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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