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Differential Calculus:
From Practice to Theory
Eugene Boman, Robert Rogers
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\(\DeclareMathOperator{\erf}{erf} \DeclareMathOperator{\arccot}{arccot} \DeclareMathOperator{\Arctan}{Arctan} \DeclareMathOperator{\Arcsin}{Arcsin} \DeclareMathOperator{\arcsec}{arcsec} \DeclareMathOperator{\arccsc}{arccsc} \DeclareMathOperator{\sech}{sech} \DeclareMathOperator{\csch}{csch} \def\abs#1{\left|#1\right|} \def\tallstrut{\rule{0mm}{1.1\baselineskip}} \def\approach#1{{\left(\rightarrow{}#1\right)}} \def\CancelToRed#1#2{\textcolor{red}{{\cancelto{#1}{#2}}}} \newcommand{\eps}{\varepsilon} \newcommand{\RR}{\mathbb {R}} \newcommand{\QQ}{\mathbb {Q}} \newcommand{\NN}{\mathbb {N}} \newcommand{\ZZ}{\mathbb {Z}} \def\DD{{\bf{}D}} \def\arccot{{\rm{}arccot}{}} \def\Arctan{{\rm{}Arctan}{}} \def\Arcsin{{\rm{}Arcsin}{}} \def\arcsec{{\rm{}arcsec}{}} \def\arccsc{{\rm{}arccsc}{}} \def\halmos{\mbox{\raggedright\rule{0.1in}{0.1in}}} \def\eval#1#2#3{\left.\strut{}#1\right|_{#2=#3}} \def\geneval#1#2#3{\left.\strut{}#1\right|_{#2#3}} \def\mineval#1#2{\left.\strut{}#1\right|_{#2}} \def\bigeval#1#2#3{\left.\strut{}#1\right|_{#2=#3}} \def\ftceval#1#2#3#4{\eval{#1}{#2}{#3}^{#4}} \def\inverse#1{#1^{-1}} \def\dx#1{\thinspace{\mathrm d}#1} \def\dfdx#1#2{\frac{\dx{#1}}{\dx{#2}}} \def\dfdxat#1#2#3{\eval{\dfdx{#1}{#2}}{#2}{#3}} \def\partialfx#1#2{\frac{\partial{#1}}{\partial{#2}}} \def\partialfxn#1#2#3{\frac{\partial^{#3}{#1}}{\partial{#2}^{#3}}} \def\dfdxn#1#2#3{\frac{\text{d}^{#3}{#1}}{\text{d}{#2}^{#3}}} \def\limit#1#2#3{{\displaystyle\lim_{#1\rightarrow #2}#3}} \def\tlimit#1#2#3{{\displaystyle\lim_{#1\rightarrow #2}}#3} \def\tlimitX#1#2#3#4{{\displaystyle\lim_{\underset{#4}{#1\rightarrow #2}}}#3} \def\rtlimit#1#2#3{{\tlimitX{#1}{#2}{#3}{#1\gt{}#2}{}}} \def\ltlimit#1#2#3{{\tlimitX{#1}{#2}{#3}{#1\lt{}#2}{}}} \def\limitX#1#2#3#4{{\displaystyle\lim_{\underset{#4}{#1\rightarrow #2}}#3}} \def\rlimit#1#2#3{{\limitX{#1}{#2}{#3}{#1\gt{}#2}{}}} \def\llimit#1#2#3{{\limitX{#1}{#2}{#3}{#1\lt{}#2}{}}} \def\limitatinf#1#2{\limit{#1}{\infty}{#2}} \def\limitatninf#1#2{\limit{#1}{-\infty}{#2}} \def\lprime#1{{#1^{\prime}_{\scriptscriptstyle L}}} \def\rprime#1{{#1^{\prime}_{\scriptscriptstyle R}}} \def\lhopeq{\stackrel{L'H}{=}} \newcommand{\ParamEqTwo}[2] { \left\{ \begin{array}{c} {#1}\\ {#2} \end{array} \right\} } \newcommand{\ParamEqThree}[3] { \left\{ \begin{array}{c} {#1}\\ {#2}\\ {#3} \end{array} \right\} } \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Acknowledgements
Epigraphs
To the Instructor: Read This First
I
Differential Calculus: From Practice
1
Calculus is a Rock
2
Introduction: Some Advice on Problem Solving
2.1
Using Letters Instead of Numbers
2.2
Substitution, or Making Things “Easy on the Eyes”
2.3
An “Easy” Problem From Geometry
2.4
Our advice, a synopsis
3
Science Before Calculus
3.1
Apologia
3.2
Some Preliminaries
3.3
The Laziness of Nature
3.4
Fermat’s Method of Adequality
3.5
Descartes’s Method of Normals
3.6
Roberval, Conic Sections, and the Dynamic Approach
3.6.1
Speed, Velocity, and Rates of Change
3.6.2
The Tangent Lines of the Conic Sections
The Tangent to a Parabola
The Tangent to an Ellipse
The Tangent to an Hyperbola
3.7
Snell’s Law and the Limitations of Adequality:
4
Differentials, Differentiation, and the Derivative
4.1
Historical Introduction
4.2
Warning: Be Sure You Understand the Purpose of This Chapter
4.3
The General Differentiation Rules
4.3.1
The Constant Rule
4.3.2
The Sum Rule
4.3.3
The Constant Multiple Rule
4.3.4
The Product Rule
4.3.5
The Power Rule
4.3.6
The Quotient Rule
5
Slopes, Tangents, and Rates of Change
5.1
Slopes and Tangents
5.2
Defining the Tangent Line
5.3
The Vomit Comet
5.4
Galileo Drops the Ball
5.5
Bringing in Calculus
5.6
The Derivative
5.7
Thinking Dynamically
5.8
Newton’s Method of Fluxions
5.9
Self–intersecting Curves and Parametric Equations
5.10
Bridges, Chains, Domes, and Telescopes
5.10.1
Bridges
5.10.2
Chains
5.10.3
Domes
5.10.4
Telescopes
6
Calculus and Trigonometry
6.1
A Trigonometric Interlude
6.1.1
Radian Measure
6.1.2
The Trigonometric Functions
6.1.3
Modeling with Trigonometric Functions
6.1.4
Phase Shifts
6.1.5
Polar Coordinates
6.2
The Differentials of the Sine and Cosine Functions
6.2.1
Circular Motion
6.2.2
Polar Coordinates and Calculus
6.2.3
The Brachistochrone Problem
6.2.4
Spin Casting, Redux
6.2.5
Simple Harmonic Oscillation
6.3
The Differentials of the Other Trigonometric Functions
6.4
The Inverse Tangent and Cotangent Functions
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
6.7.1
The Inverse Sine Function:
\(\inverse\sin(x)\)
6.7.2
The Inverse Secant and Inverse Cosecant
6.8
Curvature
7
Approximation Methods
7.1
Root Finding: Two Pre–Calculus Approaches
7.1.1
The Bisection Method
7.1.2
The Babylonian Method for Square Roots
7.2
Newton’s Method
7.2.1
The Idea Behind Newton’s Method
7.2.2
Spectacular Failure
7.2.3
Subtle Failure
7.3
Euler’s Method
7.4
Higher Derivatives, Lagrange, and Taylor
8
Exponentials and Logarithms
8.1
Initial Value Problems
8.2
The Natural Exponential
8.3
Exponential Growth and Exponential Notation
8.4
Hyperbolic Trigonometry: The Hanging Chain
8.5
The Gateway Arch
8.6
Exponential Growth
8.7
Exponential Functions and Compound Interest
8.8
John Napier Logs In: A Short Introduction to Logarithms
8.9
Logarithms, Natural and Unnatural
8.10
Applications of Logarithms
8.10.1
Radioactive Dating
8.10.2
Chillin’ with Newton: The Law of Cooling
8.11
The Derivative of the Natural Logarithm
8.12
General Logarithms and Exponentials
8.12.1
General Exponentials
8.12.2
General Logarithms
8.12.3
Common and Napierian Logarithms
8.12.4
Logarithmic Differentiation
8.13
How Euler Did It: Harmonic Oscillators, and Complex Numbers
9
Optimization: Going to Extremes
9.1
Introduction: Fermat’s Theorem
9.2
Preliminaries: Some Simple Optimizations
9.3
Reflections, Refractions, and Rainbows
9.3.1
Reflection
9.3.2
Refraction: Snell’s Law
9.3.3
Rainbows
9.3.4
The Colors of the Rainbow
9.3.5
Double Rainbows
9.4
Global vs. Local Extrema
9.5
Optimization, the Abstract Problem
9.5.1
An Illustrative Example from
\(1750\)
9.5.2
Identifying and Distinguishing Maxima and Minima
9.5.3
Transition Points, and Possible Transition Points on Open Intervals
9.5.4
Undefined Derivatives
9.5.5
Transition Points on a Closed Interval; the Problem of Endpoints
9.6
Concavity and the Second Derivative Test
9.6.1
The Second Derivative Test
9.7
Optimization Problems
9.7.1
Optimization Strategy
9.7.2
There’s More Than One Way to Optimize
First Solution:
Second Solution
9.7.3
Selected Optimization Problems
10
Graphing with Calculus
10.1
Graphing with a Formula for
\(y(x)\)
10.2
Graphing Without Formulas
10.2.1
Graphing with Incomplete Information About the Derivative
10.2.2
Graphing
\(y(x)\)
from the Graph of
\(\dfdx{y}{x} \)
10.3
Graphing with a Formula for
\(\dfdx{y}{t}\)
10.3.1
When
\(\dfdx{y}{t}\)
Depends on
\(t\)
alone
10.3.2
When
\(\dfdx{y}{t}\)
depends on
\(y\)
alone
10.3.3
When
\(\dfdx{y}{t}\)
Depends on
\(y\)
and
\(t\)
11
Modeling with Calculus
11.1
Population Dynamics
11.1.1
Modeling a Trout Farm
11.1.2
The Competing Species Model
11.2
Selected Modeling Problems
11.2.1
Epidemics, The SIR Model
11.2.2
The Tractrix
11.2.3
The Pursuit Problem
12
Limits and L’Hôpital’s Rule
12.1
Horizontal Asymptotes: Limits “at Infinity”
12.1.1
Limit Notation
12.2
The Squeeze Theorem
12.3
Vertical Asymptotes: “Infinite” Limits
12.4
Indeterminate Forms and L’Hôpital’s Rule
12.4.1
L’Hôpital’s Rule
12.4.2
L’Hôpital’s Rule and Horizontal Asymptotes
II
Differential Calculus: To Theory
13
What’s Wrong with Differentials?
13.1
Calculus and Bishop Berkeley
13.2
Secants and Tangents
14
The Differentiation Rules via Limits
14.1
The Limit Rules (Theorems)
14.1.1
The Limit of a Composition and Continuity at a Point
14.2
The General Differentiation Theorems, via Limits
14.3
The Chain Rule
14.4
The Product Rule
14.5
The Other General Differentiation Rules
14.6
Derivatives of the Trigonometric Functions, via Limits
14.7
Inverse Functions
15
The First Derivative Test, Redux
15.1
Fermat’s Theorem
15.2
Rolle’s Lemma and the Mean Value Theorem
15.3
The Proof of the First Derivative Test
16
When the Derivative Doesn’t Exist
16.1
One Sided Limits
16.2
One Sided Derivatives
17
Formal Limits
17.1
A Non-Intuitive Limit
17.2
Getting Around Infinity
17.2.1
Horizontal Asymptotes, Redux
17.2.2
Convincing Berkeley
17.2.3
Refining the Definition of a Limit
17.2.4
Vertical Asymptotes, Redux
17.3
Finite Limits at a Real Number
17.4
Limit Laws (Theorems)
17.4.1
The Limit of a Sum
17.4.2
The Squeeze Theorem
17.4.3
The Limit of a Composition
17.4.4
The Limit of a Product
17.4.5
The Function
\(f(x)=\frac1x\)
is Continuous Wherever It Is Defined
Appendices
A
List of Theorems, Lemmas, Corollaries, Examples, Exercises, and Definitions
Index
Colophon
Colophon
Colophon
This book was authored in PreTeXt.
