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8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
1/45
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Functions of Several
Variables chapter8 Functions of Several Variables Three-Dimensional Space and the Graph of
a Function of Two Variables
Partial Derivatives
Maxima and Minima; Constrained
Double Integrals
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
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Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
A Function of Several Variables
A real-valued function of f, of x, y, z, is a
rule for manufacturing a new number,
writtenf(x,y,z,), from a sequence of
independent variables (x,y,z,). Thefunction is called a real-valued function of
two variables if there are two independent
variables, a real-valued function of threevariables if there are three independent
variables, and so on.
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
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Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex.2 3( , ) 3 2f x y x y y= +
( ) ( )
2 3
(0,3) 3 0 (3) 2 3f = +25=
( ) ( )
2 3
(2, 1) 3 2 ( 1) 2 1f = + 15=
A Function of Two Variables
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
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Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Linear Function
A linear function of the variablesx1,x2, ,xn
is a function of the form
( )1 2 0 1 1, ,..., ...n n nf x x x a a x a x= + + +each ai is a constant
Ex. ( ), , 3 2 120 0.05f x y z x y z= + +
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
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Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Interaction Function
If we add to a linear function one or more
terms of the form bxixj (b constant), we get a
second-order interaction function.
Ex. ( ), , 0.08 4 9 2f x y z x xy y z= + +
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
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Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Distance Formula
( )2 2,x y
( )1 1,x y
( ) ( )2 22 1 2 1d x x y y= +
2 1y y
2 1x x
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
7/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Distance Formula
( )7,5
( )3, 2
( ) ( )
( ) ( )
2 2
1 2 1 2
2 27 ( 3) 5 ( 2)
100 49 149
d x x y y
d
d
= +
= + = + =
Ex. Find the distance between (7, 5) and
7
10
( )3, 2 .
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
8/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Equation of a Circle of
Radius rCentered at the Origin
2 2 2
x y r+ =Ex. Find an equation of the circle with
center at the origin and radius of length 4.2 2 16x y+ =
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
9/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Three-Dimensional Space (3-
space)
Point: (x,y,f(x,y))
Ex. Plot (2, 5, 4)
z
y
x
2
4
5
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
10/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Graphs of Functions of Two
Variables
Thegraph of the function of two-variables is
the set of all points (x,y,f(x,y)) in three
dimensional space where we restrict the
values of (x,y) to lie in the domain off.
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
11/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Graphs of Functions of Two
Variables
Ex.2 2( , ) 4 ( )f x y x y= +
yx
z
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
12/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Level Curves
f(x, y)is a function of two variables. Ifc
is some value of the functionf, a trace of
the graph ofz = f(x, y)= c is called a
level curve.
A contour map is created by drawing
several values ofc.
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
13/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. Sketch the level curves for the function3( , )f x y y x= forz= 1, 0, 1, 2.
3 y x c= +
C= 1
C= 0
C= 1
C= 2
Level Curves
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
14/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Analyzing the Graph of a Function
of Two VariablesIf possible, use technology to render the graph of a
given functionz=f(x,y). Analyze as follows:
Step 1 Obtain thex-,y-, andz- intercepts.
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
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Analyzing the Graph of a Function
of Two VariablesStep 2 Slice the surface along thexy-,yz-,
andxz- planes.
Setz= constant
Setx = constant
Sety = constant
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
16/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
z
yx
( , ) 2 4f x y x y= + Ex.
Plot the intercepts: (0,0,4), (2,0,0), and (0,4,0)
The Graph of a Linear Function
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
17/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Partial Derivatives
Thepartial derivative offwith respect tox is
the derivative offwith respect tox, when all
other variables are treated as constants.
Similarly, thepartial derivative offwithrespect toy is the derivative offwith respect
toy, when all other variables are treated as
constants. The partial derivatives are written, , and so on.f x f y
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
18/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. 2( , ) 3 lnf x y x y x y= +
6 lnf
xy yx
= +
2 13f
x xy y
= +
Ex.2
( , ) xy yg x y e +=
( )2
2 1 xy yg
xy ey
+ = +
Partial Derivatives
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
19/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. 4 3( , , ) 2f x y z xy z xy= +
4 3 2f
y z y
x
= +
3 34 2
fxy z x
y
= +
4 23f
xy zz
=
Partial Derivatives
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
20/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Planey = b
Geometric Interpretation of
Partial Derivatives
P
z=f(x,y)
( , )a b
f
x
is the slope of the tangent line
at the pointP(a,b,f(a,b)) along theslice throughy = b.
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
21/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. 2 3 5( , ) lnf x y x y x x y= +
Second-Order Partial Derivatives
23 3
22 20
fy x
x
= +
2 22 16
f fxy
y x x y y
= =
22
2 26
f xx y
y y
= +
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
22/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Notation for Partial Derivatives
meansxffx
meansyf
fy
2
meansxy
ff
x y
2
meansyxf
fy x
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
23/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Maxima and Minima
Letfbe a function defined on a regionR containing
(a, b). f(a, b) is a
relative maximum off if ( , ) ( , )f x y f a b
( , ) ( , )f x y f a brelative minimum off iffor all (x,y) near (a, b).
for all (x,y) near (a, b).
saddle pointoff iff has a relative minimum at (a,b)
along some slice through that point and a relative
maximum along another slice through that point.
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
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Locating Candidates for Relative
Extrema and Saddle Points in theInterior of the Domain off.
First setand solve simultaneously forx andy.
0 and 0f x f x = =
Check that the resulting points (x,y) are
in the interior of the domain off.
Points that satisfy these conditions are called
critical points and are the candidates we seek.
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
25/45Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. Determine the critical points of2 2( , ) 2f x y x x y=
2 2 0 2 0f f
x yx y
= = = =
The only critical point is (1, 0).
Critical Points
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
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Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Second Derivative Test
Let (a, b) is a critical point off.
Compute2
( , ) ( , ) ( , )xx yy xyH f a b f a b f a b =
( , )H a b ( , )xxf a b Interpretation
+
+
+
0
Relative min. at (a, b)
Relative max. at (a, b)
Test is inconclusive
Saddle point at (a, b)
8/4/2019 PP Ch 8 Functions of Several Variables Calculus III
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Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. Determine the relative extrema of the function2 2( , ) 2f x y x x y=
2 2 0 2 0x yf x f y= = = = critical point (1, 0).
( ) ( )2
(1,0) 2 2 0 4 0;H = = > ( )1,0 2 0xxf =
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