Online Textbook

0 Preliminaries

• 0.1 Fundamental Skills in Algebra
• 0.2 Numbers, Sets, and Absolute Values
• 0.3 Coordinates and Geometry
• 0.4 Defining a Function
• 0.5 Linear Functions and Equations
• 0.6 Modifying Functions; Inverse Functions
• 0.7 Quadratics
• 0.8 Trigonometry
• 0.9 Exponents and Logarithms
• 0.10 Sigma Notation

1 Limits and Continuity

• 1.1 Defining a Limit
• 1.2 Evaluating Limits Analytically
• 1.3 Squeeze Theorem and Trigonometric Limits
• 1.4 Continuity
• 1.5 Formal Definition of a Limit
• 1.6 Limits with Infinity

2 Differentiation Rules

• 2.1 Defining a Derivative
• 2.2 Differentiating Power, Exponential, and Sinusoidal Functions
• 2.3 Product Rule and Quotient Rule
• 2.4 Chain Rule
• 2.5 Implicit Differentiation and Differentiating Inverse Functions
• 2.6 Differentiating Logarithmic Functions
• 2.7 Related Rates
• 2.8 Linearization and Differentials
• 2.9 Hyperbolic Functions

3 Applications of Differentiation

• 3.1 Minimum and Maximum Values
• 3.2 Mean Value Theorem and Rolle's Theorem
• 3.3 Using the First and Second Derivatives
• 3.4 Particle Motion
• 3.5 Indeterminate Forms and L'Hopital's Rule
• 3.6 Optimization
• 3.7 Applications of Differentiation in Economics
• 3.8 Newton's Method

4 Integration

• 4.1 Antidifferentiation
• 4.2 Definite Integrals
• 4.3 Fundamental Theorem of Calculus
• 4.4 Integration by Substitution
• 4.5 Numerical Integration

5 Applications of Integration

• 5.1 Areas between Curves
• 5.2 Volumes with Cross Sections
• 5.3 Solids of Revolution
• 5.4 Shell Method
• 5.5 Work
• 5.6 Average Value of a Function

6 Integration Techniques

• 6.1 Integration by Parts
• 6.2 Trigonometric Integrals
• 6.3 Trigonometric Substitution
• 6.4 Integration by Partial Fractions
• 6.5 Improper Integrals

7 Further Applications of Integration

• 7.1 Arc Length
• 7.2 Surface Areas of Revolution
• 7.3 Consumer Surplus and Producer Surplus
• 7.4 Moments and Centers of Mass
• 7.5 Hydrostatics
• 7.6 Probability

8 Differential Equations

• 8.1 Slope Fields and Euler's Method
• 8.2 Separation of Variables
• 8.3 Exponential Models
• 8.4 Logistic Models
• 8.5 First-Order Linear Differential Equations

9 Parametric Equations and Polar Coordinates

• 9.1 Parametric Equations
• 9.2 Differentiating and Integrating Parametric Functions
• 9.3 Polar Coordinates and Functions
• 9.4 Differentiating Polar Functions
• 9.5 Areas with Polar Curves
• 9.6 Additional Calculus with Parametric and Polar

10 Infinite Sequences and Series

• 10.1 Sequences
• 10.2 Infinite Series and Divergence Test
• 10.3 Integral Test
• 10.4 Comparison Tests
• 10.5 Alternating Series
• 10.6 Absolute Convergence and the Ratio and Root Tests
• 10.7 Power Series
• 10.8 Taylor and Maclaurin Series

A Appendices