This book is intended for a first-semester course in calculus, which begins by posing a question: how do we model an epidemic mathematically? The authors use this question as a natural motivation for the study of calculus and as a context through which central calculus notions can be understood intuitively. The book’s approach to calculus is contextual and based on the principle that calculus is motivated and elucidated by its relevance to the modeling of various natural phenomena. The authors also approach calculus from a computational perspective, explaining that many natural phenomena require analysis through computer methods. As such, the book also explores some basic programming notions and skills.
Author(s): Eric Stade, Elisabeth Stade
Series: Synthesis Lectures on Mathematics & Statistics
Publisher: Springer
Year: 2023
Language: English
Pages: 283
City: Cham
Tags: Calculus; Mathematical Modeling; Dynamical Systems; Mathematical Biology; Numerical Methods; Life Sciences Calculus; Modeling Epidemics
978-3-031-24681-4
1
Preface
Contents
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1 A Context for Calculus
1.1 Introduction: Calculus and Prediction
1.2 The Spread of Disease: The SIR Model
1.2.1 Initial Setup
1.2.2 Thinking About S', I ', and R'
1.2.3 Exercises
1.3 Prediction Using SIR
1.3.1 An Example
1.3.2 Summary: Euler's Method and SIR
1.3.3 Exercises
1.4 More on the SIR Model
1.4.1 Threshold Value ST of S
1.4.2 Herd Immunity
1.4.3 Reproduction Number r(t)
1.4.4 Exercises
1.5 Using a Program
1.5.1 Computers
1.5.2 Exercises
1.6 Functions
1.6.1 Some Technical Details
1.6.2 Function Notation; Chaining, or Composing, Functions
1.6.3 Functions of Several Variables
1.6.4 Exercises
1.7 Some Families of Functions
1.7.1 Linear Functions
1.7.2 The Circular Functions
1.7.3 Functions Proportional to Their Rates of Change
1.7.4 Exercises
1.8 Summary
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2 The Derivative
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2.1 Rates of Change
2.1.1 Exercises
2.2 Local Linearity (Differentiability)
2.2.1 Continuity
2.2.2 Exercises
2.3 Formulas and Rules for Derivatives
2.3.1 Differentiation Formulas
2.3.2 Differentiation Rules
2.3.3 Exercises
2.4 The Chain Rule
2.4.1 Leibniz Notation for Derivatives
2.4.2 The Chain Rule, First Version
2.4.3 The Chain Rule, Second Version
2.4.4 Exercises
2.5 More Differentiation Rules
2.5.1 The Product Rule
2.5.2 The Quotient Rule
2.5.3 Summary of Differentiation Rules
2.5.4 Exercises
2.6 Optimization, Part I: Extreme Points of a Function
2.6.1 Extremes and Critical Points
2.6.2 Exercises
2.7 Optimization, Part II: Applications
2.7.1 The Problem of the Optimal Soup Can
2.7.2 The Solution
2.7.3 Optimization: Some Mathematical Observations
2.7.4 General Strategies for Applied Optimization
2.7.5 Exercises
2.8 Summary
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3 Differential Equations
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3.1 The (Natural) Exponential Function
3.1.1 The Equation eepicdydt= ky
3.1.2 The Equation eepicdydt= y, and the Natural Exponential Function
3.1.3 The Equation eepicdydt=ky, Again
3.1.4 Basic Properties of the (Natural) Exponential Function
3.1.5 Exercises
3.2 The Natural Logarithm Function
3.2.1 Solving the Equation ea=b for a
3.2.2 Properties of the Natural Logarithm Function
3.2.3 The Derivative of the Logarithm Function
3.2.4 Exponential Growth and Decay, Revisited
3.2.5 Exercises
3.3 Inverse Functions
3.3.1 Exercises
3.4 Modeling Populations
3.4.1 Single-Species Models: Rabbits
3.4.2 Dual-Species Models: Rabbits and Foxes
3.4.3 Exercises
3.5 Modeling Other Phenomena
3.5.1 Circadian Rhythms
3.5.2 Neural Impulses
3.5.3 Exercises
3.6 Summary
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4 Integration
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4.1 Power and Energy
4.1.1 Part A: Power Supplied at a Constant Rate
4.1.2 Part B: Power that Varies in Steps
4.1.3 Part C: Power that Varies Continuously
4.1.4 Exercises
4.2 Accumulation Functions and Definite Integrals
4.2.1 Evaluation of Definite Integrals, Part A: Integrals and Area
4.2.2 Exercises
4.3 More on Integration
4.3.1 Terminology and Notation
4.3.2 Riemann Sums Using Technology
4.3.3 Exercises
4.4 The Geometry of Definite Integrals
4.4.1 The Integral of a (sometimes) Negative Function
4.4.2 Integration Rules
4.4.3 Exercises
4.5 The Fundamental Theorem of Calculus
4.5.1 Statement and Discussion
4.5.2 Sketch of a Proof of the Fundamental Theorem of Calculus
4.5.3 More Examples and Observations
4.5.4 Exercises
4.6 Antiderivatives
4.6.1 Notation
4.6.2 Using Antiderivatives
4.6.3 Finding Antiderivatives
4.6.4 Exercises
4.7 Integration by Substitution
4.7.1 Substitution in Indefinite Integrals
4.7.2 Substitution in Definite Integrals
4.7.3 Exercises
4.8 Separation of Variables
4.8.1 The Separation of Variables Procedure
4.8.2 Diffusion Across a Cell Membrane
4.8.3 Justification
4.8.4 Exercises
4.8.5 Summary
