Precalculus wih Limites free download
GRAPHS OF PARENT FUNCTIONS
Linear Function Absolute Value Function Square Root Function x , x
ⱖ 0 ⫽
f x ⫽ mx ⫹ b f x ⫽ x f x ⫽ x <
⫺x , x
y y y
4
2
3 f (x) = x
1 (0, b) f
(x) = x
2 x x b b
(0, 0)
−2 −12
1 − , 0 − , 0
( ( ( ( m m
−1 x f f
(x) = mx + b, (x) = mx + b, −1
2
3
4 (0, 0)
−2 m m
> 0 < 0 −1
Domain: , Domain: , Domain: ⫺ ⫺ 0,
⬁ ⬁ ⬁ ⬁ ⬁
Range: , Range: Range: ⫺ 0, 0,
⬁ ⬁ ⬁ ⬁ x -intercept: Intercept: Intercept:
⫺b m, 0 0, 0 0, 0
y -intercept: Decreasing on , 0 Increasing on
0, b ⫺ 0, > ⬁ ⬁ Increasing when m Increasing on
0, < ⬁ Decreasing when m Even function
- axis symmetry
y Greatest Integer Function Quadratic (Squaring) Function Cubic Function
2
3 f f f
x ⫽ x x ⫽ ax x ⫽ x
y y y
f(x) = x [[ ]]
3
3
3
2
2
2
2
1
1 f > 0
(x) = ax , a (0, 0) x x x
−3 −2 −1
1
2 3 −2 −1
1
2
3 4 −3 −2
1
2
3 −1 −1
2
3 f < 0
(x) = ax , a f = x
(x) −2 −2 −3 −3 −3
Domain: , Domain: , Domain: , ⫺ ⫺ ⫺
⬁ ⬁ ⬁ ⬁ ⬁ ⬁
>Range: the set of integers Range : Range: , a 0, ⫺ < ⬁ ⬁ ⬁
x -intercepts: in the interval Range : , 0 Intercept:
0, 1 a ⫺ 0, 0
⬁ y -intercept: Intercept: Increasing on ,
0, 0 0, 0 ⫺ > ⬁ ⬁ Constant between each pair of Decreasing on , 0 for a Odd function
⫺
⬁ >
consecutive integers Increasing on for a Origin symmetry 0,
⬁ <
Jumps vertically one unit at Increasing on , 0 for a ⫺
⬁ <
each integer value Decreasing on for a 0,
⬁
Even function
- axis symmetry
y >
Relative minimum a , < relative maximum a , or vertex:
0, 0
⬁
⬁
1 x
(x) =
−x f (x) = a x x y f
(0, 1) f (x) = a
1 (1, 0) x
y
2 −1
1
(x) = log a x
x y f
)
⬁
, 0 傼 0,
⫺
1
⬁
,
⬁
⫺
⬁
) 0,
⬁
, 0 傼 0,
⬁
⫺
⬁
0,
⬁
−1
2
⬁
⬁
,
⬁
⫺
2 ⫺1, 1
2 ⫹ n , 0
0, 1
y -axis symmetry
Even function
x -intercepts: y -intercept:
Domain: Range: Period:
⬁
,
⫺
3
0, 0 n, 0 2 ⫺1, 1
y
x
Domain: Range: Period:
x
1
f x ⫽
, a > 0, a ⫽ 1
f x ⫽ a x
, a > 0, a ⫽ 1
3 f x ⫽ log a x
2
1
,
⫺
2
(x) = tan x x y
3 π π π 2 π − f
, 0 1, 0 0, 1
1 −2 −3
2 y
(x) = cos x x
2
− −f
3 π π π 2 π π
2
−3
3 π 2 −2
2 − f
2 y f
3 π π 2 π
1
2
2 ⫹ n
⬁ x ⫽
,
⬁
Rational (Reciprocal) Function Exponential Function Logarithmic Function
Domain: Domain: Domain: Range: Range: Range: No intercepts Intercept: Intercept: Decreasing on and Increasing on Increasing on Odd function for Vertical asymptote: y-axis Origin symmetry Decreasing on Continuous Vertical asymptote: y-axis for Reflection of graph of Horizontal asymptote: x-axis Horizontal asymptote: x-axis in the line
Continuous
(x) = sin x x
x ⫽ tan x
Domain: all Range: Period:
x ⫽ a
⬁
⫺
⬁
0,
⬁
,
⬁
⫺
⬁
0,
x
⬁ f
f
,
⬁
⫺
⫺x
x ⫽ a
x f
x ⫽ a
y ⫽ x f
x ⫽ sin x
f
x ⫽ cos x
Sine Function Cosine Function Tangent Function
- intercepts:
x -intercepts: y -intercept:
- intercept: Odd function Origin symmetry
Vertical asymptotes: Odd function Origin symmetry
x ⫽
2 ⫹ n
0, 0 n, 0 ⫺
⬁
y -axis symmetry x ⫽
2 ⫹ n
⬁
x ⫽
, ⫺1 傼 1,
⬁
0, 1 2 ⫺
2 ⫹ n
Vertical asymptotes: Even function
x
y -intercept:
x ⫽ csc x Domain: all Range: Period:
f
x ⫽ sec x
f
x ⫽ cot x
1 sin x f
2 π
(x) = csc x = x y
Domain: all Range: Period:
x ⫽ n
1
⫺1, 1 Domain: Range: Intercept: Horizontal asymptotes: Odd function Origin symmetry
,
⬁
2 ⫺
2 ,
⫺
2 0, 0
y ⫽ ±
2 0,
y
Domain: Range:
⬁ x ⫽ n
,
⬁
⫺
, 0
2 ⫹ n
3 π π π 2 − f
2
- intercepts: Vertical asymptotes: Odd function Origin symmetry
1
2
1
−1
(x) = arccos x π x y
1
f
−1
2 π x y
2 π −
2 f (x) = arctan x
x y −1 −2
x ⫽ arctan x
2 ⫺1, 1
2 ,
0, 0 ⫺
Domain: Range: Intercept: Odd function Origin symmetry
Inverse Sine Function Inverse Cosine Function Inverse Tangent Function
Domain: all Range: Period: No intercepts Vertical asymptotes: Odd function Origin symmetry
Cosecant Function Secant Function Cotangent Function
2 π π − f (x) = arcsin x f
f
1 cos x
2 − − f
(x) = sec x =
2 f
3 π 2π
2 − − x y
3 π π π 2 π
−3
1 tan x 2 −2
2 π
(x) = cot x = x y
3 π π π 2 π
x ⫽ arccos x
1
2
⬁ x ⫽ n
, ⫺1 傼 1,
⬁
2 ⫺
x ⫽ n
x ⫽ arcsin x
f
- intercept: 0,
Ron Larson The Pennsylvania State University The Behrend College With the assistance of David C. Falvo The Pennsylvania State University The Behrend College
Precalculus with Limits, Second Edition Ron Larson Publisher: Charlie VanWagner Acquiring Sponsoring Editor: Gary Whalen Development Editor: Stacy Green Assistant Editor: Cynthia Ashton Editorial Assistant: Guanglei Zhang Associate Media Editor: Lynh Pham Marketing Manager: Myriah FitzGibbon Marketing Coordinator: Angela Kim Marketing Communications Manager: Katy Malatesta Content Project Manager: Susan Miscio Senior Art Director: Jill Ort Senior Print Buyer: Diane Gibbons Production Editor: Carol Merrigan Text Designer: Walter Kopek Rights Acquiring Account Manager, Photos: Don Schlotman Photo Researcher: Prepress PMG Cover Designer: Harold Burch Cover Image: Richard Edelman/Woodstock Graphics Studio Compositor: Larson Texts, Inc.
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Contents
iv Contents
Contents v
vi Contents
Index of Applications (web) Appendix B Concepts in Statistics (web) B.1 Representing Data B.2 Measures of Central Tendency and Dispersion B.3 Least Squares Regression
Welcome to the Second Edition of Precalculus with Limits! We are proud to offer you a new and revised version of our textbook. With the Second Edition, we have listened to you, our users, and have incorporated many of your suggestions for improvement.
2nd Edition 1st Edition In this edition, we continue to offer instructors and students a text that is pedagogically sound, mathematically precise, and still comprehensible. There are many changes in the mathematics, art, and design; the more significant changes are noted here.
- New Chapter Openers Each Chapter Opener has three parts, In Mathematics, In
Real Life, and In Careers. In Mathematics describes an important mathematical
topic taught in the chapter. In Real Life tells students where they will encounter this topic in real-life situations. In Careers relates application exercises to a variety of careers.
- New Study Tips and Warning/Cautions Insightful information is given to students in two new features. The Study Tip provides students with useful information or suggestions for learning the topic. The Warning/Caution points out common mathematical errors made by students.
- New Algebra Helps Algebra Help directs students to sections of the textbook where they can review algebra skills needed to master the current topic.
- New Side-by-Side Examples Throughout the text, we present solutions to many examples from multiple perspectives—algebraically, graphically, and numerically. The side-by-side format of this pedagogical feature helps students to see that a problem can be solved in more than one way and to see that different methods yield the same result. The side-by-side format also addresses many different learning styles.
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I hope you enjoy the Second Edition of Precalculus with Limits. As always, I welcome comments and suggestions for continued improvements.
viii A Word from the Author
Acknowledgments
I would like to thank the many people who have helped me prepare the text and the supplements package. Their encouragement, criticisms, and suggestions have been invaluable.
Thank you to all of the instructors who took the time to review the changes in this edition and to provide suggestions for improving it. Without your help, this book would not be possible.
Reviewers
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Supplements xi Supplements for the Student
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1.1 Rectangular Coordinates
1.2 Graphs of Equations
1.3 Linear Equations in Two Variables
1.4 Functions
1.5 Analyzing Graphs of Functions
1.6 A Library of Parent Functions
1.7 Transformations of Functions
1.9 Inverse Functions
1.8 Combinations of Functions:
1.10 Mathematical Modeling and Variation Composite Functions In Mathematics Functions show how one variable is related to another variable.
In Real Life Functions are used to estimate values, simulate processes, and discover relation- ships. For instance, you can model the enrollment rate of children in preschool and estimate the year in which the rate will reach a certain number. Such an estimate can be used to plan measures for meeting future needs, such as hiring additional teachers and buying more books. (See Exercise 113, page 64.)
Jose Luis Pelaez/Getty Images
IN CAREERS There are many careers that use functions. Several are listed below.
- Financial analyst • Tax preparer
Exercise 95, page 51 Example 3, page 104
- Biologist • Oceanographer Exercise 73, page 91 Exercise 83, page 112
What you should learn • Plot points in the Cartesian plane.
- Use the Distance Formula to find the distance between two points.
- Use the Midpoint Formula to find the midpoint of a line segment.
- Use a coordinate plane to model and solve real-life problems.
1
1
2
3
1
2
3 −1 −2 −3
(Vertical
number line)
(Horizontal
number line)
Quadrant I Quadrant II Quadrant III Quadrant IV Origin y -axis
2 Chapter 1 Functions and Their Graphs
Why you should learn it The Cartesian plane can be used to represent relationships between two variables. For instance, in Exercise 70 on page 11, a graph represents the minimum wage in the United States from 1950 to 2009.
Directed distance Directed distance y -axis x -axis
1
3
4 −1 −2 −4
(3, 4) x
−4 −3 −1
1
3
2
4 (3, 0) (0, 0) ( 1, 2) −
( 2, 3) − − y
FIGURE
−3 −2 −1
(x, y) x y
The Cartesian Plane
Plotting Points in the Cartesian Plane
Just as you can represent real numbers by points on a real number line, you can represent ordered pairs of real numbers by points in a plane called the rectangular
coordinate system, or the Cartesian plane, named after the French mathematician René Descartes (1596–1650).
The Cartesian plane is formed by using two real number lines intersecting at right angles, as shown in Figure 1.1. The horizontal real number line is usually called the
x-axis, and the vertical real number line is usually called the y-axis. The point of
intersection of these two axes is the origin, and the two axes divide the plane into four parts called quadrants.
FIGURE
1.1 FIGURE
1.2 Each point in the plane corresponds to an ordered pair of real numbers and
called coordinates of the point. The x-coordinate represents the directed distance from the -axis to the point, and the y-coordinate represents the directed distance from the -axis to the point, as shown in Figure 1.2.
The notation denotes both a point in the plane and an open interval on the real number line. The context will tell you which meaning is intended.
Plot the points and
(x, y) x x -axis
Solution
To plot the point imagine a vertical line through on the -axis and a horizontal line through 2 on the -axis. The intersection of these two lines is the point The other four points can be plotted in a similar way, as shown in Figure 1.3.
Now try Exercise 7. ⫺1, 2 .
y ⫺ x
1 ⫺1, 2 ,
⫺2, ⫺3 . ⫺1, 2 , 3, 4 , 0, 0 , 3, 0 ,
Example 1
x, y
Directed distance from x-axis
x, y
Directed distance
from y-axis x y y ,1.3 © Ariel Skelly/Corbis
Section 1.1 Rectangular Coordinates
Number of subscribers (in millions) Subscribers to a Cellular Telecommunication Service t N
t
1994, 24.1 .86.0 109.5 128.4 140.8 158.7 182.1 207.9 233.0 255.4
69.2
55.3
44.0
33.8
24.1
1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
Year, t Subscribers, N
You could also represent the data using a bar graph or a line graph. If you have access to a graphing utility, try using it to represent graphically the data given in Example 2.
t N Example 2 T E C H N O LO G Y The scatter plot in Example 2 is only one way to represent the data graphically.
t, N
3 The beauty of a rectangular coordinate system is that it allows you to see relation-
ships between two variables. It would be difficult to overestimate the importance of Descartes’s introduction of coordinates in the plane. Today, his ideas are in common use in virtually every scientific and business-related field.
50
100
150
200
250
300
1994 1996
1
t ⫽
In Example 2, you could have let represent the year 1994. In that case, the horizontal axis would not have been broken, and the tick marks would have been labeled 1 through 14 (instead of 1994 through 2007).
1.4 Now try Exercise 25.
FIGURE
To sketch a scatter plot of the data shown in the table, you simply represent each pair of values by an ordered pair and plot the resulting points, as shown in Figure 1.4. For instance, the first pair of values is represented by the ordered pair Note that the break in the -axis indicates that the numbers between 0 and 1994 have been omitted.
Association) Solution
From 1994 through 2007, the numbers (in millions) of subscribers to a cellular telecommunication service in the United States are shown in the table, where represents the year. Sketch a scatter plot of the data. (Source: CTIA-The Wireless
Sketching a Scatter Plot
1998 2000 2002 2004 2006 Year
4 Chapter 1 Functions and Their Graphs The Pythagorean Theorem and the Distance Formula
2
2
2 a
- b = c The following famous theorem is used extensively throughout this course.
c a Pythagorean Theorem
For a right triangle with hypotenuse of length and sides of lengths and c a b , you
2
2
2
have a ⫹ b ⫽ c , as shown in Figure 1.5. (The converse is also true. That is, if
2
2
2 a ⫹ b ⫽ c , then the triangle is a right triangle.) b
FIGURE
1.5 Suppose you want to determine the distance between two points , y and
d x
1
1
2
, y in the plane. With these two points, a right triangle can be formed, as shown in x
2 Figure 1.6. The length of the vertical side of the triangle is ⫺ y , and the length of y y
2
1
the horizontal side is ⫺ x . By the Pythagorean Theorem, you can write
x
2
1 y (x , y )
1
1
1
2
2
2
⫽ ⫺ x ⫹ ⫺ y
d x y
2
1
2
1 d y − y
2
2
2
2
2
1 ⫺ x ⫹ ⫺ y ⫽ ⫺ x ⫹ ⫺ y . d ⫽ x y x y
2
1
2
1
2
1
2
1 y This result is the Distance Formula.
2 (x , y ) (x , y )
1
2
2
2 x x x
1
2 The Distance Formula
x − x
The distance between the points , y and , y in the plane is
d x x
2
1
1
1
2
2
2
2 ⫺ x ⫹ ⫺ y .
2
1
2
FIGURE 1.6 d ⫽ x y1 Example 3 Finding a Distance
Find the distance between the points and ⫺2, 1 3, 4 .
Graphical Solution Algebraic Solution
⫺2, 1
1
1
2
Let , y and , y Then apply the Use centimeter graph paper to plot the points A and x ⫽ ⫺2, 1 x ⫽ 3, 4 .
2 Distance Formula. B Carefully sketch the line segment from to A B . Then 3, 4 .
use a centimeter ruler to measure the length of the segment.
2
2 d ⫽ ⫺ x ⫹ ⫺ y Distance Formula
x y
2
1
2
1 Substitute for
2
2
⫽
3 ⫺ ⫺ 2 ⫹ 4 ⫺
1 x 1 , y , x , and y . 1 2 2
2
2 ⫽ ⫹ Simplify.
5
3 1 cm ⫽ 34 Simplify. 4 3 2 5.83 Use a calculator. 7 6 5 So, the distance between the points is about 5.83 units. You can use the Pythagorean Theorem to check that the distance is correct.
?
2
2
2 d ⫽
3 ⫹
5 Pythagorean Theorem
2
?
2
2 34 ⫽
3 ⫹ 5 Substitute for d.
FIGURE
1.7 ✓ 34 ⫽ 34 Distance checks.
The line segment measures about 5.8 centimeters, as shown in Figure 1.7. So, the distance between the points is about 5.8 units.
Section 1.1 Rectangular Coordinates
5 y
Example 4 Verifying a Right Triangle
7 (5, 7) Show that the points and are vertices of a right triangle.2, 1 , 4, 0 , 5, 7
6
5 Solution
d = 45
4
1 The three points are plotted in Figure 1.8. Using the Distance Formula, you can find the d = 50
3 lengths of the three sides as follows.
3
2
2 2 d ⫽ ⫹ ⫽ 9 ⫹ 36 ⫽
1 d
=
5
45 5 ⫺ 2 7 ⫺ 1
2 (2, 1)
2
2
1 d ⫽ ⫹ ⫽ 4 ⫹ 1 ⫽
2 (4, 0) x
5 4 ⫺ 2 0 ⫺ 1
2
2 d ⫽ ⫹ ⫽ 1 ⫹ 49 ⫽
3
1
2
3
4
5
6
50 5 ⫺ 4 7 ⫺ 0
7 Because
FIGURE
1.8
2
2
2
1
2
⫹ ⫽ 45 ⫹ 5 ⫽ 50 ⫽ d d d
3 you can conclude by the Pythagorean Theorem that the triangle must be a right triangle.
Now try Exercise 43. You can review the techniques for evaluating a radical in Appendix A.2.
The Midpoint Formula
To find the midpoint of the line segment that joins two points in a coordinate plane, you can simply find the average values of the respective coordinates of the two endpoints using the Midpoint Formula.
The Midpoint Formula
The midpoint of the line segment joining the points , y and , y is given x x by the Midpoint Formula
x ⫹ x y ⫹ y
1
2
1
2 Midpoint ⫽ , .
2
2 For a proof of the Midpoint Formula, see Proofs in Mathematics on page 122.
Example 5 Finding a Line Segment’s Midpoint
Find the midpoint of the line segment joining the points and ⫺5, ⫺3 9, 3 .
y Solution
1
1
2
2
Let and , y , y x ⫽ ⫺5, ⫺3 x ⫽ 9, 3 .
6 x ⫹ x y ⫹ y
1
2
1
2 (9, 3)
Midpoint ⫽ , Midpoint Formula
3
2
2
(2, 0) ⫺ 5 ⫹ 9 ⫺ 3 ⫹
3 x ⫽ , Substitute for x , y , x , and y . 1 1 2 2
−6 −3
3
6
9
2
2
−3 Midpoint ⫽ Simplify.
2, 0
( 5, 3) − − −6 The midpoint of the line segment is as shown in Figure 1.9.
2, 0 ,
FIGURE 1.9 Now try Exercise 47(c).6 Chapter 1 Functions and Their Graphs Applications Example 6 Finding the Length of a Pass A football quarterback throws a pass from the 28-yard line, 40 yards from the sideline.
The pass is caught by a wide receiver on the 5-yard line, 20 yards from the same sideline, as shown in Figure 1.10. How long is the pass?
Solution Football Pass You can find the length of the pass by finding the distance between the points
40, 28 and 20, 5 .
35
2
2
⫺ x ⫹ ⫺ y
30 d ⫽ x y Distance Formula
2
1
2
1 (40, 28)
25
2
2
⫽
40 ⫺ 20 ⫹ 28 ⫺ 5 x y Substitute for , y , x , and . 1 1 2 2
20
⫽ 400 ⫹ 529
15 Simplify.
10
⫽ 929 Simplify.
(20, 5) Distance (in yards)
5 30 Use a calculator.
5 10 15 20 25 30 35 40 So, the pass is about 30 yards long.
Distance (in yards)
FIGURE 1.10 Now try Exercise 57.In Example 6, the scale along the goal line does not normally appear on a football field. However, when you use coordinate geometry to solve real-life problems, you are free to place the coordinate system in any way that is convenient for the solution of the problem.
Example 7 Estimating Annual Revenue
Barnes & Noble had annual sales of approximately $5.1 billion in 2005, and $5.4 billion in 2007. Without knowing any additional information, what would you estimate the 2006 sales to have been? (Source: Barnes & Noble, Inc.)
Solution
One solution to the problem is to assume that sales followed a linear pattern. With this
Barnes & Noble Sales y
assumption, you can estimate the 2006 sales by finding the midpoint of the line segment connecting the points and 2005, 5.1 2007, 5.4 .
5.5 (2007, 5.4) x ⫹ x y ⫹ y
1
2
1
2
5.4 Midpoint ⫽ , Midpoint Formula
2
2
5.3 (2006, 5.25) Midpoint
2005 ⫹ 2007
5.1 ⫹5.4
5.2 ⫽ , Substitute for x , x , y and y . 1 2 1 2
2
2
5.1 (2005, 5.1) 5.0 ⫽ Simplify.
2006, 5.25
Sales (in billions of dollars) x
So, you would estimate the 2006 sales to have been about $5.25 billion, as shown in
2005 2006 2007
Figure 1.11. (The actual 2006 sales were about $5.26 billion.)Year
FIGURE 1.11 Now try Exercise 59.Section 1.1 Rectangular Coordinates
6
2
1
5 −2 −3 −4 y x
−2
2
3
5
7 1 −1
4
4
4
5 −2 −3 −4
(2, 3) (1, 4) − ( 1, 2) − y
2, 3 . ⫺1, 2 , 1, ⫺4 ,
Example 8 Much of computer graphics, including this computer-generated goldfish tessellation, consists of transformations of points in a coordinate plane. One type of transformation, a translation, is illustrated in Example 8. Other types include reflections, rotations, and stretches.
Paul Morrell Extending the Example Example 8 shows how to translate points in a coordinate
plane. Write a short paragraph describing how each of the following transformed points is related to the original point.
Original Point Transformed Point
ⴚx, ⴚy x, y x, ⴚy x, y ⴚx, y x, y
C LASSROOM D
ISCUSSION
3
7 Translating Points in the Plane