-
Math 2415 – Calculus IIISection 14.1 Functions of Several
Variables
• Often, a quantity may depend on more than one input value. For
example, temperature T on the surface of theearth at any given time
t, where t is fixed, depends on latitude and longitude. => T is
a function of x and y.T = f (x,y)
• Can also think of the volume of a circular cylinder as a
function of two variables: V (r,h) = πr2h
• Definition: A function f of two variables is a rule that
assigns to each ordered pair of real numbers (x,y) in aset D a
unique number denoted by f (x,y). The domain D is a subset of the
xy-plane and the range is the set off (x,y).
• Write z = f (x,y) where is the dependent variable and are
independent variables.
• NOTE: If a function f is given by a formula and no domain
specified then the domain is the set of all (x,y)such that the
expression is well-defined.
Ex: Find the domains of the following functions and evaluate f
(3,2).
a). f (x,y) =√
x+ y+1x−1
b). f (x,y) = x ln(y2− x)
• Read example 2 – wind chill depends on temperature and wind
speed.Ex: Find the domain and range of g(x,y) =
√49− x2− y2
-
Math 2415 Section 14.1 Continued
• 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) insuch that
• NOTE: The graph of a function f of one variable is a curve C
(y = f (x)) The graph of a function f of twovariables is a surface
S (z = f (x,y)). We can visualize the graph S of f as lying
directly above or below itsdomain in the xy plane. See Figure
5.
Ex: Sketch the graph of the function z = f (x,y) = 8−4x−2y
Ex: Sketch the graph of z = g(x,y) =√
16− x2− y2
Ex: Find the domain and range and sketch the graph of z = h(x,y)
= 4x2 + y2.
2
-
Math 2415 Section 14.1 Continued
• Level curves The level curves of a function f of two variables
are the curves with equationswhere
• Level curves lie in the . They are the in the of thein the
plane
• Visualize level curves being lifted to the surface at the
indicated height.
• If level curves are close together =>
• If level curves are farther apart =>
•
• Some examples of level curves are topographic maps – level
curves givetemperature functions – level curves are isothermals –
join
Ex: Given a contour map for f , estimate the values of f (1,3)
and f (4,5).
3
-
Math 2415 Section 14.1 Continued
Ex: Sketch some level curves of the function f (x,y) = x+
y+2.
Ex: Sketch some level curves of the function g(x,y) =√
25− x2− y2
Ex: Sketch some level curves of the function h(x,y) = x2
+9y2.
4
-
Math 2415 Section 14.1 Continued
•Functions of three or more variables
Ex: Find the domain of f if f (x,y,z) = ln(z− y)− zcos(xy)
• It is difficult to visualize a function of 3 or more variables
=> requires 4D or higher.
• We can look at level surfaces to visualize functions of 3
variables. These have equation f (x,y,z) = k. If apoint (x,y,z)
moves along a level surface, the value of f (x,y,z) is
Ex: Describe the level surfaces of the function f (x,y,z) = x2 +
y2 + z2.
5