(NOTE: Each chapter concludes with Key Terms and Review Problems.)
1. Preview and Review
1.1 Precalculus Skills Diagnostic Test
1.2 Preliminaries
1.2.1 The Real Numbers
1.2.2 Lines in the Plane
1.2.3 Equation of the Circle
1.2.4 Trigonometry
1.2.5 Exponentials and Logarithms
1.2.6 Complex Numbers and Quadratic Equations
1.3 Elementary Functions
1.3.1 What Is a Function?
1.3.2 Polynomial Functions
1.3.3 Rational Functions
1.3.4 Power Functions
1.3.5 Exponential Functions
1.3.6 Inverse Functions
1.3.7 Logarithmic Functions
1.3.8 Trigonometric Functions
1.4 Graphing
1.4.1 Graphing and Basic Transformations of Functions
1.4.2 The Logarithmic Scale
1.4.3 Transformations into Linear Functions
1.4.4 From a Verbal Description to a Graph (Optional)
2. Discrete-Time Models, Sequences, and Difference Equations
2.1 Exponential Growth and Decay
2.1.1 Modeling Population Growth in Discrete Time
2.1.2 Recurrence Equations
2.1.3 Visualizing Recurrence Equations
2.2 Sequences
2.2.1 What Are Sequences?
2.2.2 (Optional) Using Spreadsheets to Calculate a Recursive Sequence
2.2.3 Limits
2.2.4 Recurrence Equations
2.2.5 Using Notation to Represent Sums of Sequences
2.3 Modeling with Recurrence Equations
2.3.1 Density-Dependent Population Growth
2.3.2 Density-Dependent Population Growth: The Beverton - Holt Model
2.3.3 The Discrete Logistic Equation
2.3.4 Modeling Drug Absorption (optional)
3. Limits and Continuity
3.1 Limits
3.1.1 A Non-Rigorous Discussion of Limits
3.1.2 Pitfalls of Finding Limits
3.1.3 Limit Laws
3.2 Continuity
3.2.1 What Is Continuity?
3.2.2 Combinations of Continuous Functions
3.3 Limits at Infinity
3.4 Trigonometric Limits and the Sandwich Theorem
3.4.1 Geometric Argument for Trigonometric Limits
3.4.2 The Sandwich Theorem (Optional)
3.5 Properties of Continuous Functions
3.5.1 The Intermediate-Value Theorem and The Bisection Method
3.5.2 Using a Spreadsheet to Implement the Bisection Method (Optional)
3.5.3 A Final Remark on Continuous Functions
3.6 A Formal Definition of Limits (Optional)
4. Differentiation
4.1 Formal Definition of the Derivative
4.2 Properties of the Derivative
4.2.1 Interpreting the Derivative
4.2.2 Differentiability and Continuity
4.3 Power Rules and Basic Rules
4.4 The Product and Quotient Rules, and the Derivatives of Rational and Power Functions
4.4.1 The Product Rule
4.4.2 The Quotient Rule
4.5 Chain Rule
4.5.1 The Chain Rule
4.5.2 Proof of the Chain Rule
4.6 Implicit Functions and Implicit Differentiation
4.6.1 Implicit Differentiation
4.6.2 Related Rates
4.7 Higher Derivatives
4.8 Derivatives of Trigonometric Functions
4.9 Derivatives of Exponential Functions
4.10 Inverse Functions and Logarithms
4.10.1 Derivatives of Inverse Functions
4.10.2 The Derivative of the Logarithmic Function
4.10.3 Logarithmic Differentiation
4.11 Linear Approximation and Error Propagation
5. Applications of Differentiation
5.1 Extrema and the Mean-Value Theorem
5.1.1 The Extreme-Value Theorem
5.1.2 Local Extrema
5.1.3 The Mean-Value Theorem
5.2 Monotonicity and Concavity
5.2.1 Monotonicity
5.2.2 Concavity
5.3 Extrema and Inflection Points
5.3.1 Extrema
5.3.2 Inflection Points
5.4 Optimization
5.5 L'Hôpital's Rule
5.6 Graphing and Asymptotes
5.7 Recurrence Equations: Stability (Optional)
5.7.1 Exponential Growth
5.7.2 Stability: General Case
5.7.3 Population Growth Models
5.8 Numerical Methods: The Newton - Raphson Method (Optional)
5.9 Modeling Biological Systems Using Differential Equations (Optional)
5.9.1 Modeling Population Growth
5.9.2 Interpreting the Mathematical Model
5.9.3 Passage of Drugs Through the Human Body
5.10 Antiderivatives
6. Integration
6.1 The Definite Integral
6.1.1 The Area Problem
6.1.2 The General Theory of Riemann Integrals
6.1.3 Properties of the Riemann Integral
6.1.4 Order Properties of the Riemann Integral (Optional)
6.2 The Fundamental Theorem of Calculus
6.2.1 The Fundamental Theorem of Calculus (Part I)
6.2.2 Leibniz's Rule and a Rigorous Proof of the Fundamental Theorem of Calculus (Optional)
6.2.3 Antiderivatives and Indefinite Integrals
6.2.4 The Fundamental Theorem of Calculus (Part II)
6.3 Applications of Integration
6.3.1 Cumulative Change
6.3.2 Average Values
6.3.3 The Mean Value Theorem (Optional)
6.3.4 Areas (Optional)
6.3.5 The Volume of a Solid (Optional)
6.3.6 Rectification of Curves (Optional)
7. Integration Techniques and Computational Methods
7.1 The Substitution Rule
7.1.1 Indefinite Integrals
7.1.2 Definite Integrals
7.2 Integration by Parts and Practicing Integration
7.2.1 Integration by Parts
7.2.2 Practicing Integration
7.3 Rational Functions and Partial Fractions
7.3.1 Proper Rational Functions
7.3.2 Partial-Fraction Decomposition
7.3.3 Repeated Linear Factors
7.3.4 Irreducible Quadratic Factors (optional)
7.3.5 Summary
7.4 Improper Integrals (Optional)
7.4.1 Type 1: Unbounded Intervals
7.4.2 Type 2: Unbounded Integrand
7.4.3 A Comparison Result for Improper Integrals
7.5 Numerical Integration
7.5.1 The Midpoint Rule
7.5.2 The Trapezoidal Rule
7.5.3 Using a Spreadsheet for Numerical Integration
7.5.4 Estimating Error in a Numerical Integration (Optional)
7.6 The Taylor Approximation (optional)
7.6.1 Taylor Polynomials
7.6.2 The Taylor Polynomial about x = a
7.6.3 How Accurate Is the Approximation? (Optional)
7.7 Tables of Integrals (Optional)
8. Differential Equations
8.1 Solving Separable Differential Equations
8.1.1 Pure-Time Differential Equations
8.1.2 Autonomous Differential Equations
8.1.3 General Separable Equations
8.2 Equilibria and Their Stability
8.2.1 Equilibrium Points
8.2.2 Graphical Approach to Finding Equilibria
8.2.3 Stability of Equilibrium Points
8.2.4 Sketching Solutions Using the Vector Field Plot
8.2.5 Behavior Near an Equilibrium
8.3 Differential Equation Models
8.3.1 Compartment Models
8.3.2 An Ecological Model
8.3.3 Modeling a Chemical Reaction
8.3.4 The Evolution of Cooperation
8.3.5 Epidemic Model
8.4 Integrating Factors and Two-Compartment Models
8.4.1 Integrating Factors
8.4.2 Two-Compartment Models
9. Linear Algebra and Analytic Geometry
9.1 Linear Systems
9.1.1 Graphical Solution
9.1.2 Solving Equations Using Elimination
9.1.3 Solving Systems of Linear Equations
9.1.4 Representing Systems of Equations Using Matrices
9.2 Matrices
9.2.1 Matrix Operations
9.2.2 Matrix Multiplication
9.2.3 Inverse Matrices
9.2.4 Computing Inverse Matrices (Optional)
9.3 Linear Maps, Eigenvectors, and Eigenvalues
9.3.1 Graphical Representation
9.3.2 Eigenvalues and Eigenvectors
9.3.3 Iterated Maps (Needed for Section 10.7)
9.4 Demographic Modeling
9.4.1 Modeling with Leslie Matrices
9.4.2 Stable Age Distributions in Demographic Models
9.5 Analytic Geometry
9.5.1 Points and Vectors in Higher Dimensions
9.5.2 The Dot Product
9.5.3 Parametric Equations of Lines
10. Multivariable Calculus
10.1 Two or More Independent Variables
10.1.1 Defining a Function of Two or More Variables
10.1.2 The Graph of a Function of Two Independent Variables - Surface plot
10.1.3 Heat Maps
10.1.4 Contour plots
10.2 Limits and Continuity (optional)
10.2.1 Informal Definition of Limits
10.2.2 Continuity
10.2.3 Formal Definition of Limits
10.3 Partial Derivatives
10.3.1 Functions of Two Variables
10.3.2 Functions of More Than Two Variables
10.3.3 Higher-Order Partial Derivatives
10.4 Tangent Planes, Differentiability, and Linearization
10.4.1 Functions of Two Variables
10.4.2 Vector-Valued Functions
10.5 The Chain Rule and Implicit Differentiation (Optional)
10.5.1 The Chain Rule for Functions of Two Variables
10.5.2 Implicit Differentiation
10.6 Directional Derivatives and Gradient Vectors (Optional)
10.6.1 Deriving the Directional Derivative
10.6.2 Properties of the Gradient Vector
10.7 Maximization and Minimization of Functions (Optional)
10.7.1 Local Maxima and Minima
10.7.2 Global Extrema
10.7.3 Extrema with Constraints
10.7.4 Least-Squares Data Fitting
10.8 Diffusion (Optional)
10.9 Systems of Difference Equations (Optional)
10.9.1 A Biological Example
10.9.2 Equilibria and Stability in Systems of Linear Recurrence Equations
10.9.3 Equilibria and Stability of Nonlinear Systems of Difference Equations
11. Systems of Differential Equations
11.1 Linear Systems: Theory
11.1.1 The Direction Field
11.1.2 Solving Linear Systems
11.1.3 Equilibria and Stability
11.1.4 Systems with Complex Conjugate Eigenvalues
11.1.5 Summary of the Theory of Linear Systems
11.2 Linear Systems: Applications
11.2.1 Two-Compartment Models
11.2.2 A Mathematical Model for Love
11.2.3 The Harmonic Oscillator (Optional)
11.3 Nonlinear Autonomous Systems: Theory
11.3.1 Analytical Approach
11.3.2 Graphical Approach for 2 x 2 Systems (Optional)
11.4 Nonlinear Systems: Lotka - Volterra Model of Interspecific Interactions
11.4.1 Competition
11.4.2 A Predator--Prey Model
11.5 More Mathematical Models (Optional)
11.5.1 The Community Matrix
11.5.2 Neuron Activity
11.5.3 Enzymatic Reactions
12. Probability and Statistics
12.1 Counting
12.1.1 The Multiplication Principle
12.1.2 Permutations
12.1.3 Combinations
12.1.4 Combining the Counting Principles
12.2 What Is Probability?
12.2.1 Basic Definitions
12.2.2 Equally Likely Outcomes
12.3 Conditional Probability and Independence
12.3.1 Conditional Probability
12.3.2 The Law of Total Probability
12.3.3 Independence
12.3.4 The Bayes Formula
12.4 Discrete Random Variables and Discrete Distributions
12.4.1 Discrete Distributions
12.4.2 Mean and Variance
12.4.3 The Binomial Distribution
12.4.4 The Multinomial Distribution
12.4.5 Geometric Distribution
12.4.6 The Poisson Distribution
12.5 Continuous Distributions
12.5.1 Density Functions
12.5.2 The Normal Distribution
12.5.3 The Uniform Distribution
12.5.4 The Exponential Distribution
12.5.5 The Poisson Process
12.5.6 Aging
12.6 Limit Theorems
12.6.1 The Law of Large Numbers
12.6.2 The Central Limit Theorem
12.7 Statistical Tools
12.7.1 Describing Univariate Data
12.7.2 Estimating Parameters
12.7.3 Linear Regression
Appendix A Frequently Used Symbols
Appendix B Table of the Standard Normal Distribution
Answers to Odd-Numbered Problems
References
Photo Credits
Index
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