pp ch 8 functions of several variables calculus iii

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.

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


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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.


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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


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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= + +


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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= + +


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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


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 .



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+ =



Three-Dimensional Space (3-

space)

Point: (x,y,f(x,y))

Ex. Plot (2, 5, 4)

z

y

x

2

4

5



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.



Graphs of Functions of Two

Variables

Ex.2 2( , ) 4 ( )f x y x y= +

yx

z



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.



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



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.



Analyzing the Graph of a Function

of Two VariablesStep 2 Slice the surface along thexy-,yz-,

andxz- planes.

Setz= constant

Setx = constant

Sety = constant



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



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



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



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



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.



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

= +



Notation for Partial Derivatives

meansxffx

meansyf

fy

2

meansxy

ff

x y

2

meansyxf

fy x



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.



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.



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


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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)


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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 =

pp ch 8 functions of several variables calculus iii

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