12.1 Functions Notes

12.1 Functions Notes - 1 2.1 
F u n c t io n s 
o f
S e v e r a l 
V a r ia b l e s 
 O n e 
v a r ia b l e 


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Unformatted text preview: 1 2 .1 
F u n c t io n s 
o f
S e v e r a l 
V a r ia b l e s 
 O n e 
v a r ia b l e 
 The
statement
y
=
f(x)
or
“y
is
a
function
of
x”
means
that
y
 depends
on
x.


 x
=
independent
variable
 y=
dependent
variable
 Domain
=
{x
|
f(x)
is
defined}
 Range
=
{f(x)
|
x
is
a
member
of
the
domain}
 
 
 
 
 
 
 
 
 
 
 
 
 
 T w o 
v a r ia b l e s 
 z
=
f(x,y)
or
“
z
is
a
function
of
x
and
y”
means
that
z
depends
 on
both
x
and
y.
 x
and
y
=
independent
variables

(Values
are
assigned
 independently
to
x
and
y.)
 z
=
dependent
variable
 Domain
=
{(x,
y)
|
f(x,
y)
is
defined}
 Range
=
{f(x,
y)
|
(x,
y)
is
an
element
in
the
domain}
 
 
 
 
 
 
 
 
 
 

 
 Three
or
more
variables
can’t
be
graphed,
but
the
concepts
 stay
the
same.


For
example
w
=
f(x,
y,
z):

w
is
the
dependent
 variable
while
x,
y,
and
z
are
the
independent
variables.
 Ex.
 f ( x, y ) = y 2 ! x 2 
 Domain
=
 Range
=
 
 
 f ( x, y ) = x 2 + y 2 
 Ex.
 Domain
=
 Range
=
 
 f ( x, y ) = x 2 + y 
 Ex.
 Domain
=
 Range
=
 
 
 Ex.
 ! y$ f ( x, y) = arctan # & " x% Domain
=
 Range
=
 
 
 Ex.

Below
is
a
chart
of
elevation
above
sea
level
at
any
point
 (x,
y)
in
feet.



 x 













y 
 ‐6 
 ‐4 
 ‐2 
 0
 2
 4
 6
 ‐4 
 152
 186
 209
 218
 209
 186
 152
 ‐2 
 193
 236
 266
 277
 266
 236
 193
 0
 209
 256
 288
 300
 288
 256
 209
 2
 193
 236
 266
 277
 266
 236
 193
 4
 152
 186
 209
 218
 209
 186
 152
 
 f(4,2)
=
 The
value
of
x
when
y
=
0
so
that
the
elevation
is
300
is
 The
value
of
x
when
y
=
0
so
that
the
elevation
is
256
is
 
 
 f ( x , y ) = x 2 + y 2 .
 Ex.

Sketch
the
graph
of

 
 
 
 
 
 
 
 
 Ex.

Sketch
the
graph
of
 f ( x , y ) = y 2 ! x 2 .
 
 
 
 
 
 
 
 
 Do:

Find
the
domain
and
range
of
problems
1
and
2
and
 sketch
the
domain.
 1. x!y g ( x, y ) = x+y 
 
 
 2. 
 h( x, y) = ln ( x + y ! 1) 
 
 
 3. 
Graph
 
 
 f ( x, y) = cos y 
 L e v e l 
C u rv e s
 A
level
curve
(or
contour)
for
a
function
z
=
f(x,
y)
is
the
curve
 of
intersection
of
the
surface
z
=
f(x,
y)
and
the
horizontal
 plane
z
=
k.
 
 f ( x, y ) = x 2 + y 2 
 Ex.
 
 
 
 
 
 
 
 y g ( x, y ) = 2 x Ex.
 
 
 
 
 
 
 
 Note:
 1. The
equation
of
a
contour
(level
curve)
is
f(x,
y)
=
k,
 plotted
in
the
x‐y
coordinate
system.
 2. 
All
points
on
a
particular
contour
have
the
same
z
value.

 That
is,
f
is
constant
on
a
level
curve.
 3. Contours
cannot
intersect.
 
 Do:
1.

Draw
and
describe
the
level
curves
for
k
=
0,
1,
2
of
 f ( x, y) = x .

What
is
the
domain
and
range
of
this
 function?
 
 2.

Draw
and
describe
the
level
curves
for
k
=
0,
2,
5
for
 f ( x , y ) = 4 x 2 + y 2 + 1 .
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ...
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This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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