Summary of Chapter 8 in
Applied Calculus
Chapter 16 in
Finite Mathematics & Applied Calculus
Topic: Functions of Several Variables

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Function of Several Variables | Examples: Linear, Interaction, and Distance Functions | Three-Dimensional Space and the Graph of a Function of Two Variables | Partial Derivatives | Geometric Interpretation of Partial Derivatives | Maxima and Minima | Constrained Maxima and Minima and Applications | Double Intergrals

Function of Several Variables


A real-valued function, f, of x, y, z, ... is a rule for manufacturing a new number, written f(x, y, z, ...), from the values of a sequence of independent variables (x, y, z, ...).

The function f is called a real-valued function of two variables if there are two independent variables, a real-valued function of three variables if there are three independent variables, and so on.

As with functions of one variable, functions of several variables can be represented numerically (using a table of values), algebraically (using a formula), and sometimes graphically (using a graph).

Examples

1. f(x, y) = x - y Function of two variables
f(1, 2) = 1 - 2 = -1 Substitute 1 for x and 2 for y
f(2, -1) = 2 - (-1) = 3 Substitute 2 for x and -1 for y
f(y, x) = y - x Substitute y for x and x for y
2. h(x, y, z) = x + y + xz Function of three variables
h(2, 2, -2) = 2 + 2 + 2(-2) = 0 Substitute 2 for x, 2 for y, and -2 for z.

3. Let p(x, y) = x2 + xy - y2. Complete the following table of values of p, and press "Check".

x ®
-101
y-1
¯0
1
 

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Examples: Linear, Interaction, and Distance Functions

Linear Functions
A linear function of the variables x1, x2, ... , xn is a function of the form

    f(x1, x2, ... , xn) = a0 + a1x1 + ... + anxn
where a0, a1, a2, ..., an are constants.

Interaction Functions
If we add to a linear function one or more terms of the form bxixj (b constant), we get a second order interaction function.

Distance Functions
The distance in the plane from the point (x, y) to the point (a, b) is given by

    d(x, y) = [(x - a)2 + (y - b)2]1/2.

(Special case of above formula) The distance in the plane from the point (x, y) to the origin is given by

    d(x, y) = [x2 + y2]1/2.

The distance in 3-space from the point (x, y, z) to the point (a, b, c) is given by

    d(x, y, z) = [(x - a)2 + (y - b)2 + (z - c)2]1/2.
Examples

Linear and Interaction Functions
1. f(x, y) = 2 + 4x -y Linear function of x and y
2. C(x, y, z) = x -3y + 2z Linear function of x, y and z
3. R(x, y, z) = 4 - x -3y + 2z + 0.03xy - 0.02xz
Interaction function of x, y and z
4. P(x, y, z) = x + y2 - 0.03z  is
5. Q(x, y, z) = 1 + xy     is
6. T(x, y, z) = 4       is

Distance Functions
1. The distance between the points (3, -2) and (-1, 1) is

    d = [(-1-3)2 + (1+2)2]1/2 = 251/2 = 5.

2. The distance from (6, 8) to the origin (0, 0) is

 

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Three-Dimensional Space and the Graph of a Function of Two Variables

Points in three-dimensional space have three coordinates as shown in the following figure.

  • The x-coordinate of a point is its distance in front of the yz-plane.
    (If the x-coordinate is negative, the point is behind the yz-plane.)
  • The y-coordinate of a point is its distance to the right of the xz-plane.
    (If the y-coordinate is negative, the point is to the left of the xz-plane.)
  • The z-coordinate of a point is its height above the xy-plane.
    (If the z-coordinate is negative, the point is below the xy-plane.)

Graph of a Function of Two Variables The graph of the function f 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 of f. In other words, the graph is the set of all the points (x, y, z) with z = f(x, y).

Examples

The following figure shows where the point (1, 2, 3) is located in three-dimensional space.

The graph of f(x, y) = x2 - y2 is shown in the followig figure.

Many more can be found in the text book. If you wish to experiment by graphing surfaces on your computer, try the Surface Graphing Utility or, if you have Excel, the Excel Surface Graphing Utility.

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Partial Derivatives
The partial derivative of f with respect to x is the derivative of f with respect to x, treating all other variables as constant.

Similarly, the partial derivative of f with respect to y is the derivative of f with respect to y, treating all other variables as constant, and so on for other variables. The partial derivatives are written as ∂f/∂x, ∂f/∂y, and so on. The symbol "∂" is used (instead of "d") to remind us that there is more than one variable, and that we are holding the other variables fixed.

Interpretation
∂f

∂x
is the rate at which f changes as x changes, for a fixed (constant) y.
∂f

∂y
is the rate at which f changes as y changes, for a fixed (constant) x.

Higher Order Partial Derivatives
If f is a function of x, y, and possibly other variables, then

    2f

    ∂x2
    is defined to be


    ∂x
    ∂f

    ∂x

Similarly,

    2f

    ∂y2
    is defined to be


    ∂y
    ∂f

    ∂y
    2f

    ∂y∂x
    is defined to be


    ∂y
    ∂f

    ∂x
    2f

    ∂x∂y
    is defined to be


    ∂x
    ∂f

    ∂y

The above second order partial derivatives can also be denoted by fxx, fyy, fxy, and fyx respectively.

The last two are called mixed derivatives and will alwyas be equal to each other when all the first order partial derivatives are continuous.

Examples
1.
f(x, y) = x2 - y2
∂f

∂x
= 2x - 0 = 2x
Because y2 is treated as constant
∂f

∂y
= 0 - 2y = -2y
Because x2 is treated as constant
2.
z = x2 + xy
∂z

∂x
= 2x + y


∂x
(xy) =


∂x
(x.constant)
= constant = y
∂z

∂xy
= 0 + x


∂x
(xy) =


∂y
(constant.y)
= constant = x
3.
z = x2 + y3 + xy2
∂z

∂x
=
   
∂z

∂y
=
   
2z

∂x2
=
   
2z

∂y2
=
   
2f

∂y∂x
=
   

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Geometric Interpretation of Partial Derivatives

If f is a function of x and y, taking the partial derivative ∂f/∂x and evaluating it at (a, b) amounts to holding y constant at y = b and finding the rate of change of f at x = a. Thus, the partial derivative is the slope of the tangent line to this curve at the point where x = a and y = b, along the plane y = b. (See the figure below.)

∂z

∂y
(a, b) is the slope of the tangent line at the point P(a, b, f(a, b)) along the slice through y = b

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Maxima and Minima

If f is a function of x and y, then f has a local maximum at (a, b) if f(a, b) f(x, y) for every (x, y) in a small neighborhood of (a, b). A local minimum is defined in a similar way. f has a saddle point at (a, b) if f has a local minimum there along one slice and a local maximum along another slice.

The function illustrated below has has a relative minimum at (0, 0), a relative maximum at (1, 1), and saddle points at (1, 0) and (0, 1).

In the cases we study, all local extrema and saddle points not on the boundary of the domain of f occur as critical points, which are solutions to the equations

fx(x,y) = 0
and
fy(x,y) = 0.
Such points are called critical points of f.

Second-Derivative Test for Functions of Two Variables
Suppose f(x, y) is a function of two variables and that (a, b) is a critical point of f. (That is, fx(a, b) = 0 and fy(a, b) = 0.) Let

H = fxx(a, b)fyy(a, b) -[fxy(a, b)]2.
Then

    f has a local minimum at (a, b) if H > 0 and fxx(a,b) > 0,

    f has a local maximum at (a, b) if H > 0 and fxx(a,b) < 0, and

    f has a saddle point at (a, b) if H < 0.

If H = 0 the test tells us nothing, so we need to look at the graph to see what is going on.

Examples

1. Let f(x, y) = x2 - (y-1) 2. Then fx(x,y) = 2x; fy(x,y) = -2(y-1). To find the critical points, we solve the system

    2x = 0
    -2(y-1) = 0.
The first equation gives x = 0, and the second gives y = 1. Thus, the only critical point is (0, 1). Since the domain of f is the whole Cartesian plane, the point (0, 1) is interior, and hence a candidate for a relative extremum or saddle point.

To check which it is, first compute the second order derivtives:

    fxx(x, y) = 2
    fyy(x, y) = -2
    fxy(x, y) = fyx(x, y) = 0
Then compute
    H= fxx(0, 1)fyy(0, 1) -[fxy(0, 1)]2
    = (2)(-2) -02 =- 4
Since H is negative, we have a saddle point at (0, 1). Here is the graph of f, showing the location of the saddle point.

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Constrained Maxima and Minima and Applications

A constrained optimization problem has the form

Maximize (or minimize) f(x,y,. . . ) subject to constraints.
The constraints are either in the form of equations or restrictions on the domain of f. We can solve these problems by first solving the constraint equations for one variable and substituting into f, and then finding the maximum (or minimum) of the resulting function. In cases when the domain R of the resulting function has a boundary, we must also find the extrema of f when restricted to the boundary.

Lagrange Multipliers

To locate the candidates for local extrema of a function f(x, y, . . .) subject to the constraint g(x, y, ...) = 0, solve the following system of equations for x, y, ... and l.

fx = lgx
fy = lgy
    ...
g = 0
The unknown l is called a Lagrange multiplier. The points (x, y, . . .) that occur in solutions are then the candidates for the local extrema of the function f subject to g = 0.

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Double Integrals

Geometric Definition of the Double Integral
The double integral of f(x, y) over the region R in the xy-plane is defined as

R f(x, y) dx dy
= (volume above R and under the graph of f)
    - (volume below R and above the graph of f).

The following figure illustrates this volume (in the case that the graph of f is above the region R).


Computing Double Integrals
If R is the rectangle a x b and c y d (see figure below) then

R f(x, y) dx dy = d

c
b

a
f(x, y) dx dy
= b

a
d

c
f(x, y) dy dx


If R is the region a x b and c(x) y d(x) (see figure below) then we integrate over R according to the following equation.

R f(x, y) dx dy = b

a
d(x)

c(x)
f(x, y) dy dx


If R is the region c y d and a(y) x b(y) (see figure below) then we integrate over R according to the following equation.

R f(x, y) dx dy = d

b
b(y)

a(y)
f(x, y) dx dy

Examples

If R is the rectangle 1 x 2 and 1 y 3, then

R x dx dy =
3

1
2

1
x dx dy
=
3

1
3

2
dy
= 3


Let R be the region described by 0 x 2, 0 y x (see figure)

R x dx dy =
2

0
x

0
x dy dx
=
2

0
xy x

y=0
dx
=
2

0
x2 dx =
8

3


Let R be the same region as above, but this time described by 0 y 2, y x 2 (see figure)

R x dx dy =
2

0
2

y
x dx dy
=
2

0
x2

2
2

x=y
dy
=
2

0
2 -
y2

2
dy =
8

3

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Last Updated: February, 2001
Copyright © 2001 Stefan Waner / BrooksCole Publishers

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