CONTENTS
CHAPTER R ALGEBRA REFERENCE
R.1 Polynomials ................................................................................................................................. 1
R.2 Factoring...................................................................................................................................... 3
R.3 Rational Expressions ................................................................................................................... 4
R.4 Equations ..................................................................................................................................... 7
R.5 Inequalities................................................................................................................................... 11
R.6 Exponents .................................................................................................................................... 21
R.7 Radicals ....................................................................................................................................... 24
CHAPTER 1 FUNCTIONS
1.1 Lines and Linear Functions ......................................................................................................... 28
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1.2 The Least Squares Line ............................................................................................................... 39
1.3 Properties of Functions ................................................................................................................ 46
1.4 Quadratic Functions; Translation and Reflection ........................................................................ 54
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1.5 Polynomial and Rational Functions............................................................................................. 66
Chapter 1 Review Exercises .................................................................................................................. 76
Extended Application: Using Extrapolation to Predict Life Expectancy............................................... 88
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CHAPTER 2 EXPONENTIAL, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONS
2.1 Exponential Functions ................................................................................................................. 89
2.2 Logarithmic Functions................................................................................................................. 99
2.3 Applications: Growth and Decay................................................................................................. 109
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2.4 Trigonometric Functions ............................................................................................................. 114
Chapter 2 Review Exercises .................................................................................................................. 122
Extended Application: Power Functions ............................................................................................... 133
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CHAPTER 3 THE DERIVATIVE
3.1 Limits........................................................................................................................................... 134
3.2 Continuity .................................................................................................................................... 145
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3.3 Rates of Change........................................................................................................................... 152
3.4 Definition of the Derivative......................................................................................................... 163
3.5 Graphical Differentiation............................................................................................................. 178
Chapter 3 Review Exercises .................................................................................................................. 182
Extended Application: A Model for Drugs Administered Intravenously .............................................. 194
,CHAPTER 4 CALCULATING THE DERIVATIVE
4.1 Techniques for Finding Derivatives ............................................................................................ 196
4.2 Derivatives of Products and Quotients ........................................................................................ 205
4.3 The Chain Rule............................................................................................................................ 212
4.4 Derivatives of Exponential Functions ......................................................................................... 221
4.5 Derivatives of Logarithmic Functions ......................................................................................... 239
4.6 Derivatives of Trigonometric Functions...................................................................................... 248
Chapter 4 Review Exercises .................................................................................................................. 253
Extended Application: Managing Renewable Resources ...................................................................... 263
CHAPTER 5 GRAPHS AND THE DERIVATIVE
5.1 Increasing and Decreasing Functions .......................................................................................... 265
5.2 Relative Extrema ......................................................................................................................... 278
5.3 Higher Derivatives, Concavity, and the Second Derivative Test ................................................ 293
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5.4 Curve Sketching .......................................................................................................................... 316
Chapter 5 Review Exercises .................................................................................................................. 336
Extended Application: A Drug Concentration Model for Orally Administered Medications ............... 360
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CHAPTER 6 APPLICATIONS OF THE DERIVATIVE
6.1 Absolute Extrema ........................................................................................................................ 361
6.2 Applications of Extrema .............................................................................................................. 370
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6.3 Implicit Differentiation................................................................................................................ 383
6.4 Related Rates ............................................................................................................................... 399
6.5 Differentials: Linear Approximation ........................................................................................... 406
Chapter 6 Review Exercises .................................................................................................................. 410
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Extended Application: A Total Cost Model for a Training Program..................................................... 421
CHAPTER 7 INTEGRATION
7.1 Antiderivatives............................................................................................................................. 423
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7.2 Substitution.................................................................................................................................. 429
7.3 Area and the Definite Integral ..................................................................................................... 437
7.4 The Fundamental Theorem of Calculus....................................................................................... 451
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7.5 The Area Between Two Curves................................................................................................... 465
Chapter 7 Review Exercises .................................................................................................................. 480
Extended Application: Estimating Depletion Dates for Minerals.......................................................... 493
CHAPTER 8 FURTHER TECHNIQUES AND APPLICATIONS OF INTEGRATION
8.1 Numerical Integration.................................................................................................................. 494
8.2 Integration by Parts...................................................................................................................... 508
8.3 Volume and Average Value......................................................................................................... 517
8.4 Improper Integrals ....................................................................................................................... 525
Chapter 8 Review Exercises .................................................................................................................. 533
Extended Application: Flow Systems .................................................................................................... 545
,CHAPTER 9 MULTIVARIABLE CALCULUS
9.1 Functions of Several Variables.................................................................................................... 547
9.2 Partial Derivatives ...................................................................................................................... 555
9.3 Maxima and Minima.................................................................................................................... 571
9.4 Total Differentials and Approximations ...................................................................................... 580
9.5 Double Integrals .......................................................................................................................... 589
Chapter 9 Review Exercises .................................................................................................................. 600
Extended Application: Optimization for a Predator .............................................................................. 612
CHAPTER 10 MATRICES
10.1 Solution of Linear Systems.......................................................................................................... 614
10.2 Addition and Subtraction of Matrices.......................................................................................... 638
10.3 Multiplication of Matrices ........................................................................................................... 645
10.4 Matrix Inverses ............................................................................................................................ 654
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10.5 Eigenvalues and Eigenvectors ..................................................................................................... 675
Chapter 10 Review Exercises ............................................................................................................... 687
Extended Application: Contagion.......................................................................................................... 704
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CHAPTER 11 DIFFERENTIAL EQUATIONS
11.1 Solutions of Elementary and Separable Differential Equations .................................................. 705
11.2 Linear First-Order Differential Equations ................................................................................... 717
AN
11.3 Euler’s Method ............................................................................................................................ 725
11.4 Linear Systems of Differential Equations.................................................................................... 733
11.5 Nonlinear Systems of Differential Equations .............................................................................. 744
11.6 Applications of Differential Equations ........................................................................................ 753
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Chapter 11 Review Exercises ............................................................................................................... 759
Extended Application: Pollution of the Great Lakes ............................................................................. 772
CHAPTER 12 PROBABILITY
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12.1 Sets............................................................................................................................................... 774
12.2 Introduction to Probability........................................................................................................... 787
12.3 Conditional Probability; Independent Events; Bayes’ Theorem ................................................. 799
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12.4 Discrete Random Variables; Applications to Decision Making .................................................. 815
Chapter 12 Review Exercises ................................................................................................................ 822
Extended Application: Medical Diagnonis ............................................................................................ 835
CHAPTER 13 PROBABILITY AND CALCULUS
13.1 Continuous Probability Models ................................................................................................... 837
13.2 Expected Value and Variance of Continuous Random Variables ............................................... 845
13.3 Special Probability Density Functions......................................................................................... 856
Chapter 13 Review Exercises ................................................................................................................ 865
Extended Application: Exponential Waiting Times .............................................................................. 875
, CHAPTER 14 Discrete Dynamical Systems
14.1 Sequences .................................................................................................................................... 876
14.2 Equilibrium Points ....................................................................................................................... 884
14.3 Determining Stability................................................................................................................... 889
Chapter 14 Review Exercises ................................................................................................................ 895
Extended Application: Mathematical Modeling in a Dynamic World .................................................. 903
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CHAPTER R ALGEBRA REFERENCE
R.1 Polynomials ................................................................................................................................. 1
R.2 Factoring...................................................................................................................................... 3
R.3 Rational Expressions ................................................................................................................... 4
R.4 Equations ..................................................................................................................................... 7
R.5 Inequalities................................................................................................................................... 11
R.6 Exponents .................................................................................................................................... 21
R.7 Radicals ....................................................................................................................................... 24
CHAPTER 1 FUNCTIONS
1.1 Lines and Linear Functions ......................................................................................................... 28
M
1.2 The Least Squares Line ............................................................................................................... 39
1.3 Properties of Functions ................................................................................................................ 46
1.4 Quadratic Functions; Translation and Reflection ........................................................................ 54
EL
1.5 Polynomial and Rational Functions............................................................................................. 66
Chapter 1 Review Exercises .................................................................................................................. 76
Extended Application: Using Extrapolation to Predict Life Expectancy............................................... 88
AN
CHAPTER 2 EXPONENTIAL, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONS
2.1 Exponential Functions ................................................................................................................. 89
2.2 Logarithmic Functions................................................................................................................. 99
2.3 Applications: Growth and Decay................................................................................................. 109
IE
2.4 Trigonometric Functions ............................................................................................................. 114
Chapter 2 Review Exercises .................................................................................................................. 122
Extended Application: Power Functions ............................................................................................... 133
??
CHAPTER 3 THE DERIVATIVE
3.1 Limits........................................................................................................................................... 134
3.2 Continuity .................................................................................................................................... 145
??
3.3 Rates of Change........................................................................................................................... 152
3.4 Definition of the Derivative......................................................................................................... 163
3.5 Graphical Differentiation............................................................................................................. 178
Chapter 3 Review Exercises .................................................................................................................. 182
Extended Application: A Model for Drugs Administered Intravenously .............................................. 194
,CHAPTER 4 CALCULATING THE DERIVATIVE
4.1 Techniques for Finding Derivatives ............................................................................................ 196
4.2 Derivatives of Products and Quotients ........................................................................................ 205
4.3 The Chain Rule............................................................................................................................ 212
4.4 Derivatives of Exponential Functions ......................................................................................... 221
4.5 Derivatives of Logarithmic Functions ......................................................................................... 239
4.6 Derivatives of Trigonometric Functions...................................................................................... 248
Chapter 4 Review Exercises .................................................................................................................. 253
Extended Application: Managing Renewable Resources ...................................................................... 263
CHAPTER 5 GRAPHS AND THE DERIVATIVE
5.1 Increasing and Decreasing Functions .......................................................................................... 265
5.2 Relative Extrema ......................................................................................................................... 278
5.3 Higher Derivatives, Concavity, and the Second Derivative Test ................................................ 293
M
5.4 Curve Sketching .......................................................................................................................... 316
Chapter 5 Review Exercises .................................................................................................................. 336
Extended Application: A Drug Concentration Model for Orally Administered Medications ............... 360
EL
CHAPTER 6 APPLICATIONS OF THE DERIVATIVE
6.1 Absolute Extrema ........................................................................................................................ 361
6.2 Applications of Extrema .............................................................................................................. 370
AN
6.3 Implicit Differentiation................................................................................................................ 383
6.4 Related Rates ............................................................................................................................... 399
6.5 Differentials: Linear Approximation ........................................................................................... 406
Chapter 6 Review Exercises .................................................................................................................. 410
IE
Extended Application: A Total Cost Model for a Training Program..................................................... 421
CHAPTER 7 INTEGRATION
7.1 Antiderivatives............................................................................................................................. 423
??
7.2 Substitution.................................................................................................................................. 429
7.3 Area and the Definite Integral ..................................................................................................... 437
7.4 The Fundamental Theorem of Calculus....................................................................................... 451
??
7.5 The Area Between Two Curves................................................................................................... 465
Chapter 7 Review Exercises .................................................................................................................. 480
Extended Application: Estimating Depletion Dates for Minerals.......................................................... 493
CHAPTER 8 FURTHER TECHNIQUES AND APPLICATIONS OF INTEGRATION
8.1 Numerical Integration.................................................................................................................. 494
8.2 Integration by Parts...................................................................................................................... 508
8.3 Volume and Average Value......................................................................................................... 517
8.4 Improper Integrals ....................................................................................................................... 525
Chapter 8 Review Exercises .................................................................................................................. 533
Extended Application: Flow Systems .................................................................................................... 545
,CHAPTER 9 MULTIVARIABLE CALCULUS
9.1 Functions of Several Variables.................................................................................................... 547
9.2 Partial Derivatives ...................................................................................................................... 555
9.3 Maxima and Minima.................................................................................................................... 571
9.4 Total Differentials and Approximations ...................................................................................... 580
9.5 Double Integrals .......................................................................................................................... 589
Chapter 9 Review Exercises .................................................................................................................. 600
Extended Application: Optimization for a Predator .............................................................................. 612
CHAPTER 10 MATRICES
10.1 Solution of Linear Systems.......................................................................................................... 614
10.2 Addition and Subtraction of Matrices.......................................................................................... 638
10.3 Multiplication of Matrices ........................................................................................................... 645
10.4 Matrix Inverses ............................................................................................................................ 654
M
10.5 Eigenvalues and Eigenvectors ..................................................................................................... 675
Chapter 10 Review Exercises ............................................................................................................... 687
Extended Application: Contagion.......................................................................................................... 704
EL
CHAPTER 11 DIFFERENTIAL EQUATIONS
11.1 Solutions of Elementary and Separable Differential Equations .................................................. 705
11.2 Linear First-Order Differential Equations ................................................................................... 717
AN
11.3 Euler’s Method ............................................................................................................................ 725
11.4 Linear Systems of Differential Equations.................................................................................... 733
11.5 Nonlinear Systems of Differential Equations .............................................................................. 744
11.6 Applications of Differential Equations ........................................................................................ 753
IE
Chapter 11 Review Exercises ............................................................................................................... 759
Extended Application: Pollution of the Great Lakes ............................................................................. 772
CHAPTER 12 PROBABILITY
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12.1 Sets............................................................................................................................................... 774
12.2 Introduction to Probability........................................................................................................... 787
12.3 Conditional Probability; Independent Events; Bayes’ Theorem ................................................. 799
??
12.4 Discrete Random Variables; Applications to Decision Making .................................................. 815
Chapter 12 Review Exercises ................................................................................................................ 822
Extended Application: Medical Diagnonis ............................................................................................ 835
CHAPTER 13 PROBABILITY AND CALCULUS
13.1 Continuous Probability Models ................................................................................................... 837
13.2 Expected Value and Variance of Continuous Random Variables ............................................... 845
13.3 Special Probability Density Functions......................................................................................... 856
Chapter 13 Review Exercises ................................................................................................................ 865
Extended Application: Exponential Waiting Times .............................................................................. 875
, CHAPTER 14 Discrete Dynamical Systems
14.1 Sequences .................................................................................................................................... 876
14.2 Equilibrium Points ....................................................................................................................... 884
14.3 Determining Stability................................................................................................................... 889
Chapter 14 Review Exercises ................................................................................................................ 895
Extended Application: Mathematical Modeling in a Dynamic World .................................................. 903
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