The Principles of Calculus I, II, and III
The Principles of Calculus I, II, and III
I.1. The Algebra of Sets
I.1.1. Setting the Stage
I.1.2. The Language of Set Theory
I.1.3. Unions and Intersections
I.2. Intervals and Linear Inequalities
I.2.1. Unions and Intersections of Intervals
I.2.2. Multiple Linear Inequalities
I.3. Functions and their Basic Properties
I.3.1. Cartesian Products and Relations
I.3.2 Basic Properties of Functions
I.3.3 Comparing Functions
I.4. Functions Given by Simple Formulas
I.4.1. Formulas for Functions
I.4.2. Lines
I.4.3. An Elementary Library
I.5. Manipulating Functions
I.5.1. Restriction to Subdomains
I.5.2. The Algebra of Functions
I.5.3. Decomposing Functions
I.5.4. Computing the Range of a Function
I.6. Piecewise Functions
I.6.1. Decomposing Domains
I.6.2. Compound Piecewise Defined Functions
I.6.3. Inequalities Involving Piecewise Defined Functions
I.7. Functions on Subsets of the Plane
I.7.1. Functions on the Plane
I.7.2. Level Sets
I.7.3. Single Variable Graphs from Multivariate Functions
I.8. Linear Systems and Feasible Sets
I.8.1. Systems of Linear Equations
I.8.2. Systems of Linear Inequalities
I.8.3. Expressing Feasible Sets in Set Builder Notation
II.1. Vectors and Translation
II.1.1. Abstract Translations of the Plane
II.1.2. Vectors and the Method of Coordinates on a Plane
II.1.3. Translating Sets and Graphs
II.2. Scaling Vectors and Subsets of the Plane
II.2.1. Scaling Vectors
II.2.2. Circles and the Polar Form of a Vector
II.2.3. Scaling Subsets of the Plane
II.3. Scaling Quantities
II.3.1. Units
II.3.2. Linear Scaling
II.3.3. Simple Nonlinear Scaling
II.3.4. General Nonlinear Scaling
II.4. Movement along Lines
II.4.1. Absolute and Relative Movement
II.4.2. Parameterized Lines
II.5. Orthogonality and Reflection
II.5.1. Orthogonality of Vectors and Lines
II.5.2. Distance from Points to Lines
II.5.3. Reflecting Sets across Lines
II.6. Inverse Functions
II.6.1. Reflection and Inverse Functions
II.6.2. Restricting Domain to Guarantee Invertibility
II.7. Describing Rotation in Cartesian Coordinates
II.7.1. Abstract Motions on a Circle
II.7.2. Circle Actions and the Method of Coordinates on a Circle
II.7.3. Rotating Points around a Point
II.8. Polar Coordinates and Rotation
II.8.1. Fractions of a Circle and Measurement of Angles
II.8.2. The Sine, Cosine, and Tangent Functions
II.8.3. Angle Addition Formulae for Trigonometric Functions
II.8.4. Parameterizing Rotational Motion
II.8.5. Basic Surveying Problems
II.9. Involution
II.9.1. Reflections and Rotation by Half of a Circle
II.9.2. Inverting the Axes
III.1. Lines and Planes
III.1.1. Introductory Comments on Rigidity
III.1.2. Vectors in three Spatial Dimensions
III.1.3. Rigidity and the Determination of Lines and Planes
III.1.4. Intersections of Lines and Planes
III.2. Polynomial Functions
III.2.1. Quadratic Functions and Optimization
III.2.2. The Factor Theorem
III.2.3. Sketching Polynomials
III.3. Rational Functions
III.3.1. Sketching Reciprocals of Polynomials
III.3.2. Asymptotic Behavior
III.3.3. Sketching Rational Functions
III.4. Solving Piecewise Rational Inequalities
III.4.1. Polynomial Inequalities
III.4.2. Inequalities Involving Rational Functions
III.4.3. Inequalities Involving Piecewise Rational Functions
III.5. Orders of Intersection
III.5.1. Tangential Intersections
III.5.2. Tangency and Polynomials
III.5.3. Tangency and Rational Functions
III.6. Rigidity of Tangential Intersections
III.6.1. Decomposition and Calculation
III.6.2. The Algebraic Derivative
III.6.3. Tangency and Extremal Values
III.6.4. High Degree Intersections
IV.1. Introduction to Symmetry
IV.1.1. Invariance of Sets under a Symmetry Group
IV.1.2. Functions with Involutive Symmetry
IV.2. Translational Symmetry
IV.2.1. Periodicity
IV.2.2. Sketching Trigonometric Functions
IV.2.3. Inverse Trigonometric Functions
IV.2.4. Equations Involving Trigonometric Functions
IV.2.5. Superposition of Waves
IV.3. Symmetric Change
IV.3.1. Exponential Functions and Logarithms
IV.3.2. Models of Symmetric Change
V.3.3. The Natural Exponential and Logarithm
IV.3.4. Exponential Growth and Decay
IV.4. Symmetry of Tangential Intersections
IV.4.1. Translating Intersections
IV.4.2. Scaling Intersections
IV.4.3. An Algebraic Inverse Function Theorem
Link to: V. Finite Approximation
V.1. The Elementary Notion of Area
V.1.1. Motion and the Idea of Area
V.1.2. Polygons and Polygonal Paths
V.1.3. Triangulation and the Area of Polygons
V.2. Sequences
V.2.1. Analytical Properties of the Real Numbers
V.2.2. Sequential Limits and the Limit Laws
V.3. Measurement of a Circle
V.3.1. Fractions of a Circle
V.3.2. Length and Area
V.3.3. Limits Involving the Trigonometric Functions
V.4. Summation
V.4.1. Infinite Series and their Convergence
V.4.2. Some Convergence Tests
V.5. Limits
V.5.1. Definition and Computation of Limits
V.5.2. One Sided Limits
V.5.3. Infinite Limits
V.5.4. Limits and Paths
V.6. Continuous Functions
V.6.1. Continuity
V.6.2. Properties of Continuous Functions
V.6.3. Approximating Continuous Functions
V.7. Analysis of Error
V.7.1. Asymptotic Notation
V.7.2. Sensitivity to Perturbation
V.7.3. Composite Errors
V.8. Approximating Change
V.8.1. Average Rate of Change
V.8.2. Instantaneous Rate of Change
V.8.3. The Exponential Function
V.9. Approximating Area in the Plane
V.9.1. Rectifiable Curves
V.9.2. Areas Bounded by Closed Curves
V.9.3. Approximating Area under a Function
Link to: VI. Local Linear Approximation
VI.1. Approximation by the Tangent Line
VI.1.1. Tangency to Transcendental Functions
VI.1.2. Basic Differentiation Rules
VI.1.3. Differentiation and Decomposition
VI.1.4. Newton’s Method
VI.2. Differentiating Elementary Functions
VI.2.1. Derivatives of Inverse Functions
VI.2.2. Implicitly Defined Functions and Their Derivatives
VI.2.3. Related Rates Problems
VI.3. Rigidity and the Local Linear Approximation
VI.3.1. Extreme Values
VI.3.2. Mean Value Theorem
VI.3.3. Antiderivatives
VI.3.4. L’Hopital’s Rule
VI.4. Shape and Change
VI.4.1. Sketching and Optimization with First Order Information
VI.4.2. The Second Derivative
VI.4.3. Concavity and Curve Sketching
VI.5. Applications of the Mean Value Theorem
VI.5.1. First Order Differential Equations
VI.5.2. Solving Simple Differential Equations
VI.5.3. Uniqueness of Solutions to Certain Differential Equations
VI.6. Curves and Surfaces
VI.6.1. Particle Motion
VI.6.2. Curves on Simple Surfaces
VI.6.3. The Implicit Function Theorem
Link to: VII. Higher Order Approximation
VII.1. Sequences and Series of Functions
VII.1.1. Sequences of Functions
VII.1.2. Series of Functions and their Convergence
VII.1.3. Polynomial Approximation of Continuous Functions
VII.1.4. Power Series and the Radius of Convergence
VII.2. Approximation by Power Series
VII.2.1. Taylor Polynomials and Taylor’s Theorem
VII.2.2. Taylor Series
VII.2.3. Rigidity of Analytic Functions
VII.3. Differentiating Series
VII.3.1. Term-by-term Differentiation
VII.3.2. Application of Series
VII.4. The Geometry of Particle Motion Revisited
VII.4.1. Acceleration and Force
VII.4.2. Parameterizing Curves and Surfaces
VII.4.3. Constrained Motion
VII.4.4. Normal Forces
VIII.1. Decomposition and Integration
VIII.1.1. The Fundamental Theorem of Calculus
VIII.1.2. Areas Bounded by Functions
VIII.1.3. The Length of a Curve
VIII.1.4. Application of Symmetry Principle: Solids of Revolution
VIII.1.5. Application of Symmetry Principle: Surfaces of Revolution
VIII.1.6. Integration of Vector Valued Functions
VIII.2. Methods of Approximation
VIII.2.1. Orders of Approximation
VIII.2.2. Improper Integrals
VIII.3. Transformation of Integrals
VIII.3.1. The Method of Undetermined Coefficients
VIII.3.2. Bijections between Domains of Integration
VIII.3.3. Integration by Substitution
VIII.3.4. Coordinate Changes and Integrals over Paths
VIII.3.5. Integration by Parts
VIII.4. Creation of Functions
VIII.4.1. Using the Integral to Define Functions
VIII.4.2. The Convolution of Functions
VIII.4.3. Differentiating Integrals
VIII.4.4. Trigonometric and Hyperbolic Functions
VIII.5. Frameworks for Calculation
VIII.5.1. The Decomposition of Rational Functions
VIII.5.2. Antiderivatives of Rational Functions
VIII.5.3. Weierstrass Substitution
VIII.6. Applications to Motion and Geometry
VIII.6.1. Dynamical Evolution
VIII.6.2. Work Integrals
VIII.6.3. Areas Bounded by Paths