0
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PRELIMINARIES
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0.1
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Real Numbers, Logic and Estimation
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0.2
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Inequalities and Absolute Values
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0.3
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The Rectangular Coordinate System
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0.4
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Graphs of Equations
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1
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FUNCTIONS
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1.1
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Functions and Their Graphs
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1.2
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Operations on Functions
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1.3
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Exponential and Logarithmic Functions
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1.4
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The Trigonometric Functions & Their Inverses
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1.5
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Chapter Review
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2
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LIMITS
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2.1
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Introduction to Limits
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2.2
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Rigorous Study of Limits
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2.3
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Limit Theorems
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2.4
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Limits Involving Transcendental Functions
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2.5
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Limits at Infinity, Infinite Limits
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2.6
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Continuity of Functions
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2.7
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Chapter Review
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3
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THE DERIVATIVE
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3.1
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Two Problems with One Theme
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3.2
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The Derivative
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3.3
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Rules for Finding Derivatives
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3.4
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Derivatives of Trigonometric Functions
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3.5
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The Chain Rule
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3.6
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Higher-Order Derivatives
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3.7
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Implicit Differentiation
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3.8
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Related Rates
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3.9
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Differentials and Approximations
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3.10
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Chapter Review
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4
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APPLICATIONS OF THE DERIVATIVE
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4.1
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Maxima and Minima
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4.2
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Monotonicity and Concavity
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4.3
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Local Extrema and Extrema on Open Intervals
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4.4
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Graphing Functions Using Calculus
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4.5
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The Mean Value Theorem for Derivatives
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4.6
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Solving Equations Numerically
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4.7
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Antiderivatives
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4.8
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Introduction to Differential Equations
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5
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THE DEFINITE INTEGRAL
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5.1
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Introduction to Area
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5.2
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The Definite Integral
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5.3
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The 1st Fundamental Theorem of Calculus
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5.4
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The 2nd Fundamental Theorem of Calculus
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and the Method of Substitution
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5.5
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The Mean Value Theorem for Integrals & the Use of Symmetry
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5.6
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Numerical Integration
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5.7
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Chapter Review
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6
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APPLICATIONS OF THE INTEGRAL
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6.1
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The Area of a Plane Region
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6.2
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Volumes of Solids: Slabs, Disks, Washers
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6.3
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Volumes of Solids of Revolution: Shells
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6.4
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Length of a Plane Curve
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6.5
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Work and Fluid Pressure
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6.6
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Moments, Center of Mass
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6.8
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Probability and Random Variables
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6.8
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Chapter Review
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7
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TECHNIQUES OF INTEGRATION &
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DIFFERENTIAL EQUATIONS
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7.1
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Basic Integration Rules
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7.2
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Integration by Parts
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7.3
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Some Trigonometric Integrals
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7.4
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Rationalizing Substitutions
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7.5
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The Method of Partial Fractions
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7.6
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Strategies for Integration
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7.7
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Growth and Decay
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7.8
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First-Order Linear Differential Equations
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7.9
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Approximations for Differential Equations
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7.10
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Chapter Review
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8
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INDETERMINATE FORMS &
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IMPROPER INTEGRALS
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8.1
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Indeterminate Forms of Type 0/0
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8.2
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Other Indeterminate Forms
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8.3
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Improper Integrals: Infinite Limits of Integration
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8.4
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Improper Integrals: Infinite Integrands
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8.5
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Chapter Review
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9
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INFINITE SERIES
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9.1
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Infinite Sequences
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9.2
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Infinite Series
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9.3
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Positive Series: The Integral Test
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9.4
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Positive Series: Other Tests
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9.5
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Alternating Series, Absolute Convergence,
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and Conditional Convergence
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9.6
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Power Series
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9.7
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Operations on Power Series
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9.8
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Taylor and Maclaurin Series
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9.9
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The Taylor Approximation to a Function
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9.10
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Chapter Review
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10
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CONICS AND POLAR COORDINATES
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10.1
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The Parabola
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10.2
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Ellipses and Hyperbolas
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10.3
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Translation and Rotation of Axes
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10.4
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Parametric Representation of Curves
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10.5
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The Polar Coordinate System
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10.6
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Graphs of Polar Equations
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10.7
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Calculus in Polar Coordinates
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10.8
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Chapter Review
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11
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GEOMETRY IN SPACE, VECTORS
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11.1
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Cartesian Coordinates in Three-Space
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11.2
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Vectors
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11.3
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The Dot Product
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11.4
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The Cross Product
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11.5
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Vector Valued Functions & Curvilinear Motion
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11.6
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Lines in Three-Space
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11.7
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Curvature and Components of Acceleration
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11.8
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Surfaces in Three Space
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11.9
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Cylindrical and Spherical Coordinates
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11.10
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Chapter Review
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12
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DERIVATIVES OF FUNCTIONS OF
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TWO OR MORE VARIABLES
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12.1
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Functions of Two or More Variables
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12.2
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Partial Derivatives
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12.3
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Limits and Continuity
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12.4
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Differentiability
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12.5
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Directional Derivatives and Gradients
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12.6
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The Chain Rule
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12.7
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Tangent Planes, Approximations
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12.8
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Maxima and Minima
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12.9
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Lagrange Multipliers
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12.10
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Chapter Review
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13
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MULTIPLE INTEGRATION
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13.1
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Double Integrals over Rectangles
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13.2
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Iterated Integrals
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13.3
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Double Integrals over Nonrectangular Regions
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13.4
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Double Integrals in Polar Coordinates
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13.5
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Applications of Double Integrals
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13.6
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Surface Area
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13.7
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Triple Integrals (Cartesian Coordinates)
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13.8
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Triple Integrals (Cyl & Sph Coordinates)
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13.9
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Change of Variables in Multiple Integrals
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13.1
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Chapter Review
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14
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VECTOR CALCULUS
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14.1
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Vector Fields
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14.2
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Line Integrals
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14.3
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Independence of Path
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14.4
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Green's Theorem in the Plane
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14.5
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Surface Integrals
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14.6
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Gauss's Divergence Theorem
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14.7
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Stokes's Theorem
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14.8
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Chapter Review
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APPENDIX
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A.1
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Mathematical Induction
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A.2
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Proofs of Several Theorems
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A.3
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A Backward Look
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