Fundamentals of Precalculus

by Dugopolski, Mark

• ISBN13: 9780321122322
• ISBN10: 0321122321
• Edition: 1st
• Format: Hardcover
• Copyright: 2004-01-01
• Publisher: Addison Wesley

Summary

Fundamentals of Precalculus is designed to review the fundamental topics that are necessary for success in calculus. Containing only five chapters, this text can be covered in a one-semester or one-term course with a minimum of deleting or skipping around. A student who is well acquainted with the material in this text will have the necessary skills, understanding, and insights required in calculus.

Table of Contents

Preface	xiii
Supplements	xvii
Graphs and Functions	1	(105)
Real Numbers and Their Properties	2	(9)
The Real Numbers	2	(2)
Properties of the Real Numbers	4	(1)
Additive Inverses	5	(1)
Relations	6	(1)
Absolute Value	6	(3)
Equations Involving Absolute Value	9	(2)
Linear and Absolute Value Inequalities	11	(10)
Interval Notation	11	(1)
Linear Inequalities	12	(2)
Compound Inequalities	14	(2)
Absolute Value Inequalities	16	(2)
Modeling with Inequalities	18	(3)
Equations and Graphs in Two Variables	21	(11)
The Cartesian Coordinate System	21	(1)
The Distance Formula	22	(1)
The Midpoint Formula	23	(1)
The Circle	24	(3)
The Line	27	(2)
Using a Graph to Solve an Equation	29	(3)
Linear Equations in Two Variables	32	(11)
Slope of a Line	32	(2)
Point-Slope Form	34	(1)
Slope-Intercept Form	35	(1)
Using Slope to Graph a Line	36	(1)
The Three Forms for the Equation of a Line	37	(1)
Parallel Lines	38	(1)
Perpendicular Lines	39	(1)
Applications	40	(3)
Functions	43	(12)
The Function Concept	43	(1)
Identifying Functions	44	(3)
Domain and Range	47	(1)
Function Notation	48	(2)
The Average Rate of Change of a Function	50	(2)
Constructing Functions	52	(3)
Graphs of Relations and Functions	55	(12)
Graphing Equations	55	(3)
Semicircles	58	(1)
Piecewise Functions	59	(3)
Increasing, Decreasing, and Constant	62	(5)
Families of Functions, Transformations, and Symmetry	67	(12)
Translation	67	(3)
Reflection	70	(2)
Stretching and Shrinking	72	(2)
The Linear Family of Functions	74	(1)
Symmetry	74	(5)
Operations with Functions	79	(9)
Basic Operations with Functions	79	(3)
Composition of Functions	82	(3)
Applications	85	(3)
Inverse Functions	88	(18)
One-to-One and Invertible Functions	88	(1)
Determining Whether a Function Is One-to-One	89	(2)
Inverse Functions Using Function Notation	91	(2)
Graphs of f and f-1	93	(1)
Finding Inverse Functions Mentally	94	(5)
Highlights	99	(2)
Chapter 1 Review Exercises	101	(4)
Concepts of Calculus: Limits	105	(1)
Polynomial and Rational Functions	106	(74)
Quadratic Functions and Inequalities	107	(8)
Two Forms for a Quadratic Function	107	(2)
Opening, Vertex, and Axis of Symmetry	109	(1)
Intercepts	110	(1)
Quadratic Inequalities	111	(2)
Applications of Maximum and Minimum	113	(2)
Complex Numbers	115	(7)
Definitions	115	(2)
Addition, Subtraction, and Multiplication	117	(1)
Division of Complex Numbers	118	(1)
Roots of Negative Numbers	119	(1)
Imaginary Solutions to Quadratic Equations	120	(2)
Zeros of Polynomial Functions	122	(11)
The Remainder Theorem	122	(1)
Synthetic Division	123	(3)
The Factor Theorem	126	(1)
The Fundamental Theorem of Algebra	127	(1)
The Rational Zero Theorem	127	(6)
The Theory of Equations	133	(6)
The Number of Roots of a Polynomial Equation	133	(1)
The Conjugate Pairs Theorem	134	(1)
Descartes' Rule of Signs	135	(4)
Miscellaneous Equations	139	(10)
Factoring Higher-Degree Equations	139	(1)
Equations Involving Square Roots	140	(2)
Equations with Rational Exponents	142	(1)
Equations of Quadratic Type	143	(2)
Equations Involving Absolute Value	145	(4)
Graphs of Polynomial Functions	149	(11)
Drawing Good Graphs	149	(1)
Symmetry	149	(2)
Behavior at the x-Intercepts	151	(1)
The Leading Coefficient Test	151	(3)
Sketching Graphs of Polynomial Functions	154	(1)
Polynomial Inequalities	155	(5)
Rational Functions and Inequalities	160	(20)
Rational Functions and Their Domains	160	(1)
Horizontal and Vertical Asymptotes	161	(2)
Oblique Asymptotes	163	(1)
Sketching Graphs of Rational Functions	164	(4)
Rational Inequalities	168	(1)
Applications	169	(5)
Highlights	174	(1)
Chapter 2 Review Exercises	175	(4)
Concepts of Calculus: Instantaneous rate of change	179	(1)
Trigonometric Functions	180	(104)
Angles and Their Measurements	181	(10)
Degree Measure of Angles	181	(4)
Radian Measure of Angles	185	(3)
Arc Length	188	(3)
The Sine and Cosine Functions	191	(8)
Definition	191	(2)
Sine and Cosine of a Multiple of a 45°	193	(1)
Sine and Cosine of a Multiple of a 30°	194	(1)
Sine and Cosine of an Arc	195	(1)
Approximate Values for Sine and Cosine	196	(1)
The Fundamental Identity	196	(1)
Modeling the Motion of a Spring	197	(2)
The Graphs of the Sine and Cosine Functions	199	(14)
The Graph of y = sin(x)	199	(3)
The Graph of y = cos(x)	202	(1)
Transformations of Sine and Cosine	203	(2)
Changing the Period	205	(2)
The General Sine Wave	207	(2)
Frequency	209	(4)
The Other Trigonometric Functions and Their Graphs	213	(11)
Definitions	213	(2)
Graph of y = tan(x)	215	(2)
Graph of y = cot(x)	217	(2)
Graph of y = sec(x)	219	(1)
Graph of y = csc(x)	220	(4)
The Inverse Trigonometric Functions	224	(9)
The Inverse of the Sine Function	224	(2)
The Inverse Cosine Function	226	(1)
Inverses of Tangent, Cotangent, Secant, and Cosecant	227	(3)
Compositions of Functions	230	(3)
Right Triangle Trigonometry	233	(9)
Trigonometric Ratios	233	(1)
Right Triangles	234	(2)
Solving a Right Triangle	236	(1)
Applications	237	(5)
Identities	242	(11)
Pythagorean Identities	242	(1)
Odd and Even Identities	243	(1)
Sum and Difference Identities	243	(1)
Cofunction Identities	244	(1)
Double-Angle and Half-Angle Identities	245	(2)
Product and Sum Identities	247	(2)
The Function y = asinx + bcosx	249	(4)
Conditional Trigonometric Equations	253	(11)
Cosine Equations	253	(2)
Sine Equations	255	(1)
Tangent Equations	256	(1)
Equations Involving Multiple Angles	257	(1)
More Complicated Equations	258	(3)
Modeling Projectile Motion	261	(3)
The Law of Sines and the Law of Cosines	264	(20)
Oblique Triangles	264	(1)
The Law of Sines	264	(2)
The Ambiguous Case (SSA)	266	(3)
The Law of Cosines	269	(3)
Bearing	272	(5)
Highlights	277	(2)
Chapter 3 Review Exercises	279	(4)
Concepts of Calculus: Area of a circle and π	283	(1)
Exponential and Logarithmic Functions	284	(44)
Exponential Functions and Their Applications	285	(11)
Exponential Functions	285	(1)
Domain of Exponential Functions	285	(1)
Graphing Exponential Functions	286	(3)
The Exponential Family of Functions	289	(2)
Exponential Equations	291	(1)
The Compound Interest Model	291	(2)
Continuous Compounding and the Number e	293	(3)
Logarithmic Functions and Their Applications	296	(10)
Logarithmic Functions	296	(2)
Graphs of Logarithmic Functions	298	(1)
The Logarithmic Family of Functions	299	(1)
Logarithmic and Exponential Equations	300	(3)
Applications	303	(3)
Rules of Logarithms	306	(8)
The Product Rule for Logarithms	306	(1)
The Quotient Rule for Logarithms	307	(1)
The Power Rule for Logarithms	307	(1)
The Inverse Rules	308	(1)
Using the Rules	309	(2)
The Base Change Formula	311	(3)
More Equations and Applications	314	(14)
Logarithmic Equations	314	(1)
Exponential Equations	315	(2)
Strategy for Solving Equations	317	(1)
Radioactive Dating	317	(2)
Newton's Model for Cooling	319	(1)
Paying off a Loan	320	(3)
Highlights	323	(1)
Chapter 4 Review Exercises	324	(3)
Concepts of Calculus: Evaluating transcendental functions	327	(1)
Conic Sections, Polar Coordinates, and Parametric Equations	328	(58)
The Parabola	329	(9)
The Geometric Definition	329	(1)
Developing the Equation	330	(1)
The Standard Equation of a Parabola	331	(1)
Graphing a Parabola	332	(1)
Parabolas Opening to the Left or Right	333	(2)
Applications	335	(3)
The Ellipse and the Circle	338	(10)
Definition of Ellipse	338	(1)
The Equation of the Ellipse	339	(2)
Graphing an Ellipse Centered at the Origin	341	(2)
Translations of Ellipses	343	(1)
The Circle	344	(1)
Applications	344	(4)
The Hyperbola	348	(14)
The Definition	349	(1)
Developing the Equation	350	(1)
Graphing a Hyperbola Centered at (0,0)	351	(4)
Hyperbolas Centered at (h,k)	355	(2)
Finding the Equation of a Hyperbola	357	(1)
Classifying the Conics	358	(4)
Polar Equations	362	(10)
Polar Coordinates	362	(1)
Polar-Rectangular Conversions	363	(1)
Polar Equations	364	(3)
Converting Equations	367	(5)
Polar Equations of the Conics	372	(4)
Alternative Definition of the Conics	372	(1)
Polar Equations of the Conics	372	(2)
Equivalency of the Definitions	374	(2)
Parametric Equations	376	(10)
Graphs of Parametric Equations	376	(1)
Eliminating the Parameter	377	(1)
Writing Parametric Equations	378	(3)
Highlights	381	(1)
Chapter 5 Review Exercises	382	(3)
Concepts of Calculus: The reflection property of a parabola	385	(1)
APPENDIX A BASIC ALGEBRA REVIEW	386
A.1 Exponents and Radicals	386	(10)
Exponential Expressions	386	(1)
Negative Integral Exponents	387	(1)
Rules of Exponents	388	(1)
Roots	389	(1)
Rational Exponents	389	(2)
Radical Notation	391	(1)
The Product and Quotient Rules for Radicals	392	(1)
Simplified Form and Rationalizing the Denominator	393	(1)
Operations with Radical Expressions	394	(2)
A.2 Polynomials	396	(6)
Definitions	396	(1)
Naming and Evaluating Polynomials	397	(1)
Addition and Subtraction of Polynomials	397	(1)
Multiplication of Polynomials	398	(1)
Using FOIL	399	(1)
Special Products	399	(1)
Division of Polynomials	400	(2)
A.3 Factoring Polynomials	402	(5)
Factoring out the Greatest Common Factor	402	(1)
Factoring by Grouping	403	(1)
Factoring ax2 + bx + c	403	(1)
Factoring the Special Products	404	(1)
Factoring the Difference and Sum of Two Cubes	405	(1)
Factoring Completely	406	(1)
A.4 Rational Expressions	407
Reducing	407	(2)
Multiplication and Division	409	(1)
Building up the Denominator	410	(1)
Addition and Subtraction	411
Answers to Selected Exercises	1	(1)
Credits	1	(1)
Index	1