Table of Contents
Instructors’ Guide and Background
Chapter 1. What is Calculus? The Six Pillars
Chapter 2. Predicting the Future: The SIR Model
2.1. Worried Sick
2.2. Building the SIR Model
2.3. Analyzing the Model Numerically
2.4. Theoretical Analysis: What Goes Up Has to Stop Before It Comes Down
2.5. Covid-19 and Modified SIR Models
2.6. Same Song, Different Singer: SIR and Product Marketing
2.7. Chapter Summary
2.8. Exercises
Chapter 3. Close is Good Enough
3.1. The Idea of Approximation
3.2. Functions
3.3. Linear Functions and Their Graphs
3.4. Linear Approximations and Microscopes
3.5. Euler’s Method and Compound Interest
3.6. The SIR Model by Computer
3.7. Solving Algebraic Equations
3.8. Chapter Summary
3.9. Exercises
Chapter 4. Track the Changes
4.1. Atmospheric Carbon
4.2. Other Derivatives and Marginal Quantities
4.3. Local Linearity and Microscopes
4.4. The Derivative
4.5. A Global View
4.6. Chapter Summary
4.7. Exercises
Chapter 5. Computing and Using Derivatives (What Goes Up Has to Stop Before It Comes Down)
5.1. Building Blocks
5.2. Adding, Subtracting, Multiplying, and Dividing Functions
5.3. The Chain Rule
5.4. Optimization
5.5. The Shape of a Graph
5.6. Newton’s Method
5.7. Chapter Summary
5.8. Supplemental Material: Small Angle Approximations
5.9. Exercises
Chapter 6. Models of Growth and Oscillation
6.1. Modeling with Differential Equations
6.2. Exponential Functions and Logarithms
6.3. Simple Models of Growth and Decay
6.4. Two Models of Oscillation
6.5. More Sophisticated Population Models
6.6. Chemistry
6.7. Chapter Summary
6.8. Supplemental Material: A Crash Course in Trigonometry
6.9. Exercises
Chapter 7. The Whole Is the Sum of the Parts
7.1. Slicing and Dicing
7.2. Riemann Sums
7.3. The Definite Integral
7.4. The Accumulation Function
7.5. Chapter Summary
7.6. Exercises
Chapter 8. The Fundamental Theorem of Calculus (One Step at a Time)
8.1. Three Different Quantities
8.2. FTC2: The Integral of the Derivative
8.3. FTC1: The Derivative of the Accumulation
8.4. Anti-Derivatives and Ballistics
8.5. Computing Anti-Derivatives
8.6. Chapter Summary
8.7. Exercises
Chapter 9. Methods of Integration
9.1. Integration by Substitution
9.2. Integration by Parts
9.3. Numerical Integration
9.4. Chapter Summary
9.5. Exercises
Chapter 10. One Variable at a Time
10.1. Partial Derivatives
10.2. Linear Approximations
10.3. Double Integrals and Iterated Integrals
10.4. Chapter Summary
10.5. Exercises
Chapter 11. Taylor Series
11.1. What Does 𝜋=3.14159265⋯ Mean?
11.2. Power Series
11.3. Taylor Polynomials and Taylor Series
11.4. Sines, Cosines, Exponentials, and Logs
11.5. Tests for Convergence
11.6. Intervals of Convergence
11.7. Chapter Summary
11.8. Exercises
Index