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Mat 272 Calculus III
Updated on 10/4/07
Dr. Firoz
Chapter 14 Partial Derivatives
Section 14.1 Functions of Several Variables
Definition
: A function
f
of two variables is a rule that assigns to each ordered pair of real
numbers (
x
,
y
) in a set D a unique real number denoted by
f
(
x, y
)
.
The set D is the domain
of
f
and its range is the set of values that
f
takes on, that is { ( , )  ( , )
}
f x y
x y
D
∈
.
Definition
: If
f
is a function of two variables with domain D, then the graph of f is the set
of all points (
x
,
y
,
z
) in
3
ℝ
such that
( , )
z
f x y
=
and ( , )
x y
D
∈
.
Definition
: The
Level Curves (contour curve)
of a function
f
of two variables are the
curves with equation
( , )
f x y
k
=
where k is a constant (in the range of
f
)
Functions of three or more variables
: A function of three variables
f
, is a rule that
assigns to each ordered triple ( , , )
x y z
in a domain
3
D
⊂
ℝ
a unique real number denoted
by
( , , )
f x y z
Examples
:
1.
Find the domain of a)
1
( , )
x
y
f x y
x
y
+ +
=

and b)
1
( , )
x
y
f x y
x
y
+ +
=

Solution:
a) The domain of
f
is
{( , ) 
}
D
x y
x
y
=
≠
b) The domain of
f
is
{( , ) 
1 0,
}
D
x y
x
y
x
y
=
+ + ≥
≠
2.
Find the domain and range of
2
2
( , )
36
f x y
x
y
=


Solution: The domain of
f
is
2
2
{( , ) 
36}
D
x y
x
y
=
+
≤
that is all points inside and on
the circle of radius 6.
And the range of the function of
f
is
2
2
{ 
36
,( , )
}
z z
x
y
x y
D
=


∈
3.
Find the domain and range of
f
is
2
2
( , )
4
z
h x y
x
y
=
=
+
Solution: We have seen in chapter 13 that the function h(x, y) is an elliptic paraboloid
with vertex at (0, 0, 0), and opens upward. Horizontal traces are ellipses and vertical
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View Full Document Mat 272 Calculus III
Updated on 10/4/07
Dr. Firoz
traces are parabolas. The domain is all the ordered pairs (
x, y
)
in
2
ℝ
, that is the xy
plane. The range is the set [0, )
∞
of all nonnegative real numbers.
4.
Sketch all the level curves of the function
2
2
( , )
36
f x y
x
y
=


for
0,1,2,3
k
=
5.
Find the level surfaces of the function
2
2
2
( , , )
f x y z
x
y
z
=
+
+
Solution: Choose different numerical values of
( , , )
f x y z
and observe that
2
2
2
k
x
y
z
=
+
+
represents spheres as level surfaces. See example 15, page # 897 at
your text.
Section 14.2 Limits and Continuity
Definition
: Let
( , )
f x y
be a function of two variables whose domain D includes
points arbitrarily close to ( , )
a b
. Then the limit of
( , )
f x y
as ( , )
x y
approaches
( , )
a b
is
L
and is written as s and continuity
( , ) ( , )
lim
( , )
x y
a b
f x y
L
→
=
if for every number
0
ε
>
there is a corresponding number
0
δ
>
such that
2
2
( , )
whenever ( , )
and 0
(
)
(
)
f x y
L
x y
D
x a
y b
 <
∈
<

+ 
<
Definition: Continuous function
The function
( , )
f x y
is continuous on D if
f
is
continuous at every point ( , )
a b
in D.
Examples:
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This note was uploaded on 05/09/2008 for the course MAT 272 taught by Professor Firoz during the Spring '08 term at ASU.
 Spring '08
 Firoz
 Real Numbers, Derivative

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