1. Preview and Review

• 1.1 Precalculus Skills Diagnostic Test
• 1.2 Preliminaries
• 1.3 Elementary Functions
• 1.4 Graphing

2. Discrete-Time Models, Sequences, and Difference Equations

• 2.1 Exponential Growth and Decay
• 2.2 Sequences
• 2.3 Modeling with Recurrence Equations

3. Limits and Continuity

• 3.1 Limits
• 3.2 Continuity
• 3.3 Limits at Infinity
• 3.4 Trigonometric Limits and the Sandwich Theorem
• 3.5 Properties of 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
• 4.3 Power Rules and Basic Rules
• 4.4 The Product and Quotient Rules, and the Derivatives of Rational and Power Functions
• 4.5 Chain Rule
• 4.6 Implicit Functions and Implicit Differentiation
• 4.7 Higher Derivatives
• 4.8 Derivatives of Trigonometric Functions
• 4.9 Derivatives of Exponential Functions
• 4.10 Inverse Functions and Logarithms
• 4.11 Linear Approximation and Error Propagation

5. Applications of Differentiation

• 5.1 Extrema and the Mean-Value Theorem
• 5.2 Monotonicity and Concavity
• 5.3 Extrema and Inflection Points
• 5.4 Optimization
• 5.5 L’Hôpital’s Rule
• 5.6 Graphing and Asymptotes
• 5.7 Recurrence Equations: Stability (Optional)
• 5.8 Numerical Methods: The Newton – Raphson Method (Optional)
• 5.9 Modeling Biological Systems Using Differential Equations (Optional)
• 5.10 Antiderivatives

6. Integration

• 6.1 The Definite Integral
• 6.2 The Fundamental Theorem of Calculus
• 6.3 Applications of Integration

7. Integration Techniques and Computational Methods

• 7.1 The Substitution Rule
• 7.2 Integration by Parts and Practicing Integration
• 7.3 Rational Functions and Partial Fractions
• 7.4 Improper Integrals (Optional)
• 7.5 Numerical Integration
• 7.6 The Taylor Approximation (optional)
• 7.7 Tables of Integrals (Optional)

8. Differential Equations

• 8.1 Solving Separable Differential Equations
• 8.2 Equilibria and Their Stability
• 8.3 Differential Equation Models
• 8.4 Integrating Factors and Two-Compartment Models

9. Linear Algebra and Analytic Geometry

• 9.1 Linear Systems
• 9.2 Matrices
• 9.3 Linear Maps, Eigenvectors, and Eigenvalues
• 9.4 Demographic Modeling
• 9.5 Analytic Geometry

10. Multivariable Calculus

• 10.1 Two or More Independent Variables
• 10.2 Limits and Continuity (optional)
• 10.3 Partial Derivatives
• 10.4 Tangent Planes, Differentiability, and Linearization
• 10.5 The Chain Rule and Implicit Differentiation (Optional)
• 10.6 Directional Derivatives and Gradient Vectors (Optional)
• 10.7 Maximization and Minimization of Functions (Optional)
• 10.8 Diffusion (Optional)
• 10.9 Systems of Difference Equations (Optional)

11. Systems of Differential Equations

• 11.1 Linear Systems: Theory
• 11.2 Linear Systems: Applications
• 11.3 Nonlinear Autonomous Systems: Theory
• 11.4 Nonlinear Systems: Lotka – Volterra Model of Interspecific Interactions
• 11.5 More Mathematical Models (Optional)

12. Probability and Statistics

• 12.1 Counting
• 12.2 What Is Probability?
• 12.3 Conditional Probability and Independence
• 12.4 Discrete Random Variables and Discrete Distributions
• 12.5 Continuous Distributions
• 12.6 Limit Theorems
• 12.7 Statistical Tools

Appendices

• A: Frequently Used Symbols
• B: Table of the Standard Normal Distribution

Answers to Odd-Numbered Problems

References

Photo Credits

Index