To introduce functions of two variables. To learn what is ... · Graphs as Surfaces - 1 Picturing f(x;y) : Graphs A function of a single variable is usually visualized by means of

Unit #17 : Functions of Two Variables

Goals:

• To introduce functions of two variables.

• To learn what is meant by the graph of a function of two variables.

• To learn to visualize and even sometimes sketch the graph of a function of twovariables.

Functions of Two Variables - 1

Functions of Two Variables

Example: (Temperature in Ontario) On a particular summer day, thedaytime high temperature varied from location to location in Southern Ontarioaccording to the table on the following page.


latitude

in ◦

north

longitude in ◦ west

83 82.5 82 81.5 81 80.5 80 79.5 79 78.5 78 77.5 77 76.5

46.5 21 21 21 24

46 21 21 23 24 24.5 24.5 24.5 24

45.5 22 23 23 24 24.5 25 25 24.5 24 24

45 22 22 22 22 22 23 24 24.5 25 24.5 24

44.5 24 23 23 23 22 22 23 24 24 24

44 25 25 25 25 25 24 24 23 23 24 24

43.5 26 26 25.5 25.5 25.5 25 25 25 25

43 27 26.5 26.5 26 26 26 26 25.5

42.5 27 27 27 26.5 26.5

42 27.5 27


If you wanted to find out the temperature at a point, what information youwould need to specify?

In this example, temperature is a function of two variables. We can write it asT (x, y).


Generally speaking, we define a real-valued function of two variables as follows:

A function of two variables is a rule that takes certain pairs of numbersas inputs and assigns to each a definite output number. The set of all inputpairs is called the domain of the function and the set of resulting outputnumbers is called the range of the function.


• Inputs are pairs of real numbers =⇒• Can be represented as points in the xy-plane =⇒• Domain can be pictured as a region in the xy-plane

• Output is a single real number =⇒• Can be represented as an interval of the real line=⇒• Range can be pictured as an interval of IR

Returning to the Temperature in Ontario example, what is the range of T (x, y)?

latitudein ◦

north

longitude in ◦ west83 82.5 82 81.5 81 80.5 80 79.5 79 78.5 78 77.5 77 76.5

46.5 21 21 21 2446 21 21 23 24 24.5 24.5 24.5 2445.5 22 23 23 24 24.5 25 25 24.5 24 2445 22 22 22 22 22 23 24 24.5 25 24.5 2444.5 24 23 23 23 22 22 23 24 24 2444 25 25 25 25 25 24 24 23 23 24 2443.5 26 26 25.5 25.5 25.5 25 25 25 2543 27 26.5 26.5 26 26 26 26 25.542.5 27 27 27 26.5 26.542 27.5 27


When you travel from Toronto, at approximate latitude 43.0 ◦ and longitude 79.5 ◦,to North Bay, at approximate latitude 46.5 ◦ and longitude 79.5 ◦, the trip is almostexactly due North.

latitudein ◦

north

longitude in ◦ west83 82.5 82 81.5 81 80.5 80 79.5 79 78.5 78 77.5 77 76.5

46.5 21 21 21 2446 21 21 23 24 24.5 24.5 24.5 2445.5 22 23 23 24 24.5 25 25 24.5 24 2445 22 22 22 22 22 23 24 24.5 25 24.5 2444.5 24 23 23 23 22 22 23 24 24 2444 25 25 25 25 25 24 24 23 23 24 2443.5 26 26 25.5 25.5 25.5 25 25 25 2543 27 26.5 26.5 26 26 26 26 25.542.5 27 27 27 26.5 26.542 27.5 27

Construct a graph of these temperatures, indicating the high temperature alongthat route on that July day.


This example shows that when you have a function of two variables, you can use itto construct a function of a single variable by keeping one of the original variablesconstant.

Describe (in words) another way to construct a function of a single variablefrom T (x, y). (There are several possible answers.)

Multivariate Functions - The Ideal Gas Law - 1

Example: (Gas Law) A well-known formula in chemistry relates the pres-sure, temperature, and volume of an ideal gas as PV = nRT , where n and Rare constant for a given amount of the gas. Thus,

P = P (V, T ) =nRT

V

If we keep the temperature T constant, say at 300 K, what is the interpretationof the resulting function P (V, 300)?


Sketch the graph of the function P (V, 300). P (V, T ) =nRT

V


Sometimes in discussing functions of several variables, we can isolate the generaleffect one of the input variables has on the output. When we make these statements,we assume that all the other variables are fixed.

Question: Is P (V, T ) =nRT

Van increasing function of T ? I.e. if we increase

T , holding the other values constant, does P necessarily increase as a result?Remember that n, R, T and V are all positive values.

(a) Yes.

(b) No

.

(c) Depends.


Question: How are P (V, T ) =nRT

Vand V related?

(a) P is an increasing function of V .

(b) P is a decreasing function of V .

(c) Neither consistently/depends on other factors.


Now consider the function z = xy2.Question: How is z related to x?

(a) z is an increasing function of x.

(b) z is a decreasing function of x.



z = xy2

Question: How is z related to y?

(a) z is an increasing function of y.

(b) z is a decreasing function of y.


Multivariate Functions - Finance - 1

Example: (Finance) The monthly payments, P dollars per month, on amortgage, is given by P = f (A, r, t), where

• A dollars were borrowed,

• at an annual interest rate of r%,

• to be repaid over an amortization period of t years,

Is P an increasing function of A? of r? of t?

Multivariate Functions - Finance - 2

Interpret the statement f (250000, 4, 10) ≈ 2527.22. Give units.

Graphs as Surfaces - 1

Picturing f(x, y) : Graphs

A function of a single variable is usually visualized by means of its graphs. Recallthat for a function of one variable, the graph is defined to be the set

{ (x, y) | y = f (x) }

Such a graph can be drawn on a plane, or a space referred to as IR2. We need twodimensions to draw the graph of a function of one variable:

• one dimension for the input variable x, and

• one for the output variable y.


Real-valued functions of two variables also have graphs.

• dimensions for the two input variables, and

• for the output variable.

As a result, graphs for functions of two variables will be drawn in how manydimensions?

The graph of g(x, y) is defined as the set of points

{ (x, y, z) | z = g(x, y) }

We refer to the 3-dimensional space as IR3 since three copies of the real line, IR,are used as axes.


A typical graph for a function of two variables is a surface in IR3 and it looks likethis:


To draw this graph, you begin with the x-, y-, and z-axes. Each point on thegraph has coordinates (x, y, f (x, y)). Points on the graph lie above the xy-planeif z = f (x, y) is positive and below if z = f (x, y) is negative.

Example: Describe and then sketch the surface represented by f (x, y) = 5.


Example: Describe and then sketch the surface represented byf (x, y) = −2.


Example: Describe the graph of

f (x, y) =

{1 if y > 0−1 if y ≤ 0 .


As we have just seen, sketching graphs of 2 variables can be very difficult to do byhand. It is much more practical to use computer software to generate plots of suchfunctions. Since that is not a tool you all have access to, we are going to focus onlooking for patterns in such graphs, so we can predict structure without necessarilybeing able to draw the graphs by hand.

Graphs of Multivariate Functions - Strategies and Examples - 1

Graphs of Functions of 2 VariablesTo describe or identify graphs of 2 variables, z = f (x, y), some techniques thatwill help are the following.

• Determine if z is always positive or negative.

•When z does take on different signs, find the regions where it is consistentlythe same sign.

• Substitute x = 0 or y = 0 into the function. You will get a 1-variable functionrepresenting the cross-section of f (x, y) along one of the axes.

• Substitute in other values of x or y to get a different cross-section.

• Experiment with large values of x and y. Does the function approach the samelimit everywhere, or does it depend on the values of x and y?


Useful patterns in f (x, y)

• If f (x, y) is linear in x and y, the function represents a plane.

• If x2+y2 always appear together, the graph will be circularly symmetric aroundthe origin.

• If only x or y is present in f (x, y), the graph will be a simple 1-variable functionstretched along the missing variable’s axis.


Use the points above to identify the characteristics that tie the following func-tions with their matching graphs. If possible, indicate clearly the x and yaxes.

f (x, y) = x2 + y2


f (x, y) = e−(x2+y2)


Determine what the graph of f (x, y) =√x2 + y2 looks like based on

cross-sections through x = 0 and y = 0.


f (x, y) = 1− x− y


f (x, y) = sinx

Graphs of Multivariate Functions - Example - 1

Example: Suppose we wanted to consider the graph of the function f (x, y) =y2 − x2 in detail. We would ask some of the following questions.

Where is f (x, y) positive? Where is it negative? Where is it zero?

To get a more accurate picture of the graph of f (x, y), we can study it by cuttingit with planes. We do this by selecting a sample value of either x, y or z, andgraphing the resulting 1-variable function.


z = y2 − x2

Fix the value of x to a constant. What will the resulting z vs. y graphs looklike?


z = y2 − x2

Fix the value of y to a constant. What will the resulting z vs. x graphs looklike?


z = y2 − x2

Fix the value of z to a constant. What will the resulting y vs. x graphs looklike?


z = y2 − x2

Identifying Graphs - Practice - 1

Question: Which of the following graphs represents the function

z = 2− x2 − y2

Identifying Graphs - Practice - 2

Question: Which of the following graphs represents the function

z =1

x2 + y2

Graphs of Multivariate Functions - Further Examples - 1

Graphs of Multivariate Functions - Further Examples

Even when functions are more complicated, we can still perform some mentalexperiments that allow us to get some sense of their shape without a full sketch orcomputer-generated surface.Consider the surface defined by z = cos(x) + y2.What happens to the function when you set x = 0?


z = cos(x) + y2

What does this tell you about the intersection between the surface and theplane x = 0?

If you cut the surface at the planes x =π

4and x =

π

2, how is the resulting

function different than at x = 0?


z = cos(x) + y2

Describe the shape of the surface. Indicate the x and y axes on the graphbelow.

Relations in Several Variables - 1

Relations in several variables

The line, plane, and 3D space

• Points on the real line, x, belong to IR

• Points on the plane, (x, y), belong to IR2

• Points in 3D space, (x, y, z), belong to IR3

Some shapes in IR2 and IR3 are best described by relations, rather than func-tions. For example:In IR2, what shape is described by the relationx2 + y2 = 25?

In IR2, what shape is described by the relation (x− 2)2 + (y + 1)2 = 25?


By analogy, what shape in IR3 is described by the relation x2 + y2 + z2 = 25?

Give a formula for the distance from any point (x, y, z) to the origin.


What is the formula for a sphere of radius 5 centered at (2, 4,−1) ?

Give a formula for the distance between two arbitrary points in IR3, (a, b, c)and (x, y, z).

Documents

To introduce functions of two variables. To learn what is ... · Graphs as Surfaces - 1 Picturing f(x;y) : Graphs A function of a single variable is usually visualized by means of