978-3-031-24681-4
1
Preface
Contents
978-3-031-24681-4_1
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
978-3-031-24681-4_2
2 The Derivative
[DELETE]
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
978-3-031-24681-4_3
3 Differential Equations
[DELETE]
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
978-3-031-24681-4_4
4 Integration
[DELETE]
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