A function of two variables is defined similar to a function of one variable. It has a domain (in the plane) and a range. The graph of such a function is a surface in space and we try to sketch some.
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Lesson 5: Functions and surfaces
1. Section 9.6
Functions and Surfaces
Math 21a
February 13, 2008
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2. Outline
Functions of more than one variable
Domain and Range
Graphs
Traces
Quadric Surfaces
Review of Conic Sections
Examples of Quadric Surfaces
3. What is a function?
A function is a box which changes numbers to numbers, or vectors
to vectors, or dogs to cats, or whatever. There are lots of functions
which naturally have multiple inputs and a single output.
4. What is a function?
A function is a box which changes numbers to numbers, or vectors
to vectors, or dogs to cats, or whatever. There are lots of functions
which naturally have multiple inputs and a single output.
The temperature in this room is a function of position and
time.
The production of an economy is a function of capital (money
and goods invested) and labor
I derive utility (happiness) from eating bacon and eggs for
breakfast.
5. Definition
A function f of two variables is a rule that assigns to each
ordered pair of real numbers (x, y ) in a set D a unique real number
denoted by f (x, y ). The set D is the domain of f and its range is
the set of values that f takes on. That is { f (x, y ) | (x, y ) ∈ D }.
6. Example
Example
√
Find the domain and range of f (x, y ) = xy .
7. Example
Example
√
Find the domain and range of f (x, y ) = xy .
Solution
Working from the outside in, we see that xy must be
nonnegative, which means x ≥ 0 and y ≥ 0 or x ≤ 0 and
y ≤ 0. Thus the domain is the union of the coordinate axes,
and the first and third quadrants.
The range of f is the set of all “outputs” of f . Clearly the
range of f is restricted to the set of nonnegative numbers. To
make sure that we can get all nonnegative numbers x, notice
x = f (x 2 , 1).
9. Outline
Functions of more than one variable
Domain and Range
Graphs
Traces
Quadric Surfaces
Review of Conic Sections
Examples of Quadric Surfaces
10. Definition
If f is a function of two variables with domain D, then the graph
of f is the set of all points (x, y , z) in R3 with z = f (x, y ) and
(x, y ) ∈ D.
11. Definition
If f is a function of two variables with domain D, then the graph
of f is the set of all points (x, y , z) in R3 with z = f (x, y ) and
(x, y ) ∈ D.
Functions of one variable are easy to graph on the Cartesian plane.
Functions of two variables need a three-dimensional space. Our
goal is to understand functions of two variables and how to graph
them.
13. Example (Worksheet 2(i))
Sketch the graph of the function f (x, y ) = 3.
Example (Worksheet 2(ii))
Sketch the graph of the function f (x, y ) = 6 − 3x − 2y .
Example (Worksheet 2(iii))
Sketch the graph of the function f (x, y ) = x 2 + y 2 .
14. Traces
A trace of a surface is the intersection of it with a plane. The
result is a curve.
Multiple traces give multiple curves which help sketch the
function
choices for traces:
coordinate planes x = 0, y = 0, z = 0
parallel planes, e.g., z = k for many k