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Table of Contents
Linear Coordinate Systems. Absolute Value. Inequalities • Rectangular Coordinate Systems • Lines • Circles • Equations and their Graphs • Functions • Limits • Continuity • The Derivative • Rules for Differentiating Functions • Implicit Differentiation • Tangent and Normal Lines • Law of the Mean. Increasing and Decreasing Functions • Maximum and Minimum Values • Curve Sketching. Concavity. Symmetry • Review of Trigonometry • Differentiation of Trigonometric Functions • Inverse Trigonometric Functions • Rectilinear and Circular Motion • Related Rates • Differentials. Newton’s Method • Antiderivatives • The Definite Integral. Area under a Curve • The Fundamental Theorem of Calculus • The Natural Logarithm • Exponential and Logarithmic Functions • L’Hopital’s Rule • Exponential Growth and Decay • Applications of Integration I: Area and Arc Length • Applications of Integration II: Volume • Techniques of Integration I: Integration by Parts • Techniques of Integration II: Trigonometric Integrands and Trigonometric Substitutions • Techniques of Integration III: Integration by Partial Fractions • Miscellaneous Substitutions • Improper Integrals • Applications of Integration II: Area of a Surface of Revolution • Parametric Representation of Curves • Curvature • Plane Vectors • Curvilinear Motion • Polar Coordinates • Infinite Sequences • Infinite Series • Series with Positive Terms. The Integral Test. Comparison Tests • Alternating Series. Absolute and Conditional Convergence. The Ratio Test • Power Series • Taylor and Maclaurin Series. Taylor’s Formula with Remainder • Partial Derivatives • Total Differential. Differentiability. Chain Rules • Space Vectors • Surface and Curves in Space • Directional Derivatives. Maximum and Minimum Values • Vector Differentiation and Integration • Double and Iterated Integrals • Centroids and Moments of Inertia of Plane Areas • Double Integration Applied to Volume under a Surface and the Area of a Curved Surface • Triple Integrals • Masses of Variable Density • Differential Equations of First and Second Order