1. Preview and Review

1.1 Preliminaries

1.2 Elementary Functions

1.3 Graphing

2. Discrete Time Models, Sequences, and Difference Equations

2.1 Exponential Growth and Decay

2.2 Sequences

2.3 More Population Models

3. Limits and Continuity

3.1 Limits

3.2 Continuity

3.3 Limits at Infinity

3.4 The Sandwich Theorem and Some Trigonometric Limits

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 The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials

4.3 The Product and Quotient Rules, and the Derivatives of Rational and Power Functions

4.4 The Chain Rule and Higher Derivatives

4.5 Derivatives of Trigonometric Functions

4.6 Derivatives of Exponential Functions

4.7 Derivatives of Inverse Functions, Logarithmic Functions, and the Inverse Tangent Function

4.8 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, Inflection Points, and Graphing

5.4 Optimization

5.5 L’Hôpital’s Rule

5.6 Difference Equations: Stability (Optional)

5.7 Numerical Methods: The Newton-Raphson Method (Optional)

5.8 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

7.5 Numerical Integration

7.6 The Taylor Approximation

7.7 Tables of Integrals (Optional)

8. Differential Equations

8.1 Solving Differential Equations

8.2 Equilibria and Their Stability

8.3 Systems of Autonomous Equations (Optional)

9. Linear Algebra and Analytic Geometry

9.1 Linear Systems

9.2 Matrices

9.3 Linear Maps, Eigenvectors, and Eigenvalues

9.4 Analytic Geometry

10. Multivariable Calculus

10.1 Functions of Two or More Independent Variables

10.2 Limits and Continuity

10.3 Partial Derivatives

10.4 Tangent Planes, Differentiability, and Linearization

10.5 More about Derivatives (Optional)

10.6 Applications (Optional)

10.7 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: Applications

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

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