Download 7.3 Functions of Several Variables Tools to learn

7.3 Functions of Several Variables
Tools to learn for functions of several variables
Evaluating functions
Finding domain and range
Contour Maps (functions of 2 variables)
Using functions of several variables to model real-life
situations (how to ask and answer questions)
Definition of function of 2 variables
A function of two variables is a rule which defines a unique output,
z = f (x, y), for every input pair of real numbers, (x, y), in a
specified set of points D called the domain.
For example: f (x, y) = x3 − 3y 2 is a function which gives an
output for every pair of input values.
f (0, 0) = 03 − 3(0)2 = 0,
f (1, 2) = 13 − 3(2)2 = 1 − 3(4) = −11
f (−4, 3) = (−4)3 − 3(3)2 = 64 − 3(9) = 64 − 27 = 37.
The domain of a function
The domain of a function is very important and can be either
1
Specified by the problem, i.e. specific restrictions are given
f (x, y) = x2 + y 2 such that − 1 < x < 1, −1 < y < 1
2
Assumed to be all points for which the function is valid
p
f (x, y) = 4 − x2 − y 2
More about domains.... what are invalid points?
Recall from functions of one variable
You cannot take the square root of a negative number, so if
√
f (x, y) = 2x + 3y the assumed domain requires that
2x + 3y ≥ 0.
1
You cannot divide by zero, so if f (x, y, z) =
the
x − y 2 + 3z
assumed domain requires that x − y 2 + 3z 6= 0.
You cannot take the logarithm of 0 or a negative number, so
if f (x, y) = ln(x2 − 3y) the assumed domain requires that
x2 − 3y > 0.
The range of a function
The range of the function is the set of all possible values for the
output. So for example if
f (x, y, z) = x2 + y 2 + z 2
then we know that
z = f (x, y, z) ≥ 0.
Another example
f (x, y) =
1
x+y
Find the domain and range.
The domain is all (x, y) such that x + y 6= 0 or x 6= −y and the
range is all z except z = 0.
Contour Maps and Level Curves
Functions of two variables are often referred to as surfaces
z = f (x, y). A contour map is a collection of (x, y) traces all
drawn on the same picture.
z = x2 +
y2
4
Applications
Example: The monthly payments M for an installment loan of P
dollars taken out over t years at an annual interest rate r is given
by
Pr
12
M = f (P, r, t) =
1−
h
1
1+(r/12)
i12t .
Find the monthly payment for a home mortgage of $350, 000 take
out for 30 years at an annual interest rate of 5%.
Plugging in
350000(.05)
12
M = f (350000, .05, 30) =
1−
h
1
1+((.05)/12)
i12(30) = $1869