The Six Pillars of Calculus: Biology Edition is a conceptual and practical introduction to differential and integral calculus for use in a one- or two-semester course. By boiling calculus down to six common-sense ideas, the text invites students to make calculus an integral part of how they view the world. Each pillar is introduced by tackling and solving a challenging, realistic problem. This engaging process of discovery encourages students to wrestle with the material and understand the reasoning behind the techniques they are learning--to focus on when and why to use the tools of calculus, not just on how to apply formulas.
Modeling and differential equations are front and center. Solutions begin with numerical approximations; derivatives and integrals emerge naturally as refinements of those approximations. Students use and modify computer programs to reinforce their understanding of each algorithm.
The Biology Edition of the Six Pillars series has been extensively field-tested at the University of Texas. It features hundreds of examples and problems specifically designed for students in the life sciences. The core ideas are introduced by modeling the spread of disease, tracking changes in the amount of $\mathrm{CO}_{2}$ in the atmosphere, and optimizing blood flow in the body. Along the way, students learn about optimal drug delivery, population dynamics, chemical equilibria, and probability.
Author(s): Lorenzo Sadun
Edition: 1
Publisher: American Mathematical Society
Year: 2023
Language: English
Commentary: 2020 Mathematics Subject Classification. Primary 00-01, 26A06, 92-10, 92D30, 92D25, 00A71, 26-01.
Pages: 383
City: Providence, Rhode Island
Tags: Biomathematics; Real Functions; Functions; Calculus; Biology; Genetics; Population Dynamics; Epidemiology
Cover
Title page
Copyright
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 ?=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
Back Cover
