MULTIVARIABLE FUNCTIONS
Functions of Two Variables
• Definition: f (x, y) is a function of two variables
if a unique f value is given for each (x, y) ∈ D,
with D = {(x, y) | f (x, y) exists }, domain for f ;
the set of all f values is called the range of f .
Examples:
a) if f (x, y) = x2 + xy − cos(xy), find domain, range,
f (1, 1), f (2, 1);
p
b) if f (x, y) = x y 2 − 1, find domain, range,
f (1, 4), f (2, 0);
MULTIVARIABLE FUNCTIONS CONT.
• Graphing 2-D functions: use z = f (x, y) and plot
(x, y, z) points using standard xyz coordinate system;
octants have +++, -++, ..., - - -, sign combinations;
graph of z = f (x, y) is called a surface;
intersections with xy, xz, and yz planes are traces;
level curves are graphs of k = f (x, y), constant k.
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MULTIVARIABLE FUNCTIONS CONT.
Matlab plots surfaces and level curves (contours):
e.g. ezsurf(’sqrt(x2 + y 2 + 4)’), view([135,30])
sqrt(x2+y2+4)
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ezcontour(’sqrt(x2 + y 2 + 4)’).
sqrt(x2+y2+4)
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MULTIVARIABLE FUNCTIONS
• Some common equations and graphs:
plane: xy−, xz−, and yz− traces are triangles
ax + by + cz = d;
Examples:
a) 2x + 3y + z = 6
( ezsurf(’6-2*x-3*y’,[0 3 0 2]),view([135,30]) )
6−2 x−3 y
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MULTIVARIABLE FUNCTIONS
b) z − 2x − 3y = 6 ( ezsurf(’6+2*x+3*y’) )
6+2 x+3 y
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MULTIVARIABLE FUNCTIONS
paraboloid: xz−, and yz− traces are parabolas,
xy− traces ellipses.
x2 y 2
z = 2 + 2;
a
b
Example: z = 4x2 + y 2
Matlab: ezsurf(’4 ∗ x2 + y 2’,[-3 3 -6 6])
4 x2+y2
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MULTIVARIABLE FUNCTIONS
2
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4 x +y
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Matlab: ezsurfc(’4 ∗ x2 + y 2’,[-3 3 -6 6])
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MULTIVARIABLE FUNCTIONS
ellipsoid: xy−, xz−, and yz− traces are ellipses
x2 y 2 z 2
+ 2 + 2 = 1;
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a
b
c
2
Example: z 2 + x4 + y 2 = 1
ezsurfc(’sqrt(max(1-x^2/4-y^2,0))’,[-2 2 -1 1])
hold on, axis([-2 2 -1 1 -1 1])
ezsurfc(’-sqrt(max(1-x^2/4-y^2,0))’,[-2 2 -1 1])
title(’z^2 + 4*x^2 + y^2 = 1’)
z2 + 4*x2 + y2 = 1
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MULTIVARIABLE FUNCTIONS
hyperbolic paraboloid:
xz−, and yz− traces are parabolas.
x2 y 2
z = 2 − 2;
a
b
Example: z = 4x2 − y 2,
Matlab: ezsurfc(’4 ∗ x2 − y 2’,[-3 3 -6 6])
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4 x −y
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MULTIVARIABLE FUNCTIONS
1/2-sheet hyperboloid: xz−, yz− traces are hyperbolas
x2 y 2 z 2
+ − = ±1;
a2 b2 c2
Example: z 2 = 1 + 4x2 + y 2
ezsurf(’-sqrt(1+x^2/4+y^2)’,[-4 4 -2 2])
hold on, axis([-4 4 -2 2 -3 3])
ezsurf(’sqrt(1+x^2/4+y^2)’,[-4 4 -2 2])
sqrt(1+x2/4+y2)
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