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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 | |