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Instructions and examples for finding the domains and level curves of multivariable functions. It includes several functions with their corresponding level curves and surface plots, as well as practice problems for students to solve. The document also discusses the concept of level surfaces and their relationship to multivariable functions.
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2 2 f x y , = 9 โ x โ y.
Figure 1.
Find the domain of the function
1 g x y , sin x y 2
โ = โ + and sketch the domain onto
Figure 2.
Figure 1
Figure 1
Find the domain of (^) ( ) (^) ( ) ( ) (^) ( )
2 1 1 2 2 h x y , ln x 4 cos x y tan x y 4
โ โ = โ + + โ + โ and sketch the
domain onto Figure 3.
What does the domain of the function (^) ( ) 2 2 2
f x y z
x y z
look like?
Practice on finding domains of multivariable functions
Find and sketch the domain of each of the following functions.
a. (^) ( ) ( )
f x y , sin ln x 2 y x
b. (^) ( ) ( )
2 1 2 2 f x y , x 1 cos x y 1
โ = โ โ + โ
c. ( )
2 f x y , = โ x y + x โ y
Figure 3
The temperature (degrees Calvin) at a point inside a vat of grease is given by the formula
z T x y z x y
. The axis system for the vat has been set up so that one corner of the vat
than 2.
A level surface, (^) w = k , for this function is the set of all points in the vat that satisfy
What is the shape of the isothermal surfaces in this vat?
Finding Domains, Level Curves, Surface Plot worksheet
z sin x
2 z = x โ 1
y z
x
z = ln ( y โ x )
( )
1 z cos x y
โ = +
2
2
ln
x z
y
x
y
x
y
Figure 1: Level Curves: z =3, 2,1, 0 Figure 2: Level Curves: z = 2,1, 0, โ 1
Figure 3: Level Curves: z = 2,1, 0, โ 1 Figure 4: Level Curves: z =3, 2,1, 0
x
y
x
y
Figure 5: Level Curves:
z = โ โ Figure 6: Level Curves: z =3, 2,1, 0
x
y
x
y
