Numbers, Functions, and Graphs | |
Introduction | |
The Real Line and Coordinate Plane: Pythagoras | |
Slopes and Equations of Straight Lines | |
Circles and Parabolas: Descartes and Fermat | |
The Concept of a Function | |
Graphs of Functions | |
Introductory Trigonometry | |
The Functions Sin O and Cos O | |
The Derivative of a Function | |
What is Calculus ? | |
The Problems of Tangents | |
How to Calculate the Slope of the Tangent | |
The Definition of the Derivative | |
Velocity and Rates of Change: Newton and Leibriz | |
The Concept of a Limit: Two Trigonometric Limits | |
Continuous Functions: The Mean Value Theorem and Other Theorem | |
The Computation of Derivatives | |
Derivatives of Polynomials | |
The Product and Quotient Rules | |
Composite Functions and the Chain Rule | |
Some Trigonometric Derivatives | |
Implicit Functions and Fractional Exponents | |
Derivatives of Higher Order | |
Applications of Derivatives | |
Increasing and Decreasing Functions: Maxima and Minima | |
Concavity and Points of Inflection | |
Applied Maximum and Minimum Problems | |
More Maximum-Minimum Problems | |
Related Rates | |
Newtons Method for Solving Equations | |
Applications to Economics: Marginal Analysis | |
Indefinite Integrals and Differential Equations | |
Introduction | |
Differentials and Tangent Line Approximations | |
Indefinite Integrals: Integration by Substitution | |
Differential Equations: Separation of Variables | |
Motion Under Gravity: Escape Velocity and Black Holes | |
Definite Integrals | |
Introduction | |
The Problem of Areas | |
The Sigma Notation and Certain Special Sums | |
The Area Under a Curve: Definite Integrals | |
The Computation of Areas as Limits | |
The Fundamental Theorem of Calculus | |
Properties of Definite Integrals | |
Applications of Integration | |
Introduction: The Intuitive Meaning of Integration | |
The Area between Two Curves | |
Volumes: The Disk Method | |
Volumes: The Method of Cylindrical Shells | |
Arc Length | |
The Area of a Surface of Revolution | |
Work and Energy | |
Hydrostatic Force PART II | |
Exponential and Logarithm Functions | |
Introduction | |
Review of Exponents and Logarithms | |
The Number e and the Function y = e x | |
The Natural Logarithm Function y = ln x | |
Applications Population Growth and Radioactive Decay | |
More Applications | |
Trigonometric Functions | |
Review of Trigonometry | |
The Derivatives of the Sine and Cosine | |
The Integrals of the Sine and Cosine | |
The Derivatives of the Other Four Functions | |
The Inverse Trigonometric Functions | |
Simple Harmonic Motion | |
Hyperbolic Functions | |
Methods of Integration | |
Introduction | |
The Method of Substitution | |
Certain Trigonometric Integrals | |
Trigonometric Substitutions | |
Completing the Square | |
The Method of Partial Fractions | |
Integration by Parts | |
A Mixed Bag | |
Numerical Integration | |
Further Applications of Integration | |
The Center of Mass of a Discrete System | |
Centroids | |
The Theorems of Pappus | |
Moment of Inertia | |
Indeterminate Forms and Improper Integrals | |
Introduction. The Mean Value Theorem Revisited | |
The Interminate Form 0/0. L'Hospital's Rule | |
Other Interminate Forms | |
Improper Integrals | |
The Normal Distribution | |
Infinite Series of Constants | |
What is an Infinite Series ? | |
Convergent Sequences | |
Convergent and Divergent Series | |
General Properties of Convergent Series | |
Series on Non-negative Terms: Comparison Test | |
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