Chapter 7

Functions of Several Variables

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Figure 7.1: Three-Dimensional Coordinate System

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Figure 7.4: Distance andMidpoint Formulas

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Distance Formula in Space

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Midpoint Formula in Space

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Figure 7.6: Standard Equationof a Sphere

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Figure 7.10: General Equationof a Plane

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Figure 7.12: Planes Parallelto Coordinate Planes

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Figure 7.13: Planes Parallelto Coordinate Axes

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Figure 7.14

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Elliptic Cone

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Elliptic Paraboloid

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Hyperbolic Paraboloid

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Ellipsoid

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Hyperboloid of One Sheet

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Hyperboloid of Two Sheets

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Definition of aFunction of Two Variables

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Figure 7.19: Contour Mapsand Level Curves

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Partial Derivatives of aFunction of Two Variables

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Notation for First Partial Derivatives

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Figure 7.23: Graphical Interpretationof Partial Derivatives

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Figure 7.26: Relative Extremaof a Function of Two Variables

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Figure 7.28: First-PartialsTest for Relative Extrema

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Second-Partials Testfor Relative Extrema

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Figure 7.34: Finding aMaximum Value

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Method of Lagrange Multipliers

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Figure 7.37: Measuring theAccuracy of a Mathematical Model

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Figure 7.38: Sum of theSquared Errors

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Definition of the Sumof the Squared Errors

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The Least SquaresRegression Line

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Least SquaresRegression Quadratic

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Determining Area in thePlane by Double Integrals

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Figure 7.48: Volume ofa Solid Region

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Determining Volumeby Double Integrals

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Guidelines for Findingthe Volume of a Solid

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Average Value of aFunction over a Region

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Algebra Summary:Solving Systems of Equations