# Calculus

## 1. Review of Functions

1.1 Functions and Their Graphs 1.2 Trigonometric Functions 1.3 Other Special Functions 1.4 Inverse Functions

## 2. Limits

2.1 Rates of Change and Tangent Lines 2.2 The Definition of a Limit 2.3 Computing Limits with the Limits Laws 2.4 Continuity 2.5 One-Sided Limits 2.6 Limits Involving Infinity

## 3. Differentiation

3.1 The Derivative as Rate of Change 3.2 The Derivative at a Point 3.3 The Derivative as a Function 3.4 The Basic Rules of Differentiation 3.5 The Product and Quotient Rules 3.6 The Chain Rule 3.7 Derivatives of the Trigonometric Functions 3.8 Implicit Differentiation 3.9 Derivatives of Exponential and Logarithmic Functions 3.10 Derivatives of the Inverse Trigonometric Functions

## 4. Applications of the Derivative

4.1 Extrema for a Function 4.2 The Mean Value Theorem 4.3 First Derivatives and Increasing/Decreasing Functions 4.4 Second Derivatives and Concavity 4.5 Optimization Problems 4.6 Linear Approximation and Differentials 4.7 Newton's Method 4.8 Related Rates 4.9 L'Hopital's Rule 4.10 Antiderivatives

## 5. Integration

5.1 Estimating the Area under a Curve 5.2 The Definite Integral 5.3 The Indefinite Integral 5.4 The Fundamental Theorem of Calculus

## 6. Integration Techniques

6.1 The Basic Rules of Integration 6.2 Integration by Substitution 6.3 Integration by Parts 6.4 Integration of Trigonometric Functions 6.5 Integration by Trigonometric Substitution 6.6 Partial Fraction Decomposition 6.7 Integration Tables and Other Strategies 6.8 Numerical Integration and CAS Systems 6.9 Improper Integrals

## 7. Applications of the Integral

7.1 The Area Between Two Curves 7.2 Volumes: The Disk Method 7.3 Volumes: The Shell Method 7.4 Arc Length 7.5 Surface Area and Areas of Revolution 7.6 Net Change 7.7 Average Value 7.8 Applications in Science 7.9 Introduction to Probability

## 8. Ordinary Differential Equations

8.1 Introduction to Differential Equations: Slope Fields and Euler's Method 8.2 Exponential Growth and Decay 8.3 Separable Differential Equations 8.4 The Logistic Equation 8.5 First Order Linear Differential Equations

## 9. Sequences and Series

9.1 Sequences 9.2 Infinite Series and the nth Term Test 9.3 The Integral and p-Series Tests 9.4 The Comparison Tests 9.5 The Root and Ratio Tests 9.6 Alternating Series and Absolute Convergence 9.7 Polynomial Approximations of Functions 9.8 Power Series 9.9 Taylor Series 9.10 Convergence of Taylor Series

## 10. Parametrized Functions and Polar Coordinates

10.1 Parametric Equations of Plane Curves 10.2 Tangents, Arc Length, and Surface Area of Parametrized Regions 10.3 The Polar Coordinate System 10.4 Arc Length and Area in Polar Coordinates 10.5 Conic Sections

## 11. Vectors and the Geometry of Space

11.1 Vectors in the Plane 11.2 Vectors in 3 Dimensional Space 11.3 Spherical and Cylindrical Coordinates in 3D 11.4 The Dot Product 11.5 The Cross Product 11.6 Lines, Curves, and Planes 11.7 Surfaces

## 12. Vector Functions

12.1 Vector Functions 12.2 Differentiation and Integration of Vector Functions 12.3 Particle Motion in Space 12.4 Arc Length 12.5 Curvature

## 13. Differentiation of Functions of Several Variables

13.1 Functions of Several Variables 13.2 Limits and Continuity in Higher Dimensions 13.3 Partial Derivatives 13.4 Differentials and the Tangent Plane 13.5 The Chain Rule for Functions of Several Variables 13.6 Directional Derivatives and the Gradient 13.7 Extrema on a Surface 13.8 Lagrange Multipliers

## 14. Multiple Integration

14.1 Double Integrals over Rectangular Regions 14.2 Double Integrals over Arbitrary Regions 14.3 Double Integrals in Polar Coordinates 14.4 Surface Area 14.5 Triple Integrals 14.6 Triple Integrals in Spherical and Cylindrical Coordinates 14.7 Centroids and Moments of Inertia 14.8 Change of Variables in Multiple Integrals

## 15. Vector Analysis

15.1 Vector Fields 15.2 Line Integrals 15.3 Conservative Vector Fields and the Fundamental Theorem for Line Integrals 15.4 Green's Theorem 15.5 Parametrized Surfaces 15.6 Surface Integrals 15.7 Divergence and Curl 15.8 Stokes' Theorem 15.9 The Divergence Theorem