The Six Pillars of Calculus

Description

Contents Instructors’ Guide and Background Chapter 1. What is Calculus? The Six Pillars Chapter 2. Predicting the Future: The SIR Model 2.1. A Problem of Market Penetration 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. Epidemics 2.6. Covid-19 and the SIR model 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. The National Debt 4.2. Marginal Cost, Revenue, and Profit 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 Models 6.6. Chapter Summary 6.7. Supplemental Material: A Crash Course in Trigonometry 6.8. 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