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| Preface | |
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| Basics | |
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| Introductory Remarks | |
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| Number Systems | |
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| Coordinates in One Dimension | |
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| Coordinates in Two Dimensions | |
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| The Slope of a Line in the Plane | |
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| The Equation of a Line | |
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| Loci in the Plane | |
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| Trigonometry | |
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| Sets and Functions | |
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| Examples of Functions of a Real Variable | |
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| Graphs of Functions | |
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| Plotting the Graph of a Function | |
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| Composition of Functions | |
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| The Inverse of a Function | |
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| A Few Words About Logarithms and Exponentials | |
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| Foundations of Calculus | |
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| Limits | |
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| One-Sided Limits | |
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| Properties of Limits | |
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| Continuity | |
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| The Derivative | |
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| Rules for Calculating Derivatives | |
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| The Derivative of an Inverse | |
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| The Derivative as a Rate of Change | |
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| Applications of the Derivative | |
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| Graphing of Functions | |
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| Maximum/Minimum Problems | |
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| Related Rates | |
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| Falling Bodies | |
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| The Integral | |
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| Introduction | |
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| Antiderivatives and Indefinite Integrals | |
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| The Concept of Antiderivative | |
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| The Indefinite Integral | |
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| Area | |
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| Signed Area | |
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| The Area Between Two Curves | |
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| Rules of Integration | |
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| Linear Properties | |
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| Additivity | |
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| Indeterminate Forms | |
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| l'Hopital's Rule | |
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| Introduction | |
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| l'Hopital's Rule | |
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| Other Indeterminate Forms | |
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| Introduction | |
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| Writing a Product as a Quotient | |
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| The Use of the Logarithm | |
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| Putting Terms Over a Common Denominator | |
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| Other Algebraic Manipulations | |
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| Improper Integrals: A First Look | |
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| Introduction | |
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| Integrals with Infinite Integrands | |
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| An Application to Area | |
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| More on Improper Integrals | |
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| Introduction | |
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| The Integral on an Infinite Interval | |
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| Some Applications | |
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| Transcendental Functions | |
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| Introductory Remarks | |
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| Logarithm Basics | |
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| A New Approach to Logarithms | |
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| The Logarithm Function and the Derivative | |
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| Exponential Basics | |
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| Facts About the Exponential Function | |
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| Calculus Properties of the Exponential | |
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| The Number e | |
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| Exponentials with Arbitrary Bases | |
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| Arbitrary Powers | |
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| Logarithms with Arbitrary Bases | |
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| Calculus with Logs and Exponentials to Arbitrary Bases | |
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| Differentiation and Integration of log[subscript a] x and a[superscript x] | |
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| Graphing of Logarithmic and Exponential Functions | |
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| Logarithmic Differentiation | |
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| Exponential Growth and Decay | |
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| A Differential Equation | |
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| Bacterial Growth | |
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| Radioactive Decay | |
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| Compound Interest | |
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| Inverse Trigonometric Functions | |
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| Introductory Remarks | |
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| Inverse Sine and Cosine | |
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| The Inverse Tangent Function | |
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| Integrals in Which Inverse Trigonometric Functions Arise | |
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| Other Inverse Trigonometric Functions | |
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| An Example Involving Inverse Trigonometric Functions | |
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| Methods of Integration | |
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| Integration by Parts | |
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| Partial Fractions | |
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| Introductory Remarks | |
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| Products of Linear Factors | |
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| Quadratic Factors | |
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| Substitution | |
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| Integrals of Trigonometric Expressions | |
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| Applications of the Integral | |
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| Volumes by Slicing | |
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| Introduction | |
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| The Basic Strategy | |
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| Examples | |
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| Volumes of Solids of Revolution | |
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| Introduction | |
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| The Method of Washers | |
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| The Method of Cylindrical Shells | |
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| Different Axes | |
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| Work | |
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| Averages | |
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| Arc Length and Surface Area | |
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| Arc Length | |
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| Surface Area | |
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| Hydrostatic Pressure | |
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| Numerical Methods of Integration | |
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| The Trapezoid Rule | |
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| Simpson's Rule | |
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| Bibliography | |
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| Solutions to Exercises | |
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| Final Exam | |
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| Index | |