Lecture 6. Functions of Several Variables: Level
Curves and Limits (§11.1 and 11.2)
Fe. 12, 2012 (Sun)
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Domains and Graphs
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A several variable function is the form of z = f (x1 , x2 , ..., xn ).
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Domains and Graphs
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A several variable function is the form of z = f (x1 , x2 , ..., xn ).
√
√
Example: Consider the function f (x, y ) = x + y .
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Domains and Graphs
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A several variable function is the form of z = f (x1 , x2 , ..., xn ).
√
√
Example: Consider the function f (x, y ) = x + y .
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Domain
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Domains and Graphs
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A several variable function is the form of z = f (x1 , x2 , ..., xn ).
√
√
Example: Consider the function f (x, y ) = x + y .
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Domain
Graph with Matlab:
>> ezsurf(’sqrt (x)+sqrt(y )’, [0 10 0 10])
>> colorbar
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Domains and Graphs
I
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A several variable function is the form of z = f (x1 , x2 , ..., xn ).
√
√
Example: Consider the function f (x, y ) = x + y .
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Domain
Graph with Matlab:
>> ezsurf(’sqrt (x)+sqrt(y )’, [0 10 0 10])
>> colorbar
More Examples:
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Domains and Graphs
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A several variable function is the form of z = f (x1 , x2 , ..., xn ).
√
√
Example: Consider the function f (x, y ) = x + y .
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Domain
Graph with Matlab:
>> ezsurf(’sqrt (x)+sqrt(y )’, [0 10 0 10])
>> colorbar
More Examples:
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f (x, y ) =
√
y − x ln(y + x)
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Domains and Graphs
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A several variable function is the form of z = f (x1 , x2 , ..., xn ).
√
√
Example: Consider the function f (x, y ) = x + y .
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Domain
Graph with Matlab:
>> ezsurf(’sqrt (x)+sqrt(y )’, [0 10 0 10])
>> colorbar
More Examples:
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f (x, y ) =
f (x, y ) =
√
y − x ln(y + x)
x−3y
x+3y
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Domains and Graphs
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A several variable function is the form of z = f (x1 , x2 , ..., xn ).
√
√
Example: Consider the function f (x, y ) = x + y .
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Domain
Graph with Matlab:
>> ezsurf(’sqrt (x)+sqrt(y )’, [0 10 0 10])
>> colorbar
More Examples:
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√
f (x, y ) = y − x ln(y + x)
f (x, y ) = x−3y
x+3y
f (x, y , z) = ln(16 − 4x 2 − 4y 2 − z 2 )
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Domains and Graphs
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A several variable function is the form of z = f (x1 , x2 , ..., xn ).
√
√
Example: Consider the function f (x, y ) = x + y .
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More Examples:
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Domain
Graph with Matlab:
>> ezsurf(’sqrt (x)+sqrt(y )’, [0 10 0 10])
>> colorbar
√
f (x, y ) = y − x ln(y + x)
f (x, y ) = x−3y
x+3y
f (x, y , z) = ln(16 − 4x 2 − 4y 2 − z 2 )
Graphs: Surfaces and Higher-dimensional space surfaces.
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Domains and Graphs
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A several variable function is the form of z = f (x1 , x2 , ..., xn ).
√
√
Example: Consider the function f (x, y ) = x + y .
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More Examples:
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Domain
Graph with Matlab:
>> ezsurf(’sqrt (x)+sqrt(y )’, [0 10 0 10])
>> colorbar
√
f (x, y ) = y − x ln(y + x)
f (x, y ) = x−3y
x+3y
f (x, y , z) = ln(16 − 4x 2 − 4y 2 − z 2 )
Graphs: Surfaces and Higher-dimensional space surfaces.
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The graph of z = f (x, y ) = {(x, y , z) ∈ R3 |z = f (x, y )} =
surface in R3 .
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Domains and Graphs
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A several variable function is the form of z = f (x1 , x2 , ..., xn ).
√
√
Example: Consider the function f (x, y ) = x + y .
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More Examples:
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Domain
Graph with Matlab:
>> ezsurf(’sqrt (x)+sqrt(y )’, [0 10 0 10])
>> colorbar
√
f (x, y ) = y − x ln(y + x)
f (x, y ) = x−3y
x+3y
f (x, y , z) = ln(16 − 4x 2 − 4y 2 − z 2 )
Graphs: Surfaces and Higher-dimensional space surfaces.
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The graph of z = f (x, y ) = {(x, y , z) ∈ R3 |z = f (x, y )} =
surface in R3 .
The graph of z = f (x1 , x2 , ..., xn ) = not easy to visualize.
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Visualization
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Visualization Methods
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Visualization
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Visualization Methods
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Level curve (or Contour curve): f (x, y ) = k in xy -plane, where
k is a fixed constant.
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Visualization
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Visualization Methods
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Level curve (or Contour curve): f (x, y ) = k in xy -plane, where
k is a fixed constant.
Level surface: f (x, y , z) = k.
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Visualization
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Visualization Methods
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Level curve (or Contour curve): f (x, y ) = k in xy -plane, where
k is a fixed constant.
Level surface: f (x, y , z) = k.
Example: Graph f (x, y ) = 100 − x 2 − y 2 and plot the level
curves f (x, y ) = 0, f (x, y ) = 51, and f (x, y ) = 75,
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Visualization
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Visualization Methods
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Level curve (or Contour curve): f (x, y ) = k in xy -plane, where
k is a fixed constant.
Level surface: f (x, y , z) = k.
Example: Graph f (x, y ) = 100 − x 2 − y 2 and plot the level
curves f (x, y ) = 0, f (x, y ) = 51, and f (x, y ) = 75,
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Visualization
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Visualization Methods
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Level curve (or Contour curve): f (x, y ) = k in xy -plane, where
k is a fixed constant.
Level surface: f (x, y , z) = k.
Example: Graph f (x, y ) = 100 − x 2 − y 2 and plot the level
curves f (x, y ) = 0, f (x, y ) = 51, and f (x, y ) = 75,
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Visualization
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Visualization Methods
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Level curve (or Contour curve): f (x, y ) = k in xy -plane, where
k is a fixed constant.
Level surface: f (x, y , z) = k.
Example: Graph f (x, y ) = 100 − x 2 − y 2 and plot the level
curves f (x, y ) = 0, f (x, y ) = 51, and f (x, y ) = 75,
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Visualization
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Visualization Methods
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Level curve (or Contour curve): f (x, y ) = k in xy -plane, where
k is a fixed constant.
Level surface: f (x, y , z) = k.
Example: Graph f (x, y ) = 100 − x 2 − y 2 and plot the level
curves f (x, y ) = 0, f (x, y ) = 51, and f (x, y ) = 75,
(a) Domain
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Visualization
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Visualization Methods
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Level curve (or Contour curve): f (x, y ) = k in xy -plane, where
k is a fixed constant.
Level surface: f (x, y , z) = k.
Example: Graph f (x, y ) = 100 − x 2 − y 2 and plot the level
curves f (x, y ) = 0, f (x, y ) = 51, and f (x, y ) = 75,
(a) Domain
(b) The graph: (Elliptic) paraboloid
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Visualization
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Visualization Methods
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Level curve (or Contour curve): f (x, y ) = k in xy -plane, where
k is a fixed constant.
Level surface: f (x, y , z) = k.
Example: Graph f (x, y ) = 100 − x 2 − y 2 and plot the level
curves f (x, y ) = 0, f (x, y ) = 51, and f (x, y ) = 75,
(a) Domain
(b) The graph: (Elliptic) paraboloid
(c) Level curves with MATLAB:
>> x = −10:0.1:10; y=−10:0.1:10;
>> ezcontourf(’100−xˆ2−yˆ2’)
>> colorbar
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Another Example
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Describe thep
level surfaces of the function
f (x, y , z) = x 2 + y 2 + z 2 .
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Another Example
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Describe thep
level surfaces of the function
f (x, y , z) = x 2 + y 2 + z 2 .
It will look like
Lecture 6. Functions of Several Variables: Level Curves and Lim
Functions of Several variables: Another Example
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Describe thep
level surfaces of the function
f (x, y , z) = x 2 + y 2 + z 2 .
It will look like
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Practice: Dali’s Target
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Lecture 6. Functions of Several Variables: Level Curves and Lim
Finding Limits: Example 1
Given the function
f (x, y ) =
sin(x 2 + y 2 )
,
x2 + y2
find lim(x,y )→(0,0) f (x, y ).
x(↓);y (→)
-1.0
-0.5
-0.2
0
0.2
0.5
1.0
-1.0
0.455
0.759
0.829
0.841
0.829
0.759
0.455
-0.5
0.759
0.959
0.986
0.990
0.986
0.959
0.759
-0.2
0.829
0.986
0.999
1.000
0.999
0.986
0.829
0
0.841
0.990
1.000
1.000
0.990
0.841
0.2
0.829
0.986
0.999
1.000
0.999
0.986
0.829
0.5
0.759
0.959
0.986
0.990
0.986
0.959
0.759
1.0
0.455
0.759
0.829
0.841
0.829
0.759
0.455
Lecture 6. Functions of Several Variables: Level Curves and Lim
Finding Limits: Example 2
Given the function
g (x, y ) =
x2 − y2
,
x2 + y2
find lim(x,y )→(0,0) g (x, y ).
-1.0
-0.5
-0.2
0
0.2
0.5
1.0
-1.0
0.000
-0.600
-0.923
-1.000
-0.923
-0.600
0.000
-0.5
0.600
0.000
-0.724
-1.000
-0.724
0.000
0.600
-0.2
0.923
0.724
0.000
-1.000
0.000
0.724
0.923
0
1.000
1.000
1.000
1.000
1.000
1.000
0.2
0.923
0.724
0.000
-1.000
0.000
0.724
0.923
0.5
0.600
0.000
-0.724
-1.000
-0.724
0.000
0.600
1.0
0.000
-0.600
-0.923
-1.000
-0.923
-0.600
0.000
Lecture 6. Functions of Several Variables: Level Curves and Lim
Finding Limits: Practice Problem
Determine if lim(x,y )→(0,0) f (x, y ) exists.
(a)
f (x, y ) =
xy
x2 + y2
f (x, y ) =
xy 2
x2 + y4
(b)
Lecture 6. Functions of Several Variables: Level Curves and Lim
continuity
A fuction f of two variable is called continuous at (a, b) if
lim
f (x, y ) = f (a, b)
(x,y )→(a,b)
We say that f is continuous on D if f is continuous at every point
(a, b) in D.
Remark: A polynomial function of two variables is a sum of terms
of the form cx m y n , where c is a constant and m and n are
nonnegative integers. A rational function is a ratio of polynomials.
Example:
(a) lim(x,y )→(1,2) (x 2 y 3 − x 3 y 2 + 3x + 2y ) = 11
(b) f (x, y ) =
x 2 −y 2
x 2 +y 2
is continuous everywhere except at (0, 0).
Lecture 6. Functions of Several Variables: Level Curves and Lim