Appendix A List of Theorems, Lemmas, Corollaries, Examples, Exercises, and Definitions
2.1 Using Letters Instead of Numbers
2.2 Substitution, or Making Things “Easy on the Eyes”
Example 2.2.1 A Horrible Formula
2.3 An “Easy” Problem From Geometry
Digression DIGRESSION: Making Mistakes
2.4 Our advice, a synopsis
Problem 2.4.2 Some possibly deceptive problems
3.3 The Laziness of Nature
Example 3.3.4 Maximizing the Area of a Rectangle
Digression DIGRESSION: Variables, Constants, and Functions
3.4 Fermat’s Method of Adequality
3.5 Descartes’s Method of Normals
Problem 3.5.5 Extraneous Roots
3.6 Roberval, Conic Sections, and the Dynamic Approach
3.7 Snell’s Law and the Limitations of Adequality:
Theorem 3.7.1 Snell’s Law of Refraction
4.3 The General Differentiation Rules
Digression DIGRESSION: Differential Notation
Problem 4.3.13 Find the Pattern
Problem 4.3.14 The Product Rule for Positive Integers
Example 4.3.24 Brute Force Computation
Example 4.3.27 Making Things “Easier on the Eyes”
Problem 4.3.32 The Power Rule for Positive, Rational Powers
5.1 Slopes and Tangents
Digression DIGRESSION: Evaluation Notation
5.2 Defining the Tangent Line
Definition 5.2.4 The Line Tangent to a Curve at a Point
Definition 5.2.5 The Principle of Local Linearity
Digression DIGRESSION: Dividing by Zero
Problem 5.2.15 Find the Pattern
5.3 The Vomit Comet
Example 5.3.2 Modeling the flight path of an airliner
5.4 Galileo Drops the Ball
5.5 Bringing in Calculus
5.6 The Derivative
Digression DIGRESSION: Function Notation and Prime Notation
Problem 5.6.7 Find the Pattern
5.7 Thinking Dynamically
5.8 Newton’s Method of Fluxions
5.9 Self–intersecting Curves and Parametric Equations
Digression DIGRESSION: Parametric Functions
5.10 Bridges, Chains, Domes, and Telescopes
6.1 A Trigonometric Interlude
6.2 The Differentials of the Sine and Cosine Functions
Digression DIGRESSION: The Centripetal and Tangential Forces
Digression DIGRESSION: Neglecting Gravity
6.3 The Differentials of the Other Trigonometric Functions
6.4 The Inverse Tangent and Cotangent Functions
Digression DIGRESSION: Inverse Function Notation
Digression DIGRESSION: The Tangent Function Has No Inverse
6.5 The Witch of Agnesi and the Inverse Tangent Function
6.6 The Differentials of the Inverse Tangent and Inverse Cotangent Functions
6.7 The Other Inverse Trigonometric Functions
Definition 6.7.4 The Derivative of the Inverse Sine
Definition 6.7.6 The Derivative of the Inverse Cosine
Problem 6.7.14 Find the Pattern
6.8 Curvature
Definition 6.8.2 Curvature
Digression DIGRESSION: Hic Sunt Dracones (Here Be Dragons)
7.1 Root Finding: Two Pre–Calculus Approaches
Example 7.1.1 \(\sqrt{2}\) via Bisection
Problem 7.1.2 Bisection Method
Problem 7.1.3 Bisection Method
Problem 7.1.4 Babylonion (Heron’s) Method
Problem 7.1.5 Babylonion (Heron’s) Method
Problem 7.1.6 Babylonion (Heron’s) Method
Problem 7.1.7 Babylonion (Heron’s) Method
7.2 Newton’s Method
Problem 7.2.12 Find the Pattern
7.3 Euler’s Method
Definition 7.3.1 Initial Value Problem (IVP)
7.4 Higher Derivatives, Lagrange, and Taylor
8.1 Initial Value Problems
8.2 The Natural Exponential
Definition 8.2.2 The Natural Exponential Function
8.3 Exponential Growth and Exponential Notation
Digression DIGRESSION: The Art of Guessing Effectively
Problem 8.3.12 Find the Pattern
Problem 8.3.14 The Exponential Sum Property
8.4 Hyperbolic Trigonometry: The Hanging Chain
Problem 8.4.1 The Shape of a Hanging Chain
Definition 8.4.3 The Hyperbolic Trigonometric Functions
8.5 The Gateway Arch
8.6 Exponential Growth
Example 8.6.1 Population Growth
Problem 8.6.3 Find the Pattern
8.7 Exponential Functions and Compound Interest
8.9 Logarithms, Natural and Unnatural
Digression DIGRESSION: Exponential and Logarithmic Notation
Problem 8.9.7 Find the Pattern
8.10 Applications of Logarithms
8.11 The Derivative of the Natural Logarithm
Digression DIGRESSION: A Curious Fact
8.12 General Logarithms and Exponentials
Problem 8.12.21 The General Power Rule
8.13 How Euler Did It: Harmonic Oscillators, and Complex Numbers
9.1 Introduction: Fermat’s Theorem
Theorem 9.1.1 Fermat’s Theorem
9.2 Preliminaries: Some Simple Optimizations
Example 9.2.1 Constructing A Square on a Line
Example 9.2.4 Constructing A Square on a Circle
Problem 9.2.8 Constructing a Square on an Ellipse
Example 9.2.9 Maximizing the Area of a Rectangle, Redux: Differentials vs. Derivatives
Problem 9.2.14 Find the Pattern
Digression DIGRESSION: Triangles With Fixed Perimeters
Problem 9.2.18 The Largest Triangle with a Fixed Perimeter is Equilateral
9.3 Reflections, Refractions, and Rainbows
9.4 Global vs. Local Extrema
Example 9.4.1 Constructing a Square on a Parabola
Definition 9.4.4 Global Minimum
Digression DIGRESSION: Interval Notation
Definition 9.4.7 Local Minimum
9.5 Optimization, the Abstract Problem
Theorem 9.5.4 The First Derivative Test
Example 9.5.12 Constructing a Square on a Parabola, Redux
Definition 9.5.18 Possible Optimal Transition Points (POTPs)
Theorem 9.5.21 Extreme Value Theorem
9.6 Concavity and the Second Derivative Test
Theorem 9.6.7 Concavity
Theorem 9.6.14 The Second Derivative Test
9.7 Optimization Problems
Example 9.7.1 Maximizing the Area of a Rectangle Redux, Redux
Problem 9.7.8 Variations on a Theme
Problem 9.7.9 Variations on a Theme
Problem 9.7.13 Find the Pattern
Problem 9.7.18 Find the Pattern
Problem 9.7.20 Variations on a Theme
10.1 Graphing with a Formula for \(y(x)\)
10.2 Graphing Without Formulas
10.3 Graphing with a Formula for \(\dfdx{y}{t}\)
11.1 Population Dynamics
11.2 Selected Modeling Problems
12.1 Horizontal Asymptotes: Limits “at Infinity”
Definition 12.1.1 An Intuitive Definition of a Limit at \(\pm\infty\)
Definition 12.1.4 Horizontal Asymptotes
Theorem 12.1.6 The Limit of a Sum “at” Infinity
Theorem 12.1.7 The Limit of a Product “at” Infinity
Theorem 12.1.8 The Limit of a Quotient “at” Infinity
Example 12.1.21 (Continued from Section 10.1 )
Digression DIGRESSION: The Absolute Value Function
12.2 The Squeeze Theorem
Theorem 12.2.2 The Squeeze Theorem at Infinity
12.3 Vertical Asymptotes: “Infinite” Limits
Definition 12.3.1 Vertical Asymptotes
12.4 Indeterminate Forms and L’Hôpital’s Rule
Theorem 12.4.15 L’Hôpital’s Rule, (First Special Case)
Theorem 12.4.23 L’Hôpital’s Rule, (Second Special Case)
Theorem 12.4.25 L’Hôpital’s Rule
Example 12.4.30 More on L’Hôpital’s Rule
Example 12.4.31 The Limit \(\limit{m}{\infty}{ \left( 1+\frac{1}{m}
\right)^m}\)
13.2 Secants and Tangents
Definition 13.2.3 The Derivative
14.1 The Limit Rules (Theorems)
Theorem 14.1.1 The Limit of a Sum is the Sum of the Limits
Theorem 14.1.2 The Limit of a Product is the Product of the Limits
Theorem 14.1.3 The Limit of a Quotient is the Quotient of the Limits
Definition 14.1.6 Near
Theorem 14.1.7 The Limit of a Constant is the Constant
Definition 14.1.15 Continuity at a Point
Theorem 14.1.16 The Limit of a Composition is the Composition of the Limits
Lemma 14.1.17 Differentiability Implies Continuity
Theorem 14.1.19 The Squeeze Theorem (The Finite Case)
14.2 The General Differentiation Theorems, via Limits
Theorem 14.2.1 The Constant Rule for Differentiation
Theorem 14.2.2 The Sum Rule for Differentiation
Theorem 14.2.4 The Constant Multiple Rule for Differentiation
14.3 The Chain Rule
Theorem 14.3.2 The Chain Rule
Digression DIGRESSION: The Origins of the Chain Rule
Digression DIGRESSION: Why Assume That \(\Delta\beta\neq0\) Near Zero?
14.4 The Product Rule
Theorem 14.4.1 The Product Rule for Differentiation
14.5 The Other General Differentiation Rules
Theorem 14.5.1 The Quotient Rule for Differentiation
Problem 14.5.3 The Power Rule for Positive Integer Exponents
Problem 14.5.4 The Power Rule for Rational and Negative Exponents
Theorem 14.5.5 The Power Rule for Rational Exponents
Digression DIGRESSION: Are You a Mathematician?
14.6 Derivatives of the Trigonometric Functions, via Limits
Theorem 14.6.2 Derivative of \(\sin(x)\)
14.7 Inverse Functions
Definition 14.7.1 One-To-One Functions
Definition 14.7.2 Inverse Functions
Digression DIGRESSION: Inverse and Derivative Notation
Theorem 14.7.7 The Derivative of Inverse Functions
Example 14.7.8 The Derivative of the Inverse Sine
15.1 Fermat’s Theorem
Theorem 15.1.1 Fermat’s Theorem
15.2 Rolle’s Lemma and the Mean Value Theorem
Lemma 15.2.2 Rolle’s Lemma
Digression DIGRESSION: Theorems and Lemmas, What’s the Difference?
Theorem 15.2.4 The Mean Value Theorem
15.3 The Proof of the First Derivative Test
Theorem 15.3.1 First Derivative Test
16.1 One Sided Limits
Example 16.1.1 The Absolute Value Function
Definition 16.1.2 The Absolute Value Function
Example 16.1.10 The Heaviside Function
Definition 16.1.12 One–sided Limits
16.2 One Sided Derivatives
Definition 16.2.1 One Sided Derivatives
17.1 A Non-Intuitive Limit
17.2 Getting Around Infinity
Definition 17.2.3 Positive Function With Limit Zero at Infinity
Definition 17.2.7 Zero Limit at Infinity
Definition 17.2.13 A Limit at \(+\infty\)
Definition 17.2.19 A Limit at \(-\infty\)
Definition 17.2.22 Right–Hand, Positive, Infinite Limits
Definition 17.2.28 Right–Hand, Negative Infinite Limits
Definition 17.2.33 Positive, Infinite Limits
Problem 17.2.37 Infinite Limits: An Alternate Definition
17.3 Finite Limits at a Real Number
Definition 17.3.1 The Limit at a Point
Digression DIGRESSION: The Absolute Value, again
Digression DIGRESSION: Why We Prove Theorems
17.4 Limit Laws (Theorems)
Theorem 17.4.1 The Triangle Inequality
Theorem 17.4.2 The Limit at Infinity of a Constant Function is the Constant
Theorem 17.4.3 The Limit at Negative Infinity of a Constant Function is the Constant
Theorem 17.4.5 The Limit at a Point of a Constant Function is the Constant
Theorem 17.4.7 The Limit of a Sum at Infinity
Theorem 17.4.9 The Limit of a Sum at Negative Infinity
Theorem 17.4.11 Limit of a Sum at a Point
Theorem 17.4.12 The Squeeze Theorem at Infinity
Theorem 17.4.14 The Squeeze Theorem at Negative Infinity
Theorem 17.4.16 The Squeeze Theorem, at a Point
Theorem 17.4.18 The Limit of a Composition at Infinity
Theorem 17.4.20 The Limit of a Composition at Negative Infinity
Theorem 17.4.22 The Limit of a Composition at a Point
Theorem 17.4.23 The Limit of a Product at Infinity
Theorem 17.4.30 The Limit of a Product at Negative Infinity
Theorem 17.4.32 Limit of a Product at a Point
Theorem 17.4.35 The Limit of a Quotient is the Quotient of the Limits
Definition 17.4.37 Right-Hand Limit
