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Chapter 7

Functions of Several Variables

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Figure 7.1: Three-Dimensional Coordinate System


Figure 7.4: Distance andMidpoint Formulas


Distance Formula in Space


Midpoint Formula in Space


Figure 7.6: Standard Equationof a Sphere


Figure 7.10: General Equationof a Plane


Figure 7.12: Planes Parallelto Coordinate Planes


Figure 7.13: Planes Parallelto Coordinate Axes


Figure 7.14


Elliptic Cone


Elliptic Paraboloid


Hyperbolic Paraboloid


Ellipsoid


Hyperboloid of One Sheet


Hyperboloid of Two Sheets


Definition of aFunction of Two Variables


Figure 7.19: Contour Mapsand Level Curves


Partial Derivatives of aFunction of Two Variables


Notation for First Partial Derivatives


Figure 7.23: Graphical Interpretationof Partial Derivatives


Figure 7.26: Relative Extremaof a Function of Two Variables


Figure 7.28: First-PartialsTest for Relative Extrema


Second-Partials Testfor Relative Extrema


Figure 7.34: Finding aMaximum Value


Method of Lagrange Multipliers


Figure 7.37: Measuring theAccuracy of a Mathematical Model


Figure 7.38: Sum of theSquared Errors


Definition of the Sumof the Squared Errors


The Least SquaresRegression Line


Least SquaresRegression Quadratic


Determining Area in thePlane by Double Integrals


Figure 7.48: Volume ofa Solid Region


Determining Volumeby Double Integrals


Guidelines for Findingthe Volume of a Solid


Average Value of aFunction over a Region


Algebra Summary:Solving Systems of Equations