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Numbers, Functions, and Graphs | |

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Introduction | |

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The Real Line and Coordinate Plane: Pythagoras | |

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Slopes and Equations of Straight Lines | |

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Circles and Parabolas: Descartes and Fermat | |

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The Concept of a Function | |

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Graphs of Functions | |

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Introductory Trigonometry | |

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The Functions Sin O and Cos O | |

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The Derivative of a Function | |

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What is Calculus ? | |

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The Problems of Tangents | |

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How to Calculate the Slope of the Tangent | |

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The Definition of the Derivative | |

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Velocity and Rates of Change: Newton and Leibriz | |

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The Concept of a Limit: Two Trigonometric Limits | |

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Continuous Functions: The Mean Value Theorem and Other Theorem | |

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The Computation of Derivatives | |

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Derivatives of Polynomials | |

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The Product and Quotient Rules | |

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Composite Functions and the Chain Rule | |

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Some Trigonometric Derivatives | |

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Implicit Functions and Fractional Exponents | |

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Derivatives of Higher Order | |

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Applications of Derivatives | |

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Increasing and Decreasing Functions: Maxima and Minima | |

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Concavity and Points of Inflection | |

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Applied Maximum and Minimum Problems | |

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More Maximum-Minimum Problems | |

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Related Rates | |

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Newtons Method for Solving Equations | |

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Applications to Economics: Marginal Analysis | |

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Indefinite Integrals and Differential Equations | |

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Introduction | |

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Differentials and Tangent Line Approximations | |

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Indefinite Integrals: Integration by Substitution | |

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Differential Equations: Separation of Variables | |

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Motion Under Gravity: Escape Velocity and Black Holes | |

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Definite Integrals | |

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Introduction | |

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The Problem of Areas | |

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The Sigma Notation and Certain Special Sums | |

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The Area Under a Curve: Definite Integrals | |

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The Computation of Areas as Limits | |

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The Fundamental Theorem of Calculus | |

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Properties of Definite Integrals | |

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Applications of Integration | |

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Introduction: The Intuitive Meaning of Integration | |

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The Area between Two Curves | |

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Volumes: The Disk Method | |

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Volumes: The Method of Cylindrical Shells | |

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Arc Length | |

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The Area of a Surface of Revolution | |

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Work and Energy | |

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Hydrostatic Force PART II | |

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Exponential and Logarithm Functions | |

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Introduction | |

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Review of Exponents and Logarithms | |

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The Number e and the Function y = e x | |

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The Natural Logarithm Function y = ln x | |

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Applications Population Growth and Radioactive Decay | |

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More Applications | |

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Trigonometric Functions | |

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Review of Trigonometry | |

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The Derivatives of the Sine and Cosine | |

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The Integrals of the Sine and Cosine | |

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The Derivatives of the Other Four Functions | |

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The Inverse Trigonometric Functions | |

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Simple Harmonic Motion | |

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Hyperbolic Functions | |

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Methods of Integration | |

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Introduction | |

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The Method of Substitution | |

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Certain Trigonometric Integrals | |

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Trigonometric Substitutions | |

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Completing the Square | |

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The Method of Partial Fractions | |

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Integration by Parts | |

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A Mixed Bag | |

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Numerical Integration | |

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Further Applications of Integration | |

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The Center of Mass of a Discrete System | |

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Centroids | |

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The Theorems of Pappus | |

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Moment of Inertia | |

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Indeterminate Forms and Improper Integrals | |

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Introduction. The Mean Value Theorem Revisited | |

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The Interminate Form 0/0. L'Hospital's Rule | |

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Other Interminate Forms | |

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Improper Integrals | |

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The Normal Distribution | |

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Infinite Series of Constants | |

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What is an Infinite Series ? | |

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Convergent Sequences | |

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Convergent and Divergent Series | |

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General Properties of Convergent Series | |

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Series on Non-negative Terms: Comparison Test | |