F
unctions of
T
wo
V
ariables
Up until this point, we have studied functions that take in a single input and produce a
single output. Often times in everyday life, quantities depend on more than one variable.
For example, the body mass index (BMI) that is often used to classify whether or not a
person is overweight is given by the formula
2
Weight (in pounds)
BMI
703
[Height (in inches)]
If we were to let
x
denote a person’s weight (in pounds),
y
denote a person’s height (in
inches), and
z
denote their BMI, then we would have
2
( , )
703
x
zf
x
y
y
. In this case,
we say that
f
(
x
,
y
) is a
function of two inputs (variables)
. The
independent variables
are
x
and
y
and
z
is the
dependent variable
. That is, the output
z
(in this case, BMI)
depends on the inputs
x
and
y
(weight and height).
A natural question is how to convey the information of such a function. When we had
only a single input and a single output, we could make a onedimensional table of values
and we could plot the result in two dimensions. Now, with two inputs, we require a two
dimensional table of values and plot the result in three dimensions.
Below is a table of some BMI values.
Weight (lbs)
110
120
130
140
150
160
170
180
190
5'4"
18
20
22
24
25
27
29
31
32
5'5"
18
20
21
23
25
26
28
30
31
5'6"
17
19
21
22
24
25
27
29
30
Height
5'7"
17
18
20
22
23
25
26
28
29
(in)
5'8"
16
18
19
21
22
24
25
27
28
5'9"
16
17
19
20
22
23
25
26
28
5'10"
15
17
18
20
21
23
24
25
27
5
'
1
1
"
1
51
61
81
92
12
22
32
52
6
6'0"
14
16
17
19
20
21
23
24
25
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 Hohnhold
 Calculus, Pythagorean Theorem, Euclidean space, 3space

Click to edit the document details