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Functions of Several Variables
Lecture 21
November 6, 2006
Lecture 21
Functions of Several Variables
Functions of two variables
Lecture 21
Functions of Several Variables
Functions of two variables
A function 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 ).
Lecture 21
Functions of Several Variables
Functions of two variables
A function 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.
Lecture 21
Functions of Several Variables
Functions of two variables
A function 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.
We also write z = f (x , y )
Lecture 21
Functions of Several Variables
Functions of two variables
A function 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.
We also write z = f (x , y )
The variables x and y are independent variables and z is the
dependent variable.
Lecture 21
Functions of Several Variables
Examples
Examples
Lecture 21
Functions of Several Variables
Examples
Examples
Find the domain of the function
f (x , y ) =
Lecture 21
x
2x + 3y
2
+ y2 − 9
Functions of Several Variables
Examples
Examples
Find the domain of the function
f (x , y ) =
x
2x + 3y
2
+ y2 − 9
Find the domain and range of
f (x , y ) =
Lecture 21
p
4 − x2 − y2
Functions of Several Variables
Graphs
Denition
If f is a function of two variables with domain , then the graph
of f is the set of all points (x , y , z ) ∈ R3 such that
z = f (x , y ) and (x , y ) is in D .
Lecture 21
Functions of Several Variables
Examples
Example
Lecture 21
Functions of Several Variables
Examples
Example
A linear
function
is a function
f (x ) = ax + by + c
Lecture 21
Functions of Several Variables
Examples
Example
A linear
function
is a function
f (x ) = ax + by + c
The graph of such a function is a plane.
Lecture 21
Functions of Several Variables
Examples
Example
f (x , y ) = sin(x ) + sin(y )
Lecture 21
Functions of Several Variables
Examples
Example
f (x , y ) = sin(x ) + sin(y )
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
6
4
-6
2
-4
0
-2
0
-2
2
Lecture 21
4
-4
6 -6
Functions of Several Variables
Examples
Example
f (x , y ) = (x 2 + y 2 )e −x 2 −y 2
Lecture 21
Functions of Several Variables
Examples
Example
f (x , y ) = (x 2 + y 2 )e −x 2 −y 2
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-2
-1.5
-1
-0.5
0
0.5
Lecture 21
1
1.5
2 -2
-1.5
-1
-0.5
0
0.5
1
1.5
Functions of Several Variables
2
The Cobb-Douglas production function
Example
P (L, K ) = bLα K 1−α
Lecture 21
Functions of Several Variables
The Cobb-Douglas production function
Example
P (L, K ) = bLα K 1−α
P (L, K ) = 1.01L0.75 K 0.25
350
300
250
200
150
100
50
0
300
250
0
200
50
150
100
150
100
200
Lecture 21
250
50
3000
Functions of Several Variables
Level Curves
Denition
The level curves of a function f of two variables are the
curves with equations f (x , y ) = k , where k is constant.
Lecture 21
Functions of Several Variables
Examples
Example
f (x , y ) = sin(x ) + sin(y )
Lecture 21
Functions of Several Variables
Examples
Example
f (x , y ) = sin(x ) + sin(y )
sin(y)+sin(x)
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
6
4
-6
2
-4
0
-2
0
-2
2
Lecture 21
4
-4
6 -6
Functions of Several Variables
Examples
Example
f (x , y ) = (x 2 + y 2 )e −x 2 −y 2
Function
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-2
-1.5
-1
-0.5
0
0.5
Lecture 21
1
1.5
2 -2
-1.5
-1
-0.5
0
0.5
1
1.5
Functions of Several Variables
2
The Cobb-Douglas production function
Example
P (L, K ) = 1.01L0.75 K 0.25
1.01*x0.75*y0.25
350
300
250
200
150
100
50
0
350
300
250
200
150
100
50
0
300
250
0
200
50
150
100
150
100
200
Lecture 21
250
50
3000
Functions of Several Variables
Limits and Continuity
Denition
Lecture 21
Functions of Several Variables
Limits and Continuity
Denition
We say that a function f (x , y ) has limit L as (x , y ) approaches
a point (a, b) and we write
lim
(x ,y )→(a,b )
f (x , y ) = L
if we can make the values of f (x , y ) as close to L as we like by
taking the point (x , y ) suciently close to the point (a, b), but
not equal to (a, b).
Lecture 21
Functions of Several Variables
Limits and Continuity
Denition
We say that a function f (x , y ) has limit L as (x , y ) approaches
a point (a, b) and we write
lim
(x ,y )→(a,b )
f (x , y ) = L
if we can make the values of f (x , y ) as close to L as we like by
taking the point (x , y ) suciently close to the point (a, b), but
not equal to (a, b).
We write also f (x , y ) → L as (x , y ) → (a, b) and
lim
x →a,y →b
Lecture 21
f (x , y ) = L
Functions of Several Variables
Important
Lecture 21
Functions of Several Variables
Important
If f (x , y ) → L1 as (x , y ) → (a, b) along a path C1 and
f (x , y ) → L2 as (x , y ) → (a, b) along a path C2 , where
L1 6= L2 , then lim(x ,y )→(a,b) f (x , y ) does not exists.
Lecture 21
Functions of Several Variables
Important
If f (x , y ) → L1 as (x , y ) → (a, b) along a path C1 and
f (x , y ) → L2 as (x , y ) → (a, b) along a path C2 , where
L1 6= L2 , then lim(x ,y )→(a,b) f (x , y ) does not exists.
Example: Show that
x2 − y2
(x ,y )→(0,0) x 2 + y 2
lim
does not exist.
Lecture 21
Functions of Several Variables
Continuity
Denition
Lecture 21
Functions of Several Variables
Continuity
Denition
A function f of two variables is called
lim
(x ,y )→(a,b )
Lecture 21
continuous at
f (x , y ) = f (a, b)
Functions of Several Variables
(a, b) if
Continuity
Denition
A function f of two variables is called
lim
(x ,y )→(a,b )
continuous at
(a, b) if
f (x , y ) = f (a, b)
polynomials, rational, trigonometric, exponential,
logarithmic functions are continuous on theirs domain.
Examples:
Lecture 21
Functions of Several Variables
Examples
Examples
Lecture 21
Functions of Several Variables
Examples
Examples
Find the limit
lim
(x ,y )→(0,0)
p
Lecture 21
2x 2 − 4y
2x 2 − 4y + 1 − 1
Functions of Several Variables
Examples
Examples
Find the limit
lim
(x ,y )→(0,0)
p
2x 2 − 4y
2x 2 − 4y + 1 − 1
Find the largest set on which the function
2xy
9 − x2 − y2
is continuous.
Lecture 21
Functions of Several Variables