Section 14.1 Functions of Several Variables

Section 14.1 Functions of Several Variables

Section 14.1 Functions of Several Variables
Ruipeng Shen
March 17
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Basic Conception
Definition 1. 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}.
Example 2. For each of the following functions, evaluate f (3, 2) then find and sketch the domain
√
x+y+1
f (x, y) = x ln(y 2 − x)
(a) f (x, y) =
x−1
Solution
(a) This function is well-defined if x+y +1 ≥ 0 and x 6= 1. The value f (3, 2) =
y
-1
x+y+1=0
1
-1
x
x=1
√
Figure 1: The domain of f (x, y) =
x+y+1
x−1
(b) This function is well-defined if y 2 > x. In addition we have f (3, 2) = 0
1
√
6/2.
y
x
x=y2
Figure 2: The domain of f (x, y) = x ln(y 2 − x)
Example 3. Find the domain and range of g(x, y) =
p
16 − x2 − y 2 .
Solution This is well-defined if and only if x2 + y 2 ≤ 16. Thus the domain is a disk of radius
4. Since we know 0 ≤ 16 − x2 − y 2 ≤ 16 as (x, y) is in the domain, the range is the interval [0, 4].
y
4
x2+y2=16
4
Figure 3: The domain of f (x, y) =
2
p
16 − x2 − y 2
x
2
Graph
Definition 4. If f is a function of two variables with domain D, then the graph of f is the set
of the point (x, y, z) in R3 such that z = f (x, y) and (x, y) is in D.
Example 5. Sketch the graph of f (x, y) = 6 − 3x − 2y.
Solution This is a linear equation thus the graph is a plane. It intersects the coordinate axes at
the point (2, 0, 0), (0, 3, 0) and (0, 0, 6). They are called x-intercept, y-intercept and z-intercept,
respectively.
z
(0,0,6)
z=6-3x-2y
(0,3,0)
(2,0,0)
y
x
Figure 4: The graph of f (x, y) = 6 − 3x − 2y
Example 6. Sketch the graph of g(x, y) =
Solution
p
16 − x2 − y 2 .
In order to sketch the graph, we are looking for points (x, y, z) satisfying
p
z = 16 − x2 − y 2 =⇒ x2 + y 2 + z 2 = 16, z ≥ 0.
Thus the graph is the upper half of a sphere with radius 4.
Example 7. Find the domain and range and then sketch the graph of h(x, y) = 2x2 + y 2 .
Solution This is well-defined for all pairs (x, y), therefore the domain is the whole plane R2 .
Since x2 ≥ 0 and y 2 ≥ 0, we know h(x, y) ≥ 0 for all pairs (x, y). The range is the set of all
nonnegative real numbers. The graph is an elliptic paraboloid. We can sketch it by a computer
software as below.
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Figure 5: The graph of g(x, y) =
p
16 − x2 − y 2
Figure 6: The graph of h(x, y) = 2x2 + y 2
4
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Level Curves
In order to visualize more complicated two-variable functions, we can draw a contour map on
which points of constant elevation are joined to form contour lines, or level curves
Definition 8. The level curves of a function f of two variables are the curves with equations
f (x, y) = k, where k is a constant (in the range of f )
Figure 7: The graph of the contour map of a mountain
Example 9. Find approximate temperature of the locations in the figure below.
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Figure 8: The isothermal map
Example 10. Sketch the level curves of the function f (x, y) = 6 − 3x − 2y for the values
k = −6, −3, 0, 3, 6, 9, 12.
Solution
The level set {(x, y)|f (x, y) = k} is a line 3x + 2y = 6 − k.
5
2.5
-10
-7.5
-5
-2.5
0
2.5
5
7.5
10
k=-6
k=-3
k=0
k=3
k=6
-2.5
k=9
k=12
-5
Figure 9: The level curves of a linear function f (x, y) = 6 − 3x − 2y
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Example 11. Sketch the level curves of the function g(x, y) = sin(x + y).
Solution
For any number k ∈ [−1, 1], the level curve can be solved by
sin(x + y) = k ⇐⇒ x + y = arcsin k + 2nπ
or x + y = − arcsin k + (2n + 1)π
Here n is any integer. In the figure below we sketch a few level curves coded by different colors.
y
k=1
k=0.5
k=0
k=-0.5
k=-1
x
Figure 10: The level curves of the function z = sin(x + y)
Figure 11: The graph of the function z = sin(x + y)
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Example 12. Sketch some level curves of the function f (x, y) = 2x2 + y 2 .
Solution
The level curve is given by the equation
2x2 + y 2 = k
=⇒
x2
y2
+
= 1.
k/2
k
This is an ellipse. Thus we can sketch the graph as
z=4
z
z=3
z=2x2+y2
z=2
x2
y2
+ =1
k/2 k
y
x
Figure 12: The level curves and graph for the function z = 2x2 + y 2
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