Graphical Approach to Precalculus with Limits: A Unit Circle Approach PDF
Preview Graphical Approach to Precalculus with Limits: A Unit Circle Approach
A G r a p h i c a l A p p r o a c h t o P r e A Graphical Approach to c a Precalculus with Limits l c Hornsby Lial Rockswold u l u Fifth Edition s w i t h L i m i t s H o r n s b y e t a l . ISBN 978-1-29204-080-6 5 e 9 781292 040806 Pearson New International Edition A Graphical Approach to Precalculus with Limits Hornsby Lial Rockswold Fifth Edition International_PCL_TP.indd 1 7/29/13 11:23 AM ISBN 10: 1-292-04080-7 ISBN 13: 978-1-292-04080-6 Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affi liation with or endorsement of this book by such owners. ISBN 10: 1-292-04080-7 ISBN 10: 1-269-37450-8 ISBN 13: 978-1-292-04080-6 ISBN 13: 978-1-269-37450-7 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America Copyright_Pg_7_24.indd 1 7/29/13 11:28 AM 123345667888551932396271959167014931 P E A R S O N C U S T O M L I B R AR Y Table of Contents Chapter 1 Linear Functions, Equations, and Inequalities John Hornsby/Margaret Lial/Gary K. Rockswold 1 Chapter 2 Analysis of Graphs of Functions John Hornsby/Margaret Lial/Gary K. Rockswold 81 Chapter 3 Polynomial Functions John Hornsby/Margaret Lial/Gary K. Rockswold 159 Chapter 4 Rational, Power, and Root Functions John Hornsby/Margaret Lial/Gary K. Rockswold 255 Chapter 5 Inverse, Exponential, and Logarithmic Functions John Hornsby/Margaret Lial/Gary K. Rockswold 319 Chapter 6 Analytic Geometry John Hornsby/Margaret Lial/Gary K. Rockswold 391 Chapter 7 Systems of Equations and Inequalities; Matrices John Hornsby/Margaret Lial/Gary K. Rockswold 436 Chapter 8 The Unit Circle and the Functions of Trigonometry John Hornsby/Margaret Lial/Gary K. Rockswold 527 Chapter 9 Trigonometric Identities and Equations John Hornsby/Margaret Lial/Gary K. Rockswold 630 Chapter 10 Applications of Trigonometry and Vectors John Hornsby/Margaret Lial/Gary K. Rockswold 691 Chapter 11 Further Topics in Algebra John Hornsby/Margaret Lial/Gary K. Rockswold 764 Chapter 12 Limits, Derivatives, and Definite Integrals John Hornsby/Margaret Lial/Gary K. Rockswold 829 Chapter R Reference: Basic Algebraic Concepts John Hornsby/Margaret Lial/Gary K. Rockswold 873 I 999022723 Appendices John Hornsby/Margaret Lial/Gary K. Rockswold 907 Photo Credits John Hornsby/Margaret Lial/Gary K. Rockswold 922 Index 923 II 1 Linear Functions, Equations, and Inequalities DURING RECENTYEARS, temperatures have risen over much of the Arctic and Antarctic regions. A sustained warming of 4°F to 11°F above average twentieth- century temperatures could cause the Greenland ice sheet to melt. Sea levels may rise as much as 25 feet. (See Exercise 57 in Section 1.6.) This melting phenomenon is also occurring in the Antarctic, where the Wilkins Ice Shelf is melting at an alarming rate. During the past 50 years, the rate of warming at Antarctica has been about 0.9°F per decade. If the entire Antarctic ice cap were to melt, sea levels could rise over 200 feet. To bring attention to global warming, British adventurer and endurance swimmer Lewis Gordon Pugh dove into a region of the Antarctic that used to be covered by ice. What will the future bring? Linear functions can help us make predic- tions. In this chapter, we learn how to model many different phenomena with linear functions and equations. Source: National Snow and Ice Data Center, March 25, 2008. CHAPTER OUTLINE 1.1 Real Numbers and the Rectangular Coordinate System 1.2 Introduction to Relations and Functions 1.3 Linear Functions 1.4 Equations of Lines and Linear Models 1.5 Linear Equations and Inequalities 1.6 Applications of Linear Functions From Chapter 1 of A Graphical Approach to Precalculus with Limits: A Unit Circle Approach, Fifth Edition. John Hornsby, Margaret Lial, Gark K. Rockswold. Copyright © 2011 by Pearson Education, Inc. All rights reserved. (cid:20) 2 CHAPTER 1 Linear Functions, Equations, and Inequalities Real Numbers and the Rectangular 1.1 Coordinate System Sets of Real Numbers • The Rectangular Coordinate System • Viewing Windows • Approximations of Real Numbers • Distance and Midpoint Formulas Sets of Real Numbers There are several important sets of numbers that are used in mathematics. Some of these sets are listed in the following table. Sets of Numbers Set Description Examples Natural Numbers 51, 2, 3, 4, Á6 1, 45, 127, 103 Whole Numbers 50, 1, 2, 3, 4, Á6 0, 86, 345, 23 Integers 5Á, -2, -1, 0, 1, 2, Á6 0, -5, -102, 99 Rational Numbers Epq|p and q are integers, q Z 0F 0, -56, -2,272, 0.5 Irrational Numbers 5x|x is not rational6 22,p, -237 Real Numbers 5x|x is a decimal number6 -26,p,2, 245, 0.41 3 Whole numbersinclude the natural numbers; integersinclude the whole num- bers and the natural numbers. The result of dividing two integers (with a nonzero divi- sor) is a rational number,orfraction. Rational numbers include the natural numbers, Origin whole numbers, and integers. For example, the integer -3is a rational number because -3 it can be written as .Every rational number can be written as a repeating or termi- 1 nating decimal. For example, 0.6 = 0.66666Árepresents the rational number 2. –5–4 –3–2 –1 0 1 2 3 4 5 3 Numbers that can be written as decimal numbers are real numbers.Real numbers Graph of {–3, –1, 0, 1, 3, 5} include rational numbers and can be shown pictorially—that is, graphed—on a FIGURE 1 number line. See FIGURE 1.The point corresponding to 0 is called the origin. Every real number corresponds to one and only one point on the number line, and each point corresponds to one and only one real number. This correspondence is called a coordinate system.The number associated with a given point is called the coordinate –5–4–3–2–1 0 1 2 3 4 5 of the point. The set of all real numbers is graphed in FIGURE 2. Graph of the set of real numbers Some real numbers cannot be represented by quotients of integers or by repeating FIGURE 2 or terminating decimals. These numbers are called irrational numbers.Examples of irrational numbers include 23, 25, 23 10, and 25 20, but not 21, 24, 29, Á, which equal 1, 2, 3, Á,and hence are rational numbers. If ais a natural number but –23 √2 √5 (cid:2) 2ais not a natural number, then 2ais an irrational number. Another irrational num- ber is p, which is approximately equal to 3.14159. In FIGURE 3the numbers in the set –1Graph0 of – 1 2 , 0, 2√2,√5,3(cid:2),4 4 E-23, 0, 22, 25,p, 4F are located on a number line. Note that 22is approximately 3 equal to 1.41, so it is located between 1 and 2, slightly closer to 1. FIGURE 3 The Rectangular Coordinate System If we place two number lines at right angles, intersecting at their origins, we obtain a two-dimensional rectangular coordinate system. The number lines intersect at the (cid:21) 1.1 Real Numbers and the Rectangular Coordinate System 3 y-axis originof the system, designated 0. The horizontal line is called the x-axis,and the ver- tical line is called the y-axis. On the x-axis, positive numbers are located to the right Quadrant Quadrant of the origin, and negative numbers are located to the left. On the y-axis, positive num- II I bers are located above the origin, negative numbers below. P(a, b) b This rectangular coordinate system is also called the Cartesian coordinate system, x-axis a 0 Origin named after René Descartes (1596–1650). The plane into which the coordinate system is introduced is the coordinate plane, or xy-plane.The x-axis and y-axis divide the Quadrant Quadrant III IV plane into four regions, or quadrants,as shown in FIGURE 4.The points on the x-axis or y-axis belong to no quadrant. Rectangular coordinate system Each point P in the xy-plane corresponds to a unique ordered pair (a,b) of real FIGURE 4 numbers. We call a the x-coordinate and b the y-coordinate of point P. The point P y corresponding to the ordered pair (a,b)is often written as P(a,b), as in FIGURE 4,and referred to as “the point (a,b).” FIGURE 5illustrates how to plot the point A(3, 4). Addi- B(–5, 6) A(3, 4) tional points are labeled B–E. The coordinates of the origin are (0, 0). 4 x E(–3, 0)0 3 Viewing Windows D(4, –3) C(–2, –4) The rectangular (Cartesian) coordinate system extends indefinitely in all directions. We can show only a portion of such a system in a text figure. Similar limitations occur Plotting points in the xy-plane with the viewing “window” on a calculator screen. FIGURE 6shows a calculator screen FIGURE 5 that has been set to have a minimum x-value of -10,a maximum x-value of 10, a min- imumy-value of -10,and a maximum y-value of 10. The tick marks on the axes have 10 been set to be 1 unit apart. Thus, there are 10 tick marks on the positive x-axis. This window is called the standard viewing window. To convey information about a viewing window, we use the following abbreviations: –10 10 Xmin: minimum value of x Ymin: minimum value of y Xmax: maximum value of x Ymax: maximum value of y –10 Xscl: scale (distance between Yscl: scale (distance between Standard viewing window tick marks) on the x-axis tick marks) on the y-axis. FIGURE 6 To further condense this information, we use the following symbolism, which gives viewing information for the window in FIGURE 6. GCMTECHNOLOGY NOTE Xmax Ymin You should consult the graphing Xmin Ymax calculator manual that accompanies 3-10, 104 by 3-10, 104. this text or your owner’s manual to Xscl = 1 Yscl = 1 see how to set the viewing window on your screen. Remember that FIGURE 7 shows several other viewing windows. Notice that FIGURES 7(b) and 7(c) different settings will result in look exactly alike, and unless we are told what the settings are, we have no way of dis- different views of graphs. tinguishing between them. What are Xscl and Yscl in each figure? 3.1 10 100 –4.7 4.7 –10 10 –100 100 –3.1 –10 –100 [–4.7, 4.7] by [–3.1, 3.1] [–10, 10] by [–10, 10] [–100, 100] by [–100, 100] (a) (b) (c) FIGURE 7 (cid:22) 4 CHAPTER 1 Linear Functions, Equations, and Inequalities WHAT WENT WRONG? A student learning how to use a graphing calculator could not understand why the axes on the graph were so “thick,” as seen in FIGURE A,while those on a friend’s calculator were not, as seen in FIGURE B. 40 40 –60 60 –60 60 –40 –40 Xscl = 1, Yscl = 1 Xscl = 10, Yscl = 10 FIGURE A FIGURE B What Went Wrong? How can the student correct the problem in FIGURE Aso that the axes look like those in FIGURE B? Approximations of Real Numbers Although calculators have the capability to express numbers like 22, 23 5,and pto many decimal places, we often ask that answers be rounded. The following table reviews rounding numbers to the nearest tenth, hundredth, or thousandth. TI-84 Plus Rounding Numbers FIGURE 8 Number Nearest Tenth Nearest Hundredth Nearest Thousandth 1.3782 1.4 1.38 1.378 201.6666 201.7 201.67 201.667 0.0819 0.1 0.08 0.082 FIGURE 9 InFIGURE 8,the TI-84 Plus graphing calculator is set to round values to the near- est hundredth (two decimal places). In FIGURE 9,the numbers from the preceding table are rounded to the nearest hundredth. The symbol L indicates that two expressions are approximately equal.For exam- ple, p L 3.14,but p Z 3.14,since p = 3.141592654.Á When using pin calcula- tions, be sure to use the built-in key for prather than 3.14. See FIGURE 10. FIGURE 10 Answer to What Went Wrong? Since Xscl = 1and Yscl = 1in FIGURE A,there are 120 tick marks along the x-axis and 80 tick marks along the y-axis. The resolution of the graphing calculator screen is not high enough to show all these tick marks, so the axes appear as heavy black lines instead. The values for Xscl and Yscl need to be larger, as in FIGURE B. (cid:23) 1.1 Real Numbers and the Rectangular Coordinate System 5 GCM EXAMPLE 1 Finding Roots on a Calculator Approximate each root to the nearest thousandth. (Note: You can use the fact that 2n a = a1/nto find roots.) (a) 223 (b) 23 87 (c) 24 12 Solution (a) The screen in FIGURE 11(a) shows that an approximation for 223, to the nearest thousandth, is 4.796. It is displayed twice, once for 223and once for 231/2. (b) To the nearest thousandth, 23 87 L 4.431.See FIGURE 11(b). (c) FIGURE 11(c)indicates 24 12 L 1.861in three different ways. TECHNOLOGY NOTE In all the screens, note the inclusion of parentheses. Many graphing calculators have built-in keys for calculating square roots and menus for calculating other types of roots. (a) (b) (c) FIGURE 11 EXAMPLE 2 Approximating Expressions with a Calculator Approximate each expression to the nearest hundredth. 3.8 - 1.4 (a) (b) 3p4 - 92 (c) 2(4 - 1)2 + (-3 - 2)2 5.4 + 3.5 Solution TECHNOLOGY NOTE (a) SeeFIGURE 12(a).To the nearest hundredth, Some graphing calculators display 3.8 - 1.4 L 0.27. leading zeros in decimal numbers, 5.4 + 3.5 whereas others do not. For example, 1might be displayed as (b) Many calculators also have a special key to calculate the square of a number. To either 0.254 or .25. In this text, the nearest hundredth, 3p4 - 92 L 211.23.See FIGURE 12(b). graphing calculator screens do not (c) From FIGURE 12(c), 2(4 - 1)2 + (-3 - 2)2 L 5.83. usually show leading zeros. See FIGURES 9and12(a). Do not confuse the negation and subtraction symbols. Insert parentheses around both the numerator and the denominator. (a) (b) (c) Use parentheses FIGURE 12 carefully. (cid:24)
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