Calculus: An Intuitive and Physical Approach

CHAPTER 1 WHY CALCULUS?
1. The Historical Motivations for the Calculus
2. The Creators of the Calculus
3. The Nature of the Calculus
CHAPTER 2 THE DERIVATIVE
1. The Concept of Function
2. The Graph or Curve of a Function
3. Average and Instantaneous Speed
4. The Method of Increments
5. A Matter of Notation
6. The Method of Increments Applied to y = ax2
7. The Derived Function
8. The Differentiation of Simple Monomials
9. The Differentiation of Simple Polynomials
10. The Second Derivative
CHAPTER 3 THE ANTIDERIVED FUNCTION OR THE INTEGRAL
1. The Integral
2. Straight Line Motion in One Direction
3. Up and Down Motion
4. Motion Along an Inclined Plane
APPENDIX The Coordinate Geometry of Straight Lines
A1. The Need for Geometrical Interpretation
A2. The Distance Formula
A3. The Slope of a Straight Line
A4. The Inclination of a Line
A5. Slopes of Parallel and Perpendicular Lines
A6. The Angle Between Two Lines
A7. The Equation of a Straight Line
A8. The Distance from a Point to a Line
A9. Equation and Curve
CHAPTER 4 THE GEOMETRICAL SIGNIFICANCE OF THE DERIVATIVE
1. The Derivative as Slope
2. The Concept of Tangent to a Curve
3. Applications of the Derivative as the Slope
4. The Equation of the Parabola
5. Physical Applications of the Derivative as Slope
6. Further Discussion of the Derivative as the Slope
CHAPTER 5 THE DIFFERENTIATION AND INTEGRATION OF POWERS OF x
1. Introduction
2. The Functions xn for Positive Integral n
3. A Calculus Method of Finding Roots
4. Differentiation and Integration of xn for Fractional Values of n
CHAPTER 6 SOME THEOREMS ON DIFFERENTIATION AND ANTIDIFFERENTIATION
1. Introduction
2. Some Remarks about Functions
3. The Differentiation of Sums and Differences of Functions
4. The Differentiation of Products and Quotients of Functions
5. The Integration of Combinations of Functions
6. All Integrals Differ by a Constant
7. The Power Rule for Negative Exponents
8. The Concept of Work and an Application
CHAPTER 7 THE CHAIN RULE
1. Introduction
2. The Chain Rule
3. Application of the Chain Rule to Differentiation
4. The Differentiation of Implicit Functions
5. Equations of the Ellipse and Hyperbola
6. Differentiation of the Equations of Ellipse and Hyperbola
7. Integration Employing the Chain Rule
8. The Problem of Escape Velocity
9. Related Rates
APPENDIX Transformation of Coordinates
A1. Introduction
A2. Rotation of Axes
A3. Translation of Axes
A4. Invariants
CHAPTER 8 MAXIMA AND MINIMA
1. Introduction
2. The Geometrical Approach to Maxima and Minima
3. Analytical Treatment of Maxima and Minima
4. An Alternative Method of Determining Relative Maxima and Minima
5. Some Applications of the Method of Maxima and Minima
6. Some Applications to Economics
7. Curve Tracing
CHAPTER 9 THE DEFINITE INTEGRAL
1. Introduction
2. Area as the Limit of a Sum
3. The Definite Integral
4. The Evaluation of Definite Integrals
5. Areas Below the x-Axis
6. Areas Between Curves
7. Some Additional Properties of the Definite Integral
8. Numerical Methods for Evaluating Definite Integrals
APPENDIX The Sum of the Squares of the First n Integers
CHAPTER 10 THE TRIGONOMETRIC FUNCTIONS
1. Introduction
2. The Sinusoidal Functions
3. Some Preliminaries on Limits
4. Differentiation of the Trigonometric Functions
5. Integration of the Trigonometric Functions
6. Application of the Trigonometric Functions to Periodic Phenomena
CHAPTER 11 THE INVERSE TRIGONOMETRIC FUNCTIONS
1. The Notion of an Inverse Function
2. The Inverse Trigonometric Functions
3. The Differentiation of the Inverse Trigonometric Functions
4. Integration Involving the Inverse Trigonometric Functions
5. Change of Variable in Integration
6. Time of Motion Under Gravitational Attraction
CHAPTER 12 LOGARITHMIC AND EXPONENTIAL FUNCTIONS
1. Introduction
2. A Review of Logarithms
3. The Derived Functions of Logarithmic Functions
4. Exponential Functions and Their Derived Functions
5. Problems of Growth and Decay
6. Motion in One Direction in a Resisting Medium
7. Up and Down Motion in Resisting Media
8. Hyperbolic Functions
9. Logarithmic Differentiation
CHAPTER 13 DIFFERENTIALS AND THE LAW OF THE MEAN
1. Differentials
2. The Mean Value Theorem of the Differential Calculus
3. Indeterminate Forms
CHAPTER 14 FURTHER TECHNIQUES OF INTEGRATION
1. Introduction
2. Integration by Parts
3. Reduction Formulas
4. Integration by Partial Fractions
5. Integration by Substitution and Change of Variable
6. The Use of Tables
CHAPTER 15 SOME GEOMETRIC USES OF THE DEFINITE INTEGRAL
1. Introduction
2. Volumes of Solids: The Cylindrical Element
3. Volumes of Solids: The Shell Game
4. Lengths of Arcs of Curves
5. Curvature
6. Areas of Surfaces of Revolution
7. Remarks on Approximating Figures
CHAPTER 16 SOME PHYSICAL APPLICATIONS OF THE DEFINITE INTEGRAL
1. Introduction
2. The Calculation of Work
3. Applications to Economics
4. The Hanging Chain
5. Gravitational Attraction of Rods
6. Gravitational Attraction of Disks
7. Gravitational Attraction of Spheres
CHAPTER 17 POLAR COORDINATES
1. The Polar Coordinate System
2. The Polar Coordinate Equations of Curves
3. The Polar Coordinate Equations of the Conic Sections
4. The Relation Between Rectangular and Polar Coordinates
5. The Derivative of a Polar Coordinate Function
6. Areas in Polar Coordinates
7. Arc Length in Polar Coordinates
8. Curvature in Polar Coordinates
CHAPTER 18 RECTANGULAR PARAMETRIC EQUATIONS AND CURVILINEAR
MOTION
1. Introduction
2. The Parametric Equations of a Curve
3. Some Additional Examples of Parametric Equations
4. Projectile Motion in a Vacuum
5. Slope, Area, Arc Length, and Curvature Derived from Parametric Equations
6. An Application of Arc Length
7. Velocity and Acceleration in Curvilinear Motion
8. Tangential and Normal Acceleration in Curvilinear Motion
CHAPTER 19 POLAR PARAMETRIC EQUATIONS AND CURVILINEAR MOTION
1. Polar Parametric Equations
2. Velocity and Acceleration in the Polar Parametric Representation
3. Kepler’s Laws
4. Satellites and Projectiles
CHAPTER 20 TAYLOR’S THEOREM AND INFINITE SERIES
1. The Need to Approximate Functions
2. The Approximation of Functions by Polynomials
3. Taylor’s Formula
4. Some Applications of Taylor’s Theorem
5. The Taylor Series
6. Infinite Series of Constant Terms
7. Tests for Convergence and Divergence
8. Absolute and Conditional Convergence
9. The Ratio Test
10. Power Series
11. Return to Taylor’s Series
12. Some Applications of Taylor’s Series
13. Series as Functions
CHAPTER 21 FUNCTIONS OF TWO OR MORE VARIABLES AND THEIR GEOMETRIC REPRESENTATION
1. Functions of Two or More Variables
2. Basic Facts on Three-Dimensional Cartesian Coordinates
3. Equations of Planes
4. Equations of Straight Lines
5. Quadric or Second Degree Surfaces
6. Remarks on Further Work in Solid Analytic Geometry
CHAPTER 22 PARTIAL DIFFERENTIATION
1. Functions of Two or More Variables
2. Partial Differentiation
3. The Geometrical Meaning of the Partial Derivatives
4. The Directional Derivative
5. The Chain Rule
6. Implicit Functions
7. Differentials
8. Maxima and Minima
9. Envelopes
CHAPTER 23 MULTIPLE INTEGRALS
1. Introduction
2. Volume Under a Surface
3. Some Physical Applications of the Double Integral
4. The Double Integral
5. The Double Integral in Cylindrical Coordinates
6. Triple Integrals in Rectangular Coordinates
7. Triple Integrals in Cylindrical Coordinates
8. Triple Integrals in Spherical Coordinates
9. The Moment of Inertia of a Body
CHAPTER 24 AN INTRODUCTION TO DIFFERENTIAL EQUATIONS
1. Introduction
2. First-Order Ordinary Differential Equations
3. Second-Order Linear Homogeneous Differential Equations
4. Second-Order Linear Non-Homogeneous Differential Equations
CHAPTER 25 A RECONSIDERATION OF THE FOUNDATIONS
1. Introduction
2. The Concept of a Function
3. The Concept of the Limit of a Function
4. Some Theorems on Limits of Functions
5. Continuity and Differentiability
6. The Limit of a Sequence
7. Some Theorems on Limits of Sequences
8. The Definite Integral
9. Improper Integrals
10. The Fundamental Theorem of the Calculus
11. The Directions of Future Work