Multivariable Functions: Domain, Range, and Level Curves, Slides of Calculus for Engineers

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Lesson 3

DOMAIN AND RANGE OF

MULTI-VARIABLES FUNCTIONS

SECTION OBJECTIVES:

At the end of the lesson, the student must be able to:

• Evaluate a function of two or more variables.
• Determine the domain and range.

Definitions

Natural Domain of the Function

As with functions of one variable, the independent

variables of a function of two or more variables may be

restricted to lie in some set D, called the domain of f.

Sometimes the domain will be determined by physical

restrictions or other restrictions stated explicitly, so this

domain, called the natural domain of the function ,

consists of all points for which the formulas yields a

real value for the dependent variable.

Contour maps are useful for studying functions of two variables. If the surface 𝑧𝑧 = 𝑓𝑓(𝑥𝑥, 𝑦𝑦) is cut by a horizontal plane 𝑧𝑧 = 𝑘𝑘, then at all points on the intersection, 𝑓𝑓 (𝑥𝑥, 𝑦𝑦) = 𝑘𝑘. The projection of this intersection onto the xy-plane is called the level curve of height k or the level curve with constant k. A set of level curves for 𝑧𝑧 = 𝑓𝑓(𝑥𝑥, 𝑦𝑦) is called a contour plot or contour map of f.

• Example
• Example
• Example
• PROBLEM: Identify the level curves of 𝑓𝑓 (𝑥𝑥, 𝑦𝑦) = 𝑥𝑥^2 + 𝑦𝑦^2 ; 𝑘𝑘 = 1, 2, 3, 4,