 4. Find an expression for the area of the n-sided polygon in terms of r and θ. 5. To find a formula for the area of the circle, find the limit of the expression in step 4 as θ goes to zero. ( Hint: 152 2 • Limits Access for free at openstax.org
The technique of estimating areas of regions by using polygons is revisited in Introduction to Integration. SECTION 2.3 EXERCISESIn the following exercises, use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s). 83. 84 . 85 . 86. In the following exercises, use direct substitution to evaluate each limit. 87. 88 . 89 . 90. 91 . 92 . In the following exercises, use direct substitution to show that each limit leads to the indeterminate form Then, evaluate the limit. 93. 94 . 95 . 96. 97 . 98 . where a is a non-zero real-valued constant 99. 100 . 101 . 102. In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of Example 2.23 to simplify the function to help determine the limit. 103. 104 . 105 . 106. In the following exercises, assume that and Use these three facts and the limit laws to evaluate each limit. 107. 108 . 109 . 110. 111 . 112 . 113. 114 . 2.3 • The Limit Laws 153
[T] In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits. 115. a. b. 116 . a. b. 117 . a. b. In the following exercises, use the following graphs and the limit laws to evaluate each limit. 118. 119. 120 . 121 . 122. 123 . 124 . 125. 154 2 • Limits Access for free at openstax.org
For the following problems, evaluate the limit using the squeeze theorem. Use a calculator to graph the functions and when possible. 126. [T] True or False? If then 127 . [T] 128. where 129. [T] In physics, the magnitude of an electric field generated by a point charge at a distance r in vacuum is governed by Coulomb’s law: where E represents the magnitude of the electric field, q is the charge of the particle, r is the distance between the particle and where the strength of the field is measured, and is Coulomb’s constant: a. Use a graphing calculator to graph given that the charge of the particle is b. Evaluate What is the physical meaning of this quantity? Is it physically relevant? Why are you evaluating from the right? 130 . [T] The density of an object is given by its mass divided by its volume: a. Use a calculator to plot the volume as a function of density assuming you are examining something of mass 8 kg ( b. Evaluate and explain the physical meaning. 2.4 Continuity Learning Objectives 2.4.1 Explain the three conditions for continuity at a point. 2.4.2 Describe three kinds of discontinuities. 2.4.3 Define continuity on an interval. 2.4.4 State the theorem for limits of composite functions. 2.4.5 Provide an example of the intermediate value theorem. Many functions have the property that their graphs can be traced with a pencil without lifting the pencil from the page. Such functions are called continuous. Other functions have points at which a break in the graph occurs, but satisfy this property over intervals contained in their domains. They are continuous on these intervals and are said to have adiscontinuity at a point where a break occurs. We begin our investigation of continuity by exploring what it means for a function to have continuity at a point. Intuitively, a function is continuous at a particular point if there is no break in its graph at that point. Continuity at a Point Before we look at a formal definition of what it means for a function to be continuous at a point, let’s consider various functions that fail to meet our intuitive notion of what it means to be continuous at a point. We then create a list of conditions that prevent such failures. 2.4 • Continuity 155
Our first function of interest is shown in Figure 2.32. We see that the graph of has a hole at a. In fact, is undefined. At the very least, for to be continuous at a, we need the following condition: Figure 2.32 The function is not continuous at a because is undefined. However, as we see in Figure 2.33, this condition alone is insufficient to guarantee continuity at the point a. Although is defined, the function has a gap at a. In this example, the gap exists because does not exist. We must add another condition for continuity at a—namely, Figure 2.33 The function is not continuous at a because does not exist. However, as we see in Figure 2.34, these two conditions by themselves do not guarantee continuity at a point. The function in this figure satisfies both of our first two conditions, but is still not continuous at a. We must add a third condition to our list: Figure 2.34 The function is not continuous at a because Now we put our list of conditions together and form a definition of continuity at a point. 156 2 • Limits Access for free at openstax.org
Definition A function is continuous at a point a if and only if the following three conditions are satisfied: i. is defined ii. exists iii. A function is discontinuous at a point a if it fails to be continuous at a. The following procedure can be used to analyze the continuity of a function at a point using this definition. Problem-Solving Strategy Determining Continuity at a Point 1. Check to see if is defined. If is undefined, we need go no further. The function is not continuous at a. If is defined, continue to step 2. 2. Compute In some cases, we may need to do this by first computing and If does not exist (that is, it is not a real number), then the function is not continuous at a and the problem is solved. If exists, then continue to step 3. 3. Compare and If then the function is not continuous at a. If then the function is continuous at a. The next three examples demonstrate how to apply this definition to determine whether a function is continuous at a given point. These examples illustrate situations in which each of the conditions for continuity in the definition succeed or fail. EXAMPLE 2.26 Determining Continuity at a Point, Condition 1 Using the definition, determine whether the function is continuous at Justify the conclusion. Solution Let’s begin by trying to calculate We can see that which is undefined. Therefore, is discontinuous at 2 because is undefined. The graph of is shown in Figure 2.35. 2.4 • Continuity 157
Figure 2.35 The function is discontinuous at 2 because is undefined. EXAMPLE 2.27 Determining Continuity at a Point, Condition 2 Using the definition, determine whether the function is continuous at Justify the conclusion. Solution Let’s begin by trying to calculate Thus, is defined. Next, we calculate To do this, we must compute and and Therefore, does not exist. Thus, is not continuous at 3. The graph of is shown in Figure 2.36. 158 2 • Limits Access for free at openstax.org
Figure 2.36 The function is not continuous at 3 because does not exist. EXAMPLE 2.28 Determining Continuity at a Point, Condition 3 Using the definition, determine whether the function is continuous at Solution First, observe that Next, Last, compare and We see that Since all three of the conditions in the definition of continuity are satisfied, is continuous at 2.21 Using the definition, determine whether the function is continuous at If the function is not continuous at 1, indicate the condition for continuity at a point that fails to hold. By applying the definition of continuity and previously established theorems concerning the evaluation of limits, we can state the following theorem. 2.4 • Continuity 159
Theorem 2.8 Continuity of Polynomials and Rational Functions Polynomials and rational functions are continuous at every point in their domains. Proof Previously, we showed that if and are polynomials, for every polynomial and as long as Therefore, polynomials and rational functions are continuous on their domains. □ We now apply Continuity of Polynomials and Rational Functions to determine the points at which a given rational function is continuous. EXAMPLE 2.29 Continuity of a Rational Function For what values of x is continuous? Solution The rational function is continuous for every value of x except 2.22 For what values of x is continuous? Types of Discontinuities As we have seen in Example 2.26 and Example 2.27, discontinuities take on several different appearances. We classify the types of discontinuities we have seen thus far as removable discontinuities, infinite discontinuities, or jump discontinuities. Intuitively, a removable discontinuity is a discontinuity for which there is a hole in the graph, a jump discontinuity is a noninfinite discontinuity for which the sections of the function do not meet up, and an infinite discontinuity is a discontinuity located at a vertical asymptote. Figure 2.37 illustrates the differences in these types of discontinuities. Although these terms provide a handy way of describing three common types of discontinuities, keep in mind that not all discontinuities fit neatly into these categories. Figure 2.37 Discontinuities are classified as (a) removable, (b) jump, or (c) infinite. These three discontinuities are formally defined as follows: Definition If is discontinuous at a, then 160 2 • Limits Access for free at openstax.org
1. has a removable discontinuity at a if exists. (Note: When we state that exists, we mean that where L is a real number.) 2. has a jump discontinuity at a if and both exist, but (Note: When we state that and both exist, we mean that both are real-valued and that neither take on the values ±∞.) 3. has an infinite discontinuity at a if ∞ and/or ∞ EXAMPLE 2.30 Classifying a Discontinuity In Example 2.26, we showed that is discontinuous at Classify this discontinuity as removable, jump, or infinite. Solution To classify the discontinuity at 2 we must evaluate Since f is discontinuous at 2 and exists, f has a removable discontinuity at EXAMPLE 2.31 Classifying a Discontinuity In Example 2.27, we showed that is discontinuous at Classify this discontinuity as removable, jump, or infinite. Solution Earlier, we showed that f is discontinuous at 3 because does not exist. However, since and both exist, we conclude that the function has a jump discontinuity at 3. EXAMPLE 2.32 Classifying a Discontinuity Determine whether is continuous at −1. If the function is discontinuous at −1, classify the discontinuity as removable, jump, or infinite. Solution The function value is undefined. Therefore, the function is not continuous at −1. To determine the type of discontinuity, we must determine the limit at −1. We see that ∞ and ∞ Therefore, the function has an infinite discontinuity at −1. 2.4 • Continuity 161
2.23 For decide whether f is continuous at 1. If f is not continuous at 1, classify the discontinuity as removable, jump, or infinite. Continuity over an Interval Now that we have explored the concept of continuity at a point, we extend that idea to continuity over an interval. As we develop this idea for different types of intervals, it may be useful to keep in mind the intuitive idea that a function is continuous over an interval if we can use a pencil to trace the function between any two points in the interval without lifting the pencil from the paper. In preparation for defining continuity on an interval, we begin by looking at the definition of what it means for a function to be continuous from the right at a point and continuous from the left at a point. Continuity from the Right and from the Left A function is said to be continuous from the right at a if A function is said to be continuous from the left at a if A function is continuous over an open interval if it is continuous at every point in the interval. A function is continuous over a closed interval of the form if it is continuous at every point in and is continuous from the right at a and is continuous from the left at b. Analogously, a function is continuous over an interval of the form if it is continuous over and is continuous from the left at b. Continuity over other types of intervals are defined in a similar fashion. Requiring that and ensures that we can trace the graph of the function from the point to the point without lifting the pencil. If, for example, we would need to lift our pencil to jump from to the graph of the rest of the function over EXAMPLE 2.33 Continuity on an Interval State the interval(s) over which the function is continuous. Solution Since is a rational function, it is continuous at every point in its domain. The domain of is the set∞ ∞ Thus, is continuous over each of the intervals ∞ and ∞ EXAMPLE 2.34 Continuity over an Interval State the interval(s) over which the function is continuous. Solution From the limit laws, we know that for all values of a in We also know that exists and exists. Therefore, is continuous over the interval 2.24 State the interval(s) over which the function is continuous. The Composite Function Theorem allows us to expand our ability to compute limits. In particular, this theorem ultimately 162 2 • Limits Access for free at openstax.org
allows us to demonstrate that trigonometric functions are continuous over their domains. Theorem 2.9 Composite Function Theorem If is continuous at L and then Before we move on to Example 2.35, recall that earlier, in the section on limit laws, we showed Consequently, we know that is continuous at 0. In Example 2.35 we see how to combine this result with the composite function theorem. EXAMPLE 2.35 Limit of a Composite Cosine Function Evaluate Solution The given function is a composite of and Since and is continuous at 0, we may apply the composite function theorem. Thus, 2.25 Evaluate The proof of the next theorem uses the composite function theorem as well as the continuity of and at the point 0 to show that trigonometric functions are continuous over their entire domains. Theorem 2.10 Continuity of Trigonometric Functions Trigonometric functions are continuous over their entire domains. Proof We begin by demonstrating that is continuous at every real number. To do this, we must show that for all values of a. The proof that is continuous at every real number is analogous. Because the remaining trigonometric functions may be expressed in terms of and their continuity follows from the quotient limit law. □ As you can see, the composite function theorem is invaluable in demonstrating the continuity of trigonometric functions. 2.4 • Continuity 163
As we continue our study of calculus, we revisit this theorem many times. The Intermediate Value Theorem Functions that are continuous over intervals of the form where a and b are real numbers, exhibit many useful properties. Throughout our study of calculus, we will encounter many powerful theorems concerning such functions. The first of these theorems is the Intermediate Value Theorem. Theorem 2.11 The Intermediate Value Theorem Let f be continuous over a closed, bounded interval If z is any real number between and then there is a number c in satisfying in Figure 2.38. Figure 2.38 There is a number that satisfies EXAMPLE 2.36 Application of the Intermediate Value Theorem Show that has at least one zero. Solution Since is continuous over ∞ ∞ it is continuous over any closed interval of the form If you can find an interval such that and have opposite signs, you can use the Intermediate Value Theorem to conclude there must be a real number c in that satisfies Note that and Using the Intermediate Value Theorem, we can see that there must be a real number c in that satisfies Therefore, has at least one zero. EXAMPLE 2.37 When Can You Apply the Intermediate Value Theorem? If is continuous over and can we use the Intermediate Value Theorem to conclude that has no zeros in the interval Explain. 164 2 • Limits Access for free at openstax.org
Solution No. The Intermediate Value Theorem only allows us to conclude that we can find a value between and it doesn’t allow us to conclude that we can’t find other values. To see this more clearly, consider the function It satisfies and EXAMPLE 2.38 When Can You Apply the Intermediate Value Theorem? For and Can we conclude that has a zero in the interval Solution No. The function is not continuous over The Intermediate Value Theorem does not apply here. 2.26 Show that has a zero over the interval SECTION 2.4 EXERCISESFor the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. 131. 132 . 133 . 134. 135 . 136 . 137. 138 . For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? 139. at 140 . at 141 . at 142. at 143 . at 144. at In the following exercises, find the value(s) of k that makes each function continuous over the given interval. 145. 146 . 147. 148. 149 . 2.4 • Continuity 165
In the following exercises, use the Intermediate Value Theorem (IVT). 150. Let Over the interval there is no value of x such that although and Explain why this does not contradict the IVT. 151. A particle moving along a line has at each time t a position function which is continuous. Assume and Another particle moves such that its position is given by Explain why there must be a value c for such that 152 . [T] Use the statement “The cosine of t is equal to t cubed.” a. Write a mathematical equation of the statement. b. Prove that the equation in part a. has at least one real solution. c. Use a calculator to find an interval of length 0.01 that contains a solution. 153. Apply the IVT to determine whether has a solution in one of the intervals or Briefly explain your response for each interval. 154 . Consider the graph of the function shown in the following graph. a. Find all values for which the function is discontinuous. b. For each value in part a., state why the formal definition of continuity does not apply. c. Classify each discontinuity as either jump, removable, or infinite. 155 . Let a. Sketch the graph of f. b. Is it possible to find a value k such that which makes continuous for all real numbers? Briefly explain. 166 2 • Limits Access for free at openstax.org
156. Let for a. Sketch the graph of f. b. Is it possible to find values and such that and and that makes continuous for all real numbers? Briefly explain. 157 . Sketch the graph of a function with properties i. through vi. i. The domain of f is ∞ ∞ ii. f has an infinite discontinuity at iii. iv. v. vi. f is left continuous but not right continuous at 158. Sketch the graph of a function with properties i. through iv. i. The domain of f is ii. and exist and are equal. iii. is left continuous but not continuous at and right continuous but not continuous at iv. has a removable discontinuity at a jump discontinuity at and the following limits hold: ∞ and In the following exercises, suppose is defined for all x. For each description, sketch a graph with the indicated property. 159. Discontinuous at with and 160 . Discontinuous at but continuous elsewhere with Determine whether each of the given statements is true. Justify your response with an explanation or counterexample. 161. is continuous everywhere. 162 . If the left- and right-hand limits of as exist and are equal, then f cannot be discontinuous at 163 . If a function is not continuous at a point, then it is not defined at that point. 164. According to the IVT, has a solution over the interval 165 . If is continuous such that and have opposite signs, then has exactly one solution in 166 . The function is continuous over the interval 167. If is continuous everywhere and then there is no root of in the interval 2.4 • Continuity 167
[T] The following problems consider the scalar form of Coulomb’s law, which describes the electrostatic force between two point charges, such as electrons. It is given by the equation where is Coulomb’s constant, are the magnitudes of the charges of the two particles, and r is the distance between the two particles. 168. To simplify the calculation of a model with many interacting particles, after some threshold value we approximate F as zero. a. Explain the physical reasoning behind this assumption. b. What is the force equation? c. Evaluate the force F using both Coulomb’s law and our approximation, assuming two protons with a charge magnitude of and the Coulomb constant are 1 m apart. Also, assume How much inaccuracy does our approximation generate? Is our approximation reasonable? d. Is there any finite value of R for which this system remains continuous at R? 169 . Instead of making the force 0 at R, instead we let the force be 10−20 for Assume two protons, which have a magnitude of charge and the Coulomb constant Is there a value R that can make this system continuous? If so, find it. Recall the discussion on spacecraft from the chapter opener. The following problems consider a rocket launch from Earth’s surface. The force of gravity on the rocket is given by where m is the mass of the rocket, d is the distance of the rocket from the center of Earth, and k is a constant. 170. [T] Determine the value and units of k given that the mass of the rocket is 3 million kg. ( Hint: The distance from the center of Earth to its surface is 6378 km.) 171 . [T] After a certain distance D has passed, the gravitational effect of Earth becomes quite negligible, so we can approximate the force function by Using the value of k found in the previous exercise, find the necessary condition D such that the force function remains continuous. 172 . As the rocket travels away from Earth’s surface, there is a distance D where the rocket sheds some of its mass, since it no longer needs the excess fuel storage. We can write this function as Is there a D value such that this function is continuous, assuming Prove the following functions are continuous everywhere 173. 174 . 175 . Where is continuous? 168 2 • Limits Access for free at openstax.org
2.5 The Precise Definition of a Limit Learning Objectives 2.5.1 Describe the epsilon-delta definition of a limit. 2.5.2 Apply the epsilon-delta definition to find the limit of a function. 2.5.3 Describe the epsilon-delta definitions of one-sided limits and infinite limits. 2.5.4 Use the epsilon-delta definition to prove the limit laws. By now you have progressed from the very informal definition of a limit in the introduction of this chapter to the intuitive understanding of a limit. At this point, you should have a very strong intuitive sense of what the limit of a function means and how you can find it. In this section, we convert this intuitive idea of a limit into a formal definition using precise mathematical language. The formal definition of a limit is quite possibly one of the most challenging definitions you will encounter early in your study of calculus; however, it is well worth any effort you make to reconcile it with your intuitive notion of a limit. Understanding this definition is the key that opens the door to a better understanding of calculus. Quantifying Closeness Before stating the formal definition of a limit, we must introduce a few preliminary ideas. Recall that the distance between two points a and b on a number line is given by • The statement may be interpreted as: The distance between and L is less than ε. • The statement may be interpreted as: and the distance between x and a is less than δ. It is also important to look at the following equivalences for absolute value: • The statement is equivalent to the statement • The statement is equivalent to the statement and With these clarifications, we can state the formal epsilon-delta definition of the limit. Definition Let be defined for all over an open interval containing a. Let L be a real number. Then if, for every there exists a such that if then This definition may seem rather complex from a mathematical point of view, but it becomes easier to understand if we break it down phrase by phrase. The statement itself involves something called a universal quantifier (for every an existential quantifier (there exists a and, last, a conditional statement (if then Let’s take a look at Table 2.9, which breaks down the definition and translates each part. Definition Translation 1. For every 1. For every positive distance ε from L, 2. there exists a 2. There is a positive distance from a, 3. such that 3. such that 4. if then 4. if x is closer than to a and then is closer than ε to L. Table 2.9 Translation of the Epsilon-Delta Definition of the Limit We can get a better handle on this definition by looking at the definition geometrically. Figure 2.39 shows possible values of for various choices of for a given function a number a, and a limit L at a. Notice that as we choose smaller values of ε (the distance between the function and the limit), we can always find a small enough so that if we have chosen an x value within of a, then the value of is within ε of the limit L. 2.5 • The Precise Definition of a Limit 169
Figure 2.39 These graphs show possible values of , given successively smaller choices of ε. MEDIA Visit the following applet to experiment with finding values of for selected values of ε: The epsilon-delta definition of limit (http://www.openstax.org/l/20_epsilondelt) Example 2.39 shows how you can use this definition to prove a statement about the limit of a specific function at a specified value. EXAMPLE 2.39 Proving a Statement about the Limit of a Specific Function Prove that Solution Let The first part of the definition begins “For every This means we must prove that whatever follows is true no matter what positive value of ε is chosen. By stating “Let we signal our intent to do so. Choose The definition continues with “there exists a ” The phrase “there exists” in a mathematical statement is always a signal for a scavenger hunt. In other words, we must go and find So, where exactly did come from? There are two basic approaches to tracking down One method is purely algebraic and the other is geometric. We begin by tackling the problem from an algebraic point of view. Since ultimately we want we begin by manipulating this expression: is equivalent to which in turn is equivalent to Last, this is equivalent to Thus, it would seem that is appropriate. We may also find through geometric methods. Figure 2.40 demonstrates how this is done. 170 2 • Limits Access for free at openstax.org
Figure 2.40 This graph shows how we find geometrically. Assume When has been chosen, our goal is to show that if then To prove any statement of the form “If this, then that,” we begin by assuming “this” and trying to get “that.” Thus, Analysis In this part of the proof, we started with and used our assumption in a key part of the chain of inequalities to get to be less than ε. We could just as easily have manipulated the assumed inequality to arrive at as follows: Therefore, (Having completed the proof, we state what we have accomplished.) After removing all the remarks, here is a final version of the proof: Let Choose Assume Thus, 2.5 • The Precise Definition of a Limit 171
Therefore, The following Problem-Solving Strategy summarizes the type of proof we worked out in Example 2.39. Problem-Solving Strategy Proving That for a Specific Function 1. Let’s begin the proof with the following statement: Let 2. Next, we need to obtain a value for After we have obtained this value, we make the following statement, filling in the blank with our choice of : Choose 3. The next statement in the proof should be (at this point, we fill in our given value for a): Assume 4. Next, based on this assumption, we need to show that where and L are our function and our limit L. At some point, we need to use 5. We conclude our proof with the statement: Therefore, EXAMPLE 2.40 Proving a Statement about a Limit Complete the proof that by filling in the blanks. Let _____. Choose Assume Thus, Solution We begin by filling in the blanks where the choices are specified by the definition. Thus, we have Let Choose Assume (or equivalently, Thus, Focusing on the final line of the proof, we see that we should choose We now complete the final write-up of the proof: Let Choose Assume (or equivalently, 172 2 • Limits Access for free at openstax.org
Thus, 2.27 Complete the proof that by filling in the blanks. Let _______. Choose Assume Thus, Therefore, In Example 2.39 and Example 2.40, the proofs were fairly straightforward, since the functions with which we were working were linear. In Example 2.41, we see how to modify the proof to accommodate a nonlinear function. EXAMPLE 2.41 Proving a Statement about the Limit of a Specific Function (Geometric Approach) Prove that Solution 1. Let The first part of the definition begins “For every so we must prove that whatever follows is true no matter what positive value of ε is chosen. By stating “Let we signal our intent to do so. 2. Without loss of generality, assume Two questions present themselves: Why do we want and why is it okay to make this assumption? In answer to the first question: Later on, in the process of solving for we will discover that involves the quantity Consequently, we need In answer to the second question: If we can find that “works” for then it will “work” for any as well. Keep in mind that, although it is always okay to put an upper bound on ε, it is never okay to put a lower bound (other than zero) on ε. 3. Choose Figure 2.41 shows how we made this choice of Figure 2.41 This graph shows how we find δ geometrically for a given ε for the proof in Example 2.41. 4. We must show: If then so we must begin by assuming We don’t really need (in other words, for this proof. Since it is okay to drop 2.5 • The Precise Definition of a Limit 173
Hence, Recall that Thus, and consequently We also use here. We might ask at this point: Why did we substitute for on the left- hand side of the inequality and on the right-hand side of the inequality? If we look at Figure 2.41, we see that corresponds to the distance on the left of 2 on the x-axis and corresponds to the distance on the right. Thus, We simplify the expression on the left: Then, we add 2 to all parts of the inequality: We square all parts of the inequality. It is okay to do so, since all parts of the inequality are positive: We subtract 4 from all parts of the inequality: Last, 5. Therefore, 2.28 Find δ corresponding to for a proof that The geometric approach to proving that the limit of a function takes on a specific value works quite well for some functions. Also, the insight into the formal definition of the limit that this method provides is invaluable. However, we may also approach limit proofs from a purely algebraic point of view. In many cases, an algebraic approach may not only provide us with additional insight into the definition, it may prove to be simpler as well. Furthermore, an algebraic approach is the primary tool used in proofs of statements about limits. For Example 2.42, we take on a purely algebraic approach. EXAMPLE 2.42 Proving a Statement about the Limit of a Specific Function (Algebraic Approach) Prove that Solution Let’s use our outline from the Problem-Solving Strategy: 1. Let 2. Choose This choice of may appear odd at first glance, but it was obtained by taking a look at our ultimate desired inequality: This inequality is equivalent to At this point, the temptation simply to choose is very strong. Unfortunately, our choice of must depend on ε only and no other variable. If we can replace by a numerical value, our problem can be resolved. This is the place 174 2 • Limits Access for free at openstax.org
where assuming comes into play. The choice of here is arbitrary. We could have just as easily used any other positive number. In some proofs, greater care in this choice may be necessary. Now, since and we are able to show that Consequently, At this point we realize that we also need Thus, we choose 3. Assume Thus, Since we may conclude that Thus, by subtracting 4 from all parts of the inequality, we obtain Consequently, This gives us Therefore, 2.29 Complete the proof that Let choose assume Since we may conclude that Thus, Hence, You will find that, in general, the more complex a function, the more likely it is that the algebraic approach is the easiest to apply. The algebraic approach is also more useful in proving statements about limits. Proving Limit Laws We now demonstrate how to use the epsilon-delta definition of a limit to construct a rigorous proof of one of the limit laws. The triangle inequality is used at a key point of the proof, so we first review this key property of absolute value. Definition The triangle inequality states that if a and b are any real numbers, then Proof We prove the following limit law: If and then Let Choose so that if then Choose so that if then Choose Assume Thus, Hence, □ We now explore what it means for a limit not to exist. The limit does not exist if there is no real number L for 2.5 • The Precise Definition of a Limit 175
which Thus, for all real numbers L, To understand what this means, we look at each part of the definition of together with its opposite. A translation of the definition is given in Table 2.10. Definition Opposite 1. For every 1. There exists so that 2. there exists a so that 2. for every 3. if then 3. There is an x satisfying so that Table 2.10 Translation of the Definition of and its Opposite Finally, we may state what it means for a limit not to exist. The limit does not exist if for every real number L, there exists a real number so that for all there is an x satisfying so that Let’s apply this in Example 2.43 to show that a limit does not exist. EXAMPLE 2.43 Showing That a Limit Does Not Exist Show that does not exist. The graph of is shown here: Solution Suppose that L is a candidate for a limit. Choose Let Either or If then let Thus, and On the other hand, if then let Thus, and 176 2 • Limits Access for free at openstax.org
Thus, for any value of L, One-Sided and Infinite Limits Just as we first gained an intuitive understanding of limits and then moved on to a more rigorous definition of a limit, we now revisit one-sided limits. To do this, we modify the epsilon-delta definition of a limit to give formal epsilon-delta definitions for limits from the right and left at a point. These definitions only require slight modifications from the definition of the limit. In the definition of the limit from the right, the inequality replaces which ensures that we only consider values of x that are greater than (to the right of) a. Similarly, in the definition of the limit from the left, the inequality replaces which ensures that we only consider values ofx that are less than (to the left of) a. Definition Limit from the Right: Let be defined over an open interval of the form where Then, if for every there exists a such that if then Limit from the Left: Let be defined over an open interval of the form where Then, if for every there exists a such that if then EXAMPLE 2.44 Proving a Statement about a Limit From the Right Prove that Solution Let Choose Since we ultimately want we manipulate this inequality to get or, equivalently, making a clear choice. We may also determine geometrically, as shown in Figure 2.42. Figure 2.42 This graph shows how we find δ for the proof in Example 2.44. 2.5 • The Precise Definition of a Limit 177
Assume Thus, Hence, Finally, Therefore, 2.30 Find corresponding to ε for a proof that We conclude the process of converting our intuitive ideas of various types of limits to rigorous formal definitions by pursuing a formal definition of infinite limits. To have ∞ we want the values of the function to get larger and larger as x approaches a. Instead of the requirement that for arbitrarily small ε when for small enough we want for arbitrarily large positive M when for small enough Figure 2.43 illustrates this idea by showing the value of for successively larger values of M. Figure 2.43 These graphs plot values of for M to show that ∞ Definition Let be defined for all in an open interval containing a. Then, we have an infinite limit ∞ if for every there exists such that if then Let be defined for all in an open interval containing a. Then, we have a negative infinite limit ∞ if for every there exists such that if then SECTION 2.5 EXERCISESIn the following exercises, write the appropriate - definition for each of the given statements. 176. 177 . 178 . 179. 178 2 • Limits Access for free at openstax.org
The following graph of the function f satisfies In the following exercises, determine a value of that satisfies each statement. 180 . If then 181. If then The following graph of the function f satisfies In the following exercises, determine a value of that satisfies each statement. 182 . If then 183. If then 2.5 • The Precise Definition of a Limit 179
The following graph of the function f satisfies In the following exercises, for each value of ε, find a value of such that the precise definition of limit holds true. 184 . 185. [T] In the following exercises, use a graphing calculator to find a number such that the statements hold true. 186. whenever 187 . In the following exercises, use the precise definition of limit to prove the given limits. 188. 189 . 190 . 191. 192 . In the following exercises, use the precise definition of limit to prove the given one-sided limits. 193. 194 . 195. In the following exercises, use the precise definition of limit to prove the given infinite limits. 196. ∞ 197 . ∞ 198 . ∞ 199. An engineer is using a machine to cut a flat square of Aerogel of area 144 cm2. If there is a maximum error tolerance in the area of 8 cm2, how accurately must the engineer cut on the side, assuming all sides have the same length? How do these numbers relate to ε, a, and L? 200 . Use the precise definition of limit to prove that the following limit does not exist: 201 . Using precise definitions of limits, prove that does not exist, given that is the ceiling function. ( Hint: Try any 180 2 • Limits Access for free at openstax.org
202. Using precise definitions of limits, prove that does not exist: ( Hint: Think about how you can always choose a rational number but 203. Using precise definitions of limits, determine for (Hint: Break into two cases, x rational and x irrational.) 204 . Using the function from the previous exercise, use the precise definition of limits to show that does not exist for For the following exercises, suppose that and both exist. Use the precise definition of limits to prove the following limit laws: 205. 206 . for any real constant c (Hint: Consider two cases: and 207. ( Hint: 2.5 • The Precise Definition of a Limit 181
Chapter Review Key Terms average velocity the change in an object’s position divided by the length of a time period; the average velocity of an object over a time interval (if or if , with a position given by that is constant multiple law for limits the limit law continuity at a point A function is continuous at a point a if and only if the following three conditions are satisfied: (1) is defined, (2) exists, and (3) continuity from the left A function is continuous from the left at b if continuity from the right A function is continuous from the right at a if continuity over an interval a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a function is continuous over a closed interval of the form if it is continuous at every point in and it is continuous from the right at a and from the left at b difference law for limits the limit law differential calculus the field of calculus concerned with the study of derivatives and their applications discontinuity at a point A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point epsilon-delta definition of the limit if for every there exists a such that if then infinite discontinuity An infinite discontinuity occurs at a point a if ∞ or ∞ infinite limit A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a instantaneous velocity The instantaneous velocity of an object with a position function that is given by is the value that the average velocities on intervals of the form and approach as the values of t move closer to provided such a value exists integral calculus the study of integrals and their applications Intermediate Value Theorem Let f be continuous over a closed bounded interval if z is any real number between and then there is a number c in satisfying intuitive definition of the limit If all values of the function approach the real number L as the values of approach a, approaches L jump discontinuity A jump discontinuity occurs at a point a if and both exist, but limit the process of letting x or t approach a in an expression; the limit of a function as x approaches a is the value that approaches as x approaches a limit laws the individual properties of limits; for each of the individual laws, let and be defined for all over some open interval containing a; assume that L and M are real numbers so that and let c be a constant multivariable calculus the study of the calculus of functions of two or more variables one-sided limit A one-sided limit of a function is a limit taken from either the left or the right power law for limits the limit law for every positive integer n product law for limits the limit law quotient law for limits the limit law for removable discontinuity A removable discontinuity occurs at a point a if is discontinuous at a, but exists root law for limits the limit law for all L if n is odd and for if n is even secant A secant line to a function at a is a line through the point and another point on the function; the slope of the secant line is given by 182 2 • Chapter Review Access for free at openstax.org
squeeze theorem states that if for all over an open interval containing a and where L is a real number, then sum law for limits The limit law tangent A tangent line to the graph of a function at a point is the line that secant lines through approach as they are taken through points on the function with x-values that approach a; the slope of the tangent line to a graph at a measures the rate of change of the function at a triangle inequality If a and b are any real numbers, then vertical asymptote A function has a vertical asymptote at if the limit as x approaches a from the right or left is infinite Key Equations Slope of a Secant Line Average Velocity over Interval Intuitive Definition of the Limit Two Important Limits One-Sided Limits Infinite Limits from the Left ∞ ∞ Infinite Limits from the Right ∞ ∞ Two-Sided Infinite Limits ∞ ∞ and ∞ ∞ ∞ and ∞ Basic Limit Results Important Limits Key Concepts 2.1 A Preview of Calculus • Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the derivative. Calculating a derivative requires finding a limit. • Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the x-axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the integral. The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function. • Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in 2 • Chapter Review 183
space and finding volumes of solids. 2.2 The Limit of a Function • A table of values or graph may be used to estimate a limit. • If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist. • If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value. • We may use limits to describe infinite behavior of a function at a point. 2.3 The Limit Laws • The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time. • For polynomials and rational functions, • You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction. • The squeeze theorem allows you to find the limit of a function if the function is always greater than one function and less than another function with limits that are known. 2.4 Continuity • For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point. • Discontinuities may be classified as removable, jump, or infinite. • A function is continuous over an open interval if it is continuous at every point in the interval. It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints. • The composite function theorem states: If is continuous at L and then • The Intermediate Value Theorem guarantees that if a function is continuous over a closed interval, then the function takes on every value between the values at its endpoints. 2.5 The Precise Definition of a Limit • The intuitive notion of a limit may be converted into a rigorous mathematical definition known as the epsilon-delta definition of the limit. • The epsilon-delta definition may be used to prove statements about limits. • The epsilon-delta definition of a limit may be modified to define one-sided limits. Review ExercisesTrue or False. In the following exercises, justify your answer with a proof or a counterexample. 208. A function has to be continuous at if the exists. 209 . You can use the quotient rule to evaluate 210 . If there is a vertical asymptote at for the function then f is undefined at the point 184 2 • Chapter Review Access for free at openstax.org
211. If does not exist, then f is undefined at the point 212 . Using the graph, find each limit or explain why the limit does not exist. a. b. c. d. In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 213. 214 . 215 . 216. 217 . 218 . 219. 220 . 221 . 222. In the following exercises, use the squeeze theorem to prove the limit. 223. 224 . 225 . Determine the domain such that the function is continuous over its domain. In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous. 226. 227 . 2 • Chapter Review 185
In the following exercises, use the precise definition of limit to prove the limit. 228. 229 . 230 . A ball is thrown into the air and the vertical position is given by Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw. 231. A particle moving along a line has a displacement according to the function where x is measured in meters and t is measured in seconds. Find the average velocity over the time period 232 . From the previous exercises, estimate the instantaneous velocity at by checking the average velocity within 186 2 • Chapter Review Access for free at openstax.org
Figure 3.1 The Hennessey Venom GT can go from 0 to 200 mph in 14.51 seconds. (credit: modification of work by Codex41, Flickr) Chapter Outline 3.1 Defining the Derivative 3.2 The Derivative as a Function 3.3 Differentiation Rules 3.4 Derivatives as Rates of Change 3.5 Derivatives of Trigonometric Functions 3.6 The Chain Rule 3.7 Derivatives of Inverse Functions 3.8 Implicit Differentiation 3.9 Derivatives of Exponential and Logarithmic Functions Introduction The Hennessey Venom GT is one of the fastest cars in the world. In 2014, it reached a record-setting speed of 270.49 mph. It can go from 0 to 200 mph in 14.51 seconds. The techniques in this chapter can be used to calculate the acceleration the Venom achieves in this feat (see Example 3.8.) Calculating velocity and changes in velocity are important uses of calculus, but it is far more widespread than that. Calculus is important in all branches of mathematics, science, and engineering, and it is critical to analysis in business and health as well. In this chapter, we explore one of the main tools of calculus, the derivative, and show convenient ways to calculate derivatives. We apply these rules to a variety of functions in this chapter so that we can then explore applications of these techniques. 3 DERIVATIVES 3 • Introduction 187
3.1 Defining the Derivative Learning Objectives 3.1.1 Recognize the meaning of the tangent to a curve at a point. 3.1.2 Calculate the slope of a tangent line. 3.1.3 Identify the derivative as the limit of a difference quotient. 3.1.4 Calculate the derivative of a given function at a point. 3.1.5 Describe the velocity as a rate of change. 3.1.6 Explain the difference between average velocity and instantaneous velocity. 3.1.7 Estimate the derivative from a table of values. Now that we have both a conceptual understanding of a limit and the practical ability to compute limits, we have established the foundation for our study of calculus, the branch of mathematics in which we compute derivatives and integrals. Most mathematicians and historians agree that calculus was developed independently by the Englishman Isaac Newton and the German Gottfried Leibniz whose images appear in Figure 3.2. When we credit Newton and Leibniz with developing calculus, we are really referring to the fact that Newton and Leibniz were the first to understand the relationship between the derivative and the integral. Both mathematicians benefited from the work of predecessors, such as Barrow, Fermat, and Cavalieri. The initial relationship between the two mathematicians appears to have been amicable; however, in later years a bitter controversy erupted over whose work took precedence. Although it seems likely that Newton did, indeed, arrive at the ideas behind calculus first, we are indebted to Leibniz for the notation that we commonly use today. Figure 3.2 Newton and Leibniz are credited with developing calculus independently. Tangent Lines We begin our study of calculus by revisiting the notion of secant lines and tangent lines. Recall that we used the slope of a secant line to a function at a point to estimate the rate of change, or the rate at which one variable changes in relation to another variable. We can obtain the slope of the secant by choosing a value of near and drawing a line through the points and as shown in Figure 3.3. The slope of this line is given by an equation in the form of a difference quotient: 188 3 • Derivatives Access for free at openstax.org
We can also calculate the slope of a secant line to a function at a value a by using this equation and replacing with where is a value close to 0. We can then calculate the slope of the line through the points and In this case, we find the secant line has a slope given by the following difference quotient with increment Definition Let be a function defined on an interval containing If is in then (3.1) is a difference quotient. Also, if is chosen so that is in then (3.2) is a difference quotient with increment MEDIA View the development of the derivative (http://www.openstax.org/l/20_calcapplets) with this applet. These two expressions for calculating the slope of a secant line are illustrated in Figure 3.3. We will see that each of these two methods for finding the slope of a secant line is of value. Depending on the setting, we can choose one or the other. The primary consideration in our choice usually depends on ease of calculation. Figure 3.3 We can calculate the slope of a secant line in either of two ways. In Figure 3.4(a) we see that, as the values of approach the slopes of the secant lines provide better estimates of the rate of change of the function at Furthermore, the secant lines themselves approach the tangent line to the function at which represents the limit of the secant lines. Similarly, Figure 3.4(b) shows that as the values of get closer to the secant lines also approach the tangent line. The slope of the tangent line at is the rate of change of the function at as shown in Figure 3.4(c). 3.1 • Defining the Derivative 189
Figure 3.4 The secant lines approach the tangent line (shown in green) as the second point approaches the first. MEDIA You can use this site (http://www.openstax.org/l/20_diffmicros) to explore graphs to see if they have a tangent line at a point. In Figure 3.5 we show the graph of and its tangent line at in a series of tighter intervals about As the intervals become narrower, the graph of the function and its tangent line appear to coincide, making the values on the tangent line a good approximation to the values of the function for choices of close to In fact, the graph of itself appears to be locally linear in the immediate vicinity of Figure 3.5 For values of close to the graph of and its tangent line appear to coincide. 190 3 • Derivatives Access for free at openstax.org
Formally we may define the tangent line to the graph of a function as follows. Definition Let be a function defined in an open interval containing The tangent line to at is the line passing through the point having slope (3.3) provided this limit exists. Equivalently, we may define the tangent line to at to be the line passing through the point having slope (3.4) provided this limit exists. Just as we have used two different expressions to define the slope of a secant line, we use two different forms to define the slope of the tangent line. In this text we use both forms of the definition. As before, the choice of definition will depend on the setting. Now that we have formally defined a tangent line to a function at a point, we can use this definition to find equations of tangent lines. EXAMPLE 3.1 Finding a Tangent Line Find an equation of the line tangent to the graph of at Solution First find the slope of the tangent line. In this example, use Equation 3.3. Next, find a point on the tangent line. Since the line is tangent to the graph of at it passes through the point We have so the tangent line passes through the point Using the point-slope equation of the line with the slope and the point we obtain the line Simplifying, we have The graph of and its tangent line at are shown in Figure 3.6. Figure 3.6 The tangent line to at 3.1 • Defining the Derivative 191
EXAMPLE 3.2 The Slope of a Tangent Line Revisited Use Equation 3.4 to find the slope of the line tangent to the graph of at Solution The steps are very similar to Example 3.1. See Equation 3.4 for the definition. We obtained the same value for the slope of the tangent line by using the other definition, demonstrating that the formulas can be interchanged. EXAMPLE 3.3 Finding the Equation of a Tangent Line Find an equation of the line tangent to the graph of at Solution We can use Equation 3.3, but as we have seen, the results are the same if we use Equation 3.4. We now know that the slope of the tangent line is To find an equation of the tangent line, we also need a point on the line. We know that Since the tangent line passes through the point we can use the point-slope equation of a line to find an equation of the tangent line. Thus the tangent line has the equation The graphs of and are shown in Figure 3.7. 192 3 • Derivatives Access for free at openstax.org