<
Section
4.1
Three-Dimensional
Co
o
rdinate
Systems
Goals:
1
Plot
p
oints
in
a
three-dimensional
co
ordinate
system.
2
Use
the
distance
fo
rmula.
3
Recognize
the
equation
of
a
sphere
and
find
its
radius
and
center.
4
Graph
an
implicit
function
with
a
free
va
riable.
Question
4.1.1
Ho
w
Do
Ca
rtesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
d
isplaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
251
Question
4.1.1
Ho
w
Do
Ca
rtesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
d
isplaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
251
Question
4.1.1
Ho
w
Do
Ca
rtesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
displaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
251
Question
4.1.1
Ho
w
Do
Ca
rtesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
displaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
251
Question
4.1.1
Ho
w
Do
Ca
rtesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
displaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
251
Question
4.1.1
Ho
w
Do
Ca
rtesian
Co
o
rdinates
Extend
to
Higher
Dimensions?
Recall
ho
w
w
e
constructed
the
Ca
rtesian
plane.
x
y
y
−
4
−
4
−
3
−
3
−
2
−
2
−
1
−
1
1
1
2
2
3
3
4
4
(2
,
0)
(2
,
3)
1
Assign
o
rigin
and
t
w
o
directions
(
x
,
y
).
2
y
is
90
degrees
anticlo
ckwise
from
x
.
3
Axes
consist
of
the
p
oints
displaced
in
only
one
direction.
4
Co
o
rdinates
refer
to
displacement
from
the
o
rigin
in
each
direction.
5
Either
displacement
can
happ
en
first.
6
Each
p
oint
has
exactly
one
o
rdered
pair
that
refers
to
it.
251
Question
4.1.1
How
Do
Cartesian
Co
ordinates
Extend
to
Higher
Dimensions?
In
a
three-dimensional
Ca
rtesian
co
o
rdinate
system.
W
e
can
extrap
olate
from
t
w
o
dimensions.
Click to Load Applet
1
Assign
o
rigin
and
three
directions
(
x
,
y
,
z
).
2
Each
axis
mak
es
a
90
degree
angle
with
the
other
t
w
o.
3
The
z
direction
is
determined
b
y
the
right-hand
rule.
252
Question
4.1.2
Ho
w
Do
W
e
Establish
Which
Direction
Is
Positive
in
Each
Axis?
The
right
hand
rule
sa
ys
that
if
y
ou
mak
e
the
fingers
of
y
our
right
hand
follo
w
the
(counterclo
ckwise)
unit
circle
in
the
xy
-plane,
then
y
our
thumb
indicates
the
direction
of
the
p
ositive
z
-axis.
Click to Load Applet
Figure:
The
counterclo
ckwise
unit
circle
in
the
xy
-plane
253
Example
4.1.3
Dra
wing
a
Lo
cation
in
Three-Dimensional
Co
o
rdinates
The
p
oint
(2
,
3
,
5)
is
the
p
oint
displaced
from
the
origin
by
2
in
the
x
direction
3
in
the
y
direction
5
in
the
z
direction.
Ho
w
do
w
e
dra
w
a
reasonable
diagram
of
where
this
p
oint
lies?
254
Example
4.1.3
Dra
wing
a
Lo
cation
in
Three-Dimensional
Co
o
rdinates
The
p
oint
(2
,
3
,
5)
is
the
p
oint
displaced
from
the
origin
by
2
in
the
x
direction
3
in
the
y
direction
5
in
the
z
direction.
Ho
w
do
w
e
dra
w
a
reasonable
diagram
of
where
this
p
oint
lies?
Click to Load Applet
254
Example
4.1.3
Drawing
a
Lo
cation
in
Three-Dimensional
Coordinates
Ho
w
can
w
e
dra
w
a
reasonable
diagram
of
(
−
5
,
1
,
−
4)?
255
Example
4.1.3
Drawing
a
Lo
cation
in
Three-Dimensional
Coordinates
Ho
w
can
w
e
dra
w
a
reasonable
diagram
of
(
−
5
,
1
,
−
4)?
Click to Load Applet
255
Question
4.1.4
Ho
w
Do
W
e
Measure
Distance
in
Three-Space?
Theo
rem
The
distance
from
the
o
rigin
to
the
p
oint
(
x
,
y
,
z
)
is
given
b
y
the
Pythago
rean
Theo
rem
D
=
p
x
2
+
y
2
+
z
2
Click to Load Applet
256
Question
4.1.4
How
Do
We
Measure
Distance
in
Three-Space?
Theo
rem
The
distance
from
the
p
oint
(
x
1
,
y
1
,
z
1
)
to
the
p
oint
(
x
2
,
y
2
,
z
2
)
is
given
b
y
D
=
q
(
x
1
−
x
2
)
2
+
(
y
1
−
y
2
)
2
+
(
z
1
−
z
2
)
2
257
Question
4.1.5
What
Is
a
Graph?
Definition
The
graph
of
an
implicit
equation
is
the
set
of
p
oints
whose
co
ordinates
satisfy
that
equation.
In
other
w
o
rds,
the
t
w
o
sides
a
re
equal
when
w
e
plug
the
co
o
rdinates
in
fo
r
x
,
y
and
z
.
Example
The
graph
of
x
2
+
(
y
−
4)
2
+
(
z
+
1)
2
=
9
is
the
set
of
p
oints
that
a
re
distance
3
from
the
p
oint
(0
,
4
,
−
1)
Click to Load Applet
258
Example
4.1.6
Graphing
an
Equation
with
Two
F
ree
V
a
riables
Sk
etch
the
graph
of
the
equation
y
=
3.
259
Example
4.1.6
Graphing
an
Equation
with
Two
Free
Variables
In
addition
to
co
o
rdinate
axes,
3-dimensional
space
has
3
co
ordinate
planes
.
1
The
graph
of
z
=
0
is
the
xy
-plane.
2
The
graph
of
x
=
0
is
the
yz
-plane.
3
The
graph
of
y
=
0
is
the
xz
-plane.
Click to Load Applet
Figure:
The
co
o
rdinate
planes
in
3-dimensional
space.
260
Example
4.1.7
Graphing
an
Equation
with
One
F
ree
V
a
riable
Sk
etch
the
graph
of
the
equation
z
=
x
2
−
3.
261
Question
4.1.8
What
Do
the
Graphs
of
Implicit
Equations
Lo
ok
Lik
e
Generally?
Notice
that
the
graph
of
an
implicit
equation
in
the
plane
is
generally
one-dimensional
(a
curve),
whereas
the
graph
of
an
implicit
equation
in
three-space
is
generally
t
w
o-dimensional
(a
surface).
Figure:
The
curve
y
=
x
2
−
3
Click to Load Applet
Figure:
The
surface
z
=
x
2
−
3
262
Question
4.1.9
What
Is
the
Slop
e-Intercept
Equation
of
a
Plane?
Unlik
e
a
line,
a
non-vertical
plane
has
tw
o
slop
es.
One
measures
rise
over
run
in
the
x
-direction,
the
other
in
the
y
-direction.
Click to Load Applet
Figure:
A
plane
with
slop
es
in
the
x
and
y
directions.
263
Question
4.1.9
What
Is
the
Slope-Intercept
Equation
of
a
Plane?
Equation
A
plane
with
z
intercept
(0
,
0
,
b
)
and
slop
es
m
x
and
m
y
in
the
x
and
y
directions
has
equation
z
=
m
x
x
+
m
y
y
+
b
.
264
Example
4.1.10
W
riting
the
Equation
of
a
Plane
W
rite
the
equation
of
a
plane
with
intercepts
(4
,
0
,
0),
(0
,
6
,
0)
and
(0
,
0
,
8).
265
Example
4.1.10
Writing
the
Equation
of
a
Plane
Main
Idea
Given
three
p
oints
in
a
plane
A
=
(
x
1
,
y
1
,
z
1
),
B
=
(
x
2
,
y
2
,
z
2
)
and
C
=
(
x
3
,
y
3
,
z
3
)
1
If
t
w
o
p
oints
sha
re
an
x
-co
ordinate,
we
can
directly
compute
m
y
and
vice
versa.
2
F
ailing
that,
w
e
can
set
up
a
system
of
equations
and
solve
fo
r
m
x
,
m
y
and
b
.
266
Question
4.1.11
Ho
w
Do
We
Extrap
olate
to
Even
Higher
Dimensions?
W
e
can
use
a
co
o
rdinate
system
to
describ
e
a
space
with
mo
re
than
3
dimensions.
k
-dimensional
space
can
b
e
defined
as
the
set
of
p
oints
of
the
fo
rm
P
=
(
x
1
,
x
2
,
.
.
.
,
x
k
)
.
Theo
rem
The
distance
from
the
o
rigin
to
P
=
(
x
1
,
x
2
,
.
.
.
,
x
k
)
in
k
-space
is
q
x
2
1
+
x
2
2
+
·
·
·
+
x
2
k
There
is
no
right
hand
rule
fo
r
higher
dimensions,
b
ecause
w
e
can’t
dra
w
these
spaces
anyw
a
y
.
267
Section
4.1
Summa
ry
Questions
Q1
What
displacements
a
re
rep
resented
b
y
the
notation
(
a
,
b
,
c
)?
Q2
What
is
the
right
hand
rule
and
what
do
es
it
tell
you
ab
out
a
three-dimensional
co
o
rdinate
system?
Q3
In
three-space,
what
is
the
y
-axis?
What
are
the
coordinates
of
a
general
p
oint
on
it?
Q4
In
three
space,
what
is
the
xz
-plane?
What
are
the
co
ordinates
of
a
general
p
oint
on
it?
What
is
its
equation?
Q5
Ho
w
do
w
e
use
a
free
va
riable
to
sk
etch
a
graph?
Q6
Ho
w
do
w
e
recognize
the
equation
of
a
sphere?
268
Section
4.1
Q11
Dra
w
diagrams
of
p
oints
with
the
follo
wing
co
o
rdinates.
a
(6
,
1
,
2)
b
(
−
3
,
0
,
0)
c
(2
,
−
1
,
4)
d
(0
,
3
,
5)
269
Section
4.1
Q11
Click to Load Applet
269
Section
4.1
Q50
The
graph
of
x
2
+
y
2
=
0
in
R
2
is
a
p
oint,
not
a
curve.
Use
this
idea
to
write
an
equation
fo
r
the
intersection
of
the
graphs
f
(
x
,
y
,
z
)
=
c
and
g
(
x
,
y
,
z
)
=
d
.
What
do
you
exp
ect
the
dimension
of
this
intersection
to
b
e?
270
Section
4.1
Q34
Gabb
y
is
trying
to
find
the
equation
of
a
plane
P
,
but
she
do
esn’t
kno
w
any
p
oints
on
the
xz
-plane
or
yz
-plane.
Instead
she
knows
that
P
contains
the
p
oints:
A
=
(1
,
3
,
6)
B
=
(5
,
3
,
4)
C
=
(7
,
5
,
10)
Using
p
oints
A
and
B
,
she
decides
that
m
x
=
4
−
6
5
−
1
=
−
1
2
.
Using
p
oints
A
and
C
,
she
decides
that
m
y
=
10
−
6
5
−
3
=
2.
a
Which
of
Gabb
y’s
conclusions
do
y
ou
agree
with
and
which
do
y
ou
disagree
with?
Why?
b
Ho
w
could
y
ou
fix
the
one
that
is
wrong?
271
Section
4.1
Q36
Recall
that
we
can
write
the
equation
of
a
line
in
R
2
in
p
oint-slop
e
form:
y
−
y
0
=
m
(
x
−
x
0
)
where
m
is
the
slop
e
and
(
x
0
,
y
0
)
is
a
kno
wn
p
oint.
This
w
as
esp
ecially
useful
in
single-va
riable
calculus
fo
r
writing
equations
of
tangent
lines.
a
Ho
w
w
ould
you
exp
ect
to
write
the
equation
of
the
plane
P
through
(2
,
4
,
−
6)
with
slop
es
m
x
=
1
2
and
m
y
=
−
3?
b
Do
es
y
our
answ
er
to
a
actually
pass
through
(2
,
4
,
−
6)?
Ho
w
do
y
ou
kno
w?
c
Is
y
our
answ
er
to
a
actually
the
equation
of
a
plane?
Ho
w
do
y
ou
kno
w?
Do
es
it
have
the
co
rrect
slop
es?
d
W
rite
a
general
exp
re
ssion
fo
r
p
oint-slop
e
fo
rm
fo
r
a
plane.
272
Section
4.1
Q36
a
Ho
w
w
ould
you
exp
ect
to
write
the
equation
of
the
plane
P
through
(2
,
4
,
−
6)
with
slop
es
m
x
=
1
2
and
m
y
=
−
3?
272
Section
4.1
Q36
b
Do
es
y
our
answ
er
to
a
actually
pass
through
(2
,
4
,
−
6)?
Ho
w
do
y
ou
kno
w?
272
Section
4.1
Q36
c
Is
y
our
answ
er
to
a
actually
the
equation
of
a
plane?
Ho
w
do
y
ou
kno
w?
Do
es
it
have
the
co
rrect
slop
es?
272
Section
4.1
Q36
d
W
rite
a
general
exp
re
ssion
fo
r
p
oint-slop
e
fo
rm
fo
r
a
plane.
272
Section
4.2
F
unctions
of
Several
V
ariables
Goals:
1
Convert
an
implicit
function
to
an
explicit
function.
2
Calculate
the
domain
of
a
multiva
riable
function.
3
Calculate
level
curves
and
cross
sections
.
Question
4.2.1
What
Is
a
F
unction
of
More
than
One
Va
riable?
Definition
A
function
of
t
w
o
va
riables
is
a
rule
that
assigns
a
numb
er
(the
output
)
to
each
o
rdered
pair
of
real
numb
ers
(
x
,
y
)
in
its
domain
.
The
output
is
denoted
f
(
x
,
y
).
Some
functions
can
b
e
defined
algeb
raically
.
If
f
(
x
,
y
)
=
p
36
−
4
x
2
−
y
2
then
f
(1
,
4)
=
p
36
−
4
·
1
2
−
4
2
=
4
.
274
Example
4.2.2
The
Domain
of
a
Function
Identify
the
domain
of
f
(
x
,
y
)
=
p
36
−
4
x
2
−
y
2
.
Figure:
The
domain
of
a
function
275
Example
4.2.2
The
Domain
of
a
Function
Identify
the
domain
of
f
(
x
,
y
)
=
p
36
−
4
x
2
−
y
2
.
Figure:
The
domain
of
a
function
275
Application
4.2.3
T
emp
erature
Maps
Many
useful
functions
cannot
b
e
defined
algeb
raically
.
There
is
a
function
T
(
x
,
y
)
which
gives
the
temp
erature
at
each
latitude
and
longitude
(
x
,
y
)
on
ea
rth.
T
(
−
71
.
06
,
42
.
36)
=
50
T
(
−
84
.
38
,
33
.
75)
=
59
T
(
−
83
.
74
,
42
.
28)
=
41
Figure:
A
temp
erature
map
276
Application
4.2.4
Digital
Images
A
digital
image
can
b
e
defined
b
y
a
b
rightness
function
B
(
x
,
y
).
y
x
687
1024
B
(339
,
773)
=
158
B
(340
,
773)
=
127
Figure:
An
image
rep
resented
as
a
b
rightness
function
B
on
each
pixel
277
Question
4.2.5
What
Is
the
Graph
of
a
Two-V
a
riable
F
unction?
Definition
The
graph
of
a
function
f
(
x
,
y
)
is
the
set
of
all
p
oints
(
x
,
y
,
z
)
that
satisfy
z
=
f
(
x
,
y
)
.
The
height
z
ab
ove
a
p
oint
(
x
,
y
)
represents
the
value
of
the
function
at
(
x
,
y
).
278
Question
4.2.5
What
Is
the
Graph
of
a
Two-V
ariable
Function?
In
this
figure,
f
(1
,
4)
is
equal
to
the
height
of
the
graph
ab
ove
(1
,
4
,
0).
Click to Load Applet
Figure:
The
graph
z
=
p
36
−
4
x
2
−
y
2
279
Question
4.2.6
Ho
w
Do
We
Visualize
a
Graph
in
Three-Space?
Definition
A
level
set
of
a
function
f
(
x
,
y
)
is
the
graph
of
the
equation
f
(
x
,
y
)
=
c
fo
r
some
constant
c
.
Fo
r
a
function
of
tw
o
variables
this
graph
lies
in
the
xy
-plane
and
is
called
a
level
curve
.
Example
Consider
the
function
f
(
x
,
y
)
=
p
36
−
4
x
2
−
y
2
.
The
level
curve
p
36
−
4
x
2
−
y
2
=
4
simplifies
to
4
x
2
+
y
2
=
20.
This
is
an
ellipse.
Other
level
curves
have
the
fo
rm
p
36
−
4
x
2
−
y
2
=
c
o
r
4
x
2
+
y
2
=
36
−
c
2
.
These
a
re
la
rger
or
smaller
ellipses.
280
Question
4.2.6
How
Do
We
Visualize
a
Graph
in
Three-Space?
Level
curves
tak
e
their
shap
e
from
the
intersection
of
z
=
f
(
x
,
y
)
and
z
=
c
.
Seeing
many
level
curves
at
once
can
help
us
visualize
the
shap
e
of
the
graph.
Click to Load Applet
Figure:
The
graph
z
=
f
(
x
,
y
)
,
the
planes
z
=
c
,
and
the
level
curves
281
Example
4.2.7
Dra
wing
Level
Curves
Where
a
re
the
level
curves
on
this
temp
erature
map?
Figure:
A
temp
erature
map
282
Example
4.2.8
Using
Level
Curves
to
Describ
e
a
Graph
What
features
can
w
e
discern
from
the
level
curves
of
this
top
ographical
map?
Figure:
A
top
ographical
map
283
Example
4.2.9
A
Cross
Section
Definition
The
intersection
of
a
plane
with
a
graph
is
a
cross
section
.
A
level
curve
is
a
t
yp
e
of
cross
section,
but
not
all
cross
sections
a
re
level
curves.
Find
the
cross
section
of
z
=
p
36
−
4
x
2
−
y
2
at
the
plane
y
=
1.
284
Example
4.2.9
A
Cross
Section
Click to Load Applet
Figure:
The
y
=
1
cross
section
of
z
=
p
36
−
4
x
2
−
y
2
285
Example
4.2.10
Converting
an
Implicit
Equation
to
a
Function
Definition
W
e
sometimes
call
an
equation
in
x
,
y
and
z
an
implicit
equation
.
Often
in
o
rder
to
graph
these,
w
e
convert
them
to
explicit
functions
of
the
fo
rm
z
=
f
(
x
,
y
)
W
rite
the
equation
of
a
pa
rab
oloid
x
2
−
y
+
z
2
=
0
as
one
o
r
mo
re
explicit
functions
so
it
can
b
e
graphed.
Then
find
the
level
curves.
286
Example
4.2.10
Converting
an
Implicit
Equation
to
a
Function
Click to Load Applet
Figure:
Level
curves
of
x
2
−
y
+
z
2
=
0
287
Question
4.2.11
Ho
w
Do
es
this
Apply
to
F
unctions
of
More
Va
riables?
W
e
can
define
functions
of
three
va
riables
as
w
ell.
Denoting
them
f
(
x
,
y
,
z
).
Fo
r
even
more
variables,
we
use
x
1
through
x
n
.
The
definitions
of
this
section
can
b
e
extrap
olated
as
follows.
V
ariables
2
3
n
F
unction
f
(
x
,
y
)
f
(
x
,
y
,
z
)
f
(
x
1
,
.
.
.
,
x
n
)
Domain
subset
of
R
2
subset
of
R
3
subset
of
R
n
Graph
z
=
f
(
x
,
y
)
in
R
3
w
=
f
(
x
,
y
,
z
)
in
R
4
x
n
+1
=
f
(
x
1
,
.
.
.
,
x
n
)
in
R
n
+1
Level
Sets
level
curve
in
R
2
level
surface
in
R
3
level
set
in
R
n
288
Question
4.2.11
How
Do
es
this
Apply
to
Functions
of
More
Va
riables?
Observation
W
e
might
hop
e
to
solve
an
implicit
equation
of
n
variables
to
obtain
an
explicit
function
of
n
−
1
variables.
How
ever,
we
can
also
treat
it
as
a
level
set
of
an
explicit
function
of
n
variables
(whose
graph
lives
in
n
+
1
dimensional
space).
x
2
+
y
2
+
z
2
=
25
F
(
x
,
y
,
z
)
=
x
2
+
y
2
+
z
2
F
(
x
,
y
,
z
)
=
25
f
(
x
,
y
)
=
±
p
25
−
x
2
−
y
2
Both
viewp
oints
will
b
e
useful
in
the
future.
289
Section
4.2
Summa
ry
Questions
Q1
What
do
es
the
height
of
the
graph
z
=
f
(
x
,
y
)
represent?
Q2
What
is
the
distinction
b
et
w
een
a
level
set
and
a
cross
section?
Q3
What
a
re
level
sets
in
R
2
and
R
3
called?
Q4
What
is
the
difference
b
et
w
een
an
implicit
equation
and
explicit
function?
290
Section
4.2
Q50
Consider
the
implicit
equation
zx
=
y
a
Rewrite
this
equation
as
an
explicit
function
z
=
f
(
x
,
y
).
b
What
is
the
domain
of
f
?
c
Solve
fo
r
and
sk
etch
a
few
level
sets
of
f
.
d
What
do
the
level
sets
tell
y
ou
ab
out
the
graph
z
=
f
(
x
,
y
)?
291
Section
4.2
Q50
Click to Load Applet
291
Section
4.3
Limits
and
Continuity
Goals:
1
Understand
the
definition
of
a
limit
of
a
multiva
riable
function.
2
Use
the
Squeeze
Theo
rem
3
Apply
the
definition
of
continuit
y
.
Question
4.3.1
What
Is
the
Limit
of
a
Function?
Definition
W
e
write
lim
(
x
,
y
)
→
(
a
,
b
)
f
(
x
,
y
)
=
L
if
w
e
can
mak
e
the
values
of
f
sta
y
a
rbitrarily
close
to
L
b
y
restricting
to
a
sufficiently
small
neighb
o
rho
o
d
of
(
a
,
b
).
Proving
a
limit
exists
requires
a
fo
rmula
o
r
rule.
F
o
r
any
amount
of
closeness
required
(
ϵ
),
y
ou
must
b
e
able
to
p
ro
duce
a
radius
δ
around
(
a
,
b
)
sufficiently
small
to
keep
|
f
(
x
,
y
)
−
L
|
<
ϵ
.
293
Example
4.3.2
A
Limit
That
Do
es
Not
Exist
Sho
w
that
lim
(
x
,
y
)
→
(0
,
0)
x
2
−
y
2
x
2
+
y
2
do
es
not
exist.
Click to Load Applet
294
Example
4.3.3
Another
Limit
That
Do
es
Not
Exist
Sho
w
that
lim
(
x
,
y
)
→
(0
,
0)
xy
x
2
+
y
2
do
es
not
exist.
Click to Load Applet
295
Example
4.3.4
Y
et
Another
Limit
That
Do
es
Not
Exist
Sho
w
that
lim
(
x
,
y
)
→
(0
,
0)
xy
2
x
2
+
y
4
do
es
not
exist.
Click to Load Applet
296
Question
4.3.5
What
T
o
ols
Apply
to
Multi-V
ariable
Limits?
The
limit
la
ws
from
single-va
riable
limits
transfer
comfo
rtably
to
multi-va
riable
functions.
1
Sum/Difference
Rule
2
Constant
Multiple
Rule
3
Pro
duct/Quotient
Rule
The
Squeeze
Theo
rem
If
g
<
f
<
h
in
some
neighb
orhoo
d
of
(
a
,
b
)
and
lim
(
x
,
y
)
→
(
a
,
b
)
g
(
x
,
y
)
=
lim
(
x
,
y
)
→
(
a
,
b
)
h
(
x
,
y
)
=
L
,
then
lim
(
x
,
y
)
→
(
a
,
b
)
f
(
x
,
y
)
=
L
.
297
Question
4.3.6
What
Is
a
Continuous
Function?
Definition
W
e
sa
y
f
(
x
,
y
)
is
continuous
at
(
a
,
b
)
if
lim
(
x
,
y
)
→
(
a
,
b
)
f
(
x
,
y
)
=
f
(
a
,
b
)
.
Theo
rem
P
olynomials,
ro
ots,
trig
functions,
exp
onential
functions
and
loga
rithms
a
re
continuous
on
their
domains.
Sums,
differences,
p
ro
ducts,
quotients
and
comp
ositions
of
continuous
functions
a
re
continuous
on
their
domains.
In
each
of
our
examples,
the
function
w
as
a
quotient
of
p
olynomials,
but
(0
,
0)
w
as
not
in
the
domain.
298
Question
4.3.6
What
Is
a
Continuous
Function?
Rema
rk
Limits,
continuit
y
and
these
theo
rems
can
all
b
e
extrap
olated
to
functions
of
mo
re
va
riables.
299
Section
4.3
Summa
ry
Questions
Q1
Why
is
it
ha
rder
to
verify
a
limit
of
a
multiva
riable
function?
Q2
What
do
y
ou
need
to
check
in
o
rder
to
determine
whether
a
function
is
continuous?
300
Section
4.4
P
artial
Derivatives
Goals:
1
Calculate
pa
rtial
derivatives
.
2
Realize
when
not
to
calculate
pa
rtial
derivatives.
Question
4.4.1
What
Is
the
Rate
of
Change
of
a
Multivariable
Function?
Motivational
Example
The
fo
rce
due
to
gravit
y
b
et
w
een
tw
o
objects
dep
ends
on
their
masses
and
on
the
distance
b
et
w
een
them.
Supp
ose
at
a
distance
of
8
,
000km
the
force
b
etw
een
t
w
o
particula
r
objects
is
100
newtons
and
at
a
distance
of
10
,
000km,
the
fo
rce
is
64
newtons.
Ho
w
much
do
w
e
exp
ect
the
fo
rce
b
et
w
een
these
objects
to
increase
o
r
decrease
p
er
kilometer
of
distance?
302
Question
4.4.1
What
Is
the
Rate
of
Change
of
a
Multivariable
Function?
Derivatives
of
a
single-va
riable
function
w
ere
a
w
ay
of
measuring
the
change
in
a
function.
Recall
the
follo
wing
facts
ab
out
f
′
(
x
).
1
Average
rate
of
change
is
realized
as
the
slop
e
of
a
secant
line:
f
(
x
)
−
f
(
x
0
)
x
−
x
0
2
The
derivative
f
′
(
x
)
is
defined
as
a
limit
of
slop
es:
f
′
(
x
)
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
3
The
derivative
is
the
instantaneous
rate
of
change
of
f
at
x
.
4
The
derivative
f
′
(
x
0
)
is
realized
geometrically
as
the
slop
e
of
the
tangent
line
to
y
=
f
(
x
)
at
x
0
.
5
The
equation
of
that
tangent
line
can
b
e
written
in
p
oint-slop
e
fo
rm:
y
−
y
0
=
f
′
(
x
0
)(
x
−
x
0
)
303
Question
4.4.1
What
Is
the
Rate
of
Change
of
a
Multivariable
Function?
A
pa
rtial
derivative
measures
the
rate
of
change
of
a
multiva
riable
function
as
one
va
riable
changes,
but
the
others
remain
constant.
Definition
The
pa
rtial
derivatives
of
a
t
w
o-va
riable
function
f
(
x
,
y
)
are
the
functions
f
x
(
x
,
y
)
=
lim
h
→
0
f
(
x
+
h
,
y
)
−
f
(
x
,
y
)
h
and
f
y
(
x
,
y
)
=
lim
h
→
0
f
(
x
,
y
+
h
)
−
f
(
x
,
y
)
h
.
304
Question
4.4.1
What
Is
the
Rate
of
Change
of
a
Multivariable
Function?
Notation
The
pa
rtial
derivative
of
a
function
can
b
e
denoted
a
va
riet
y
of
w
ays.
Here
a
re
some
equivalent
notations
f
x
∂
f
∂
x
∂
z
∂
x
∂
∂
x
f
D
x
f
305
Example
4.4.2
Computing
a
Pa
rtial
Derivative
Find
∂
∂
y
(
y
2
−
x
2
+
3
x
sin
y
).
Main
Idea
T
o
compute
a
pa
rtial
derivative
f
y
,
p
erfo
rm
single-va
riable
differentiation.
T
reat
y
as
the
indep
endent
va
riable
and
x
as
a
constant.
306
Synthesis
4.4.3
Interp
reting
Derivatives
from
Level
Sets
Belo
w
a
re
the
le
vel
curves
f
(
x
,
y
)
=
c
for
some
values
of
c
.
Can
we
tell
whether
f
x
(
−
4
,
1
.
25)
and
f
y
(
−
4
,
1
.
25)
a
re
p
ositive
o
r
negative?
Figure:
Some
level
curves
of
f
(
x
,
y
)
307
Question
4.4.4
What
Is
the
Geometric
Significance
of
a
Pa
rtial
Derivative?
The
pa
rtial
derivative
f
x
(
x
0
,
y
0
)
is
realized
geometrically
as
the
slop
e
of
the
line
tangent
to
z
=
f
(
x
,
y
)
at
(
x
0
,
y
0
,
z
0
)
and
traveling
in
the
x
direction.
Since
y
is
held
constant,
this
tangent
line
lives
in
y
=
y
0
.
Click to Load Applet
Figure:
The
tangent
line
to
z
=
f
(
x
,
y
)
in
the
x
direction
308
Example
4.4.5
Derivative
Rules
and
P
a
rtial
Derivatives
Find
f
x
fo
r
the
follo
wing
functions
f
(
x
,
y
):
a
f
=
√
xy
(on
the
domain
x
>
0
,
y
>
0)
b
f
=
y
x
c
f
=
√
x
+
y
d
f
=
sin
(
xy
)
309
Question
4.4.6
What
If
We
Have
Mo
re
than
Two
V
a
riables?
W
e
can
also
calculate
pa
rtial
derivatives
of
functions
of
mo
re
variables.
All
va
riables
but
one
a
re
held
to
b
e
constants.
F
o
r
example
if
f
(
x
,
y
,
z
)
=
x
2
−
xy
+
cos(
yz
)
−
5
z
3
then
w
e
can
calculate
∂
f
∂
y
:
310
Example
4.4.7
A
F
unction
of
Three
V
a
riables
F
o
r
an
ideal
gas,
w
e
have
the
la
w
P
=
nRT
V
,
where
P
is
p
ressure,
n
is
the
numb
er
of
moles
of
gas
molecules,
T
is
the
temp
erature,
and
V
is
the
volume.
a
Calculate
∂
P
∂
V
.
b
Calculate
∂
P
∂
T
.
c
(Science
Question)
Supp
ose
w
e’re
heating
a
sealed
gas
contained
in
a
glass
container.
Do
es
∂
P
∂
T
tell
us
ho
w
quickly
the
p
ressure
is
increasing
p
er
degree
of
temp
erature
increase?
311
Question
4.4.8
Ho
w
Do
Higher
Order
Derivatives
Wo
rk?
T
aking
a
pa
rtial
derivative
of
a
pa
rtial
derivative
gives
us
a
higher
o
rder
pa
rtial
derivative.
W
e
use
the
follo
wing
notation.
Notation
(
f
x
)
x
=
f
xx
=
∂
2
f
∂
x
2
W
e
need
not
use
the
same
va
riable
each
time
Notation
(
f
x
)
y
=
f
xy
=
∂
∂
y
∂
∂
x
f
=
∂
2
f
∂
y
∂
x
312
Example
4.4.9
A
Higher
Order
P
a
rtial
Derivative
If
f
(
x
,
y
)
=
sin(3
x
+
x
2
y
)
calculate
f
xy
.
313
Question
4.4.10
Do
es
Differentiation
Order
Matter?
No.
Sp
ecifically
,
the
follo
wing
is
due
to
Clairaut:
Theo
rem
If
f
is
defined
on
a
neighb
o
rho
o
d
of
(
a
,
b
)
and
the
functions
f
xy
and
f
yx
a
re
b
oth
continuous
on
that
neighb
o
rho
o
d,
then
f
xy
(
a
,
b
)
=
f
yx
(
a
,
b
).
This
readily
generalizes
to
la
rger
numb
ers
of
va
riables,
and
higher
o
rder
derivatives.
F
o
r
example
f
xyyz
=
f
zyxy
.
314
Section
4.4
Summa
ry
Questions
Q1
What
is
the
role
of
each
va
riable
when
w
e
compute
a
pa
rtial
derivative?
Q2
What
do
es
the
pa
rtial
derivative
f
y
(
a
,
b
)
mean
geometrically?
Q3
Can
y
ou
think
of
an
example
where
the
pa
rtial
derivative
do
es
not
accurately
mo
del
the
change
in
a
function?
Q4
What
is
Clairaut’s
Theo
rem?
315
Section
4.4
Q10
In
the
diagram
from
this
example,
use
a
p
oint
on
the
c
=
30
level
set
to
app
ro
ximate
f
y
(4
,
−
1
.
25).
Figure:
Some
level
curves
of
f
(
x
,
y
)
316
Section
4.4
Q20
Supp
ose
Jinteki
Co
rp
o
ration
mak
es
widgets
which
is
sells
fo
r
$100
each.
It
commands
a
small
enough
p
o
rtion
of
the
ma
rk
et
that
its
p
ro
duction
level
do
es
not
affect
the
demand
(p
rice)
fo
r
its
p
ro
ducts.
If
W
is
the
numb
er
of
widgets
p
ro
duced
and
C
is
their
op
erating
cost,
Jinteki’s
p
rofit
is
mo
deled
b
y
P
=
100
W
−
C
.
Since
∂
P
∂
W
=
100
do
es
this
mean
that
increasing
p
ro
duction
can
b
e
exp
ected
to
increase
profit
at
a
rate
of
$100
p
er
widget?
317
Section
4.4
Q28
Ho
w
many
third
pa
rtial
derivatives
do
es
a
t
w
o-va
riable
function
have?
Assuming
these
derivatives
a
re
continuous,
which
of
them
a
re
equal
acco
rding
to
Clairaut’s
theo
rem?
318
Section
4.5
Linea
r
App
roximations
Goals:
1
Calculate
the
equation
of
a
tangent
plane
.
2
Rewrite
the
tangent
plane
formula
as
a
linea
rization
o
r
differential
.
3
Use
linea
rizations
to
estimate
values
of
a
function.
4
Use
a
differential
to
estimate
the
erro
r
in
a
calculation.
Question
4.5.1
What
Is
a
T
angent
Plane?
Definition
A
tangent
plane
at
a
p
oint
P
=
(
x
0
,
y
0
,
z
0
)
on
a
surface
is
a
plane
containing
the
tangent
lines
to
the
surface
through
P
.
Click to Load Applet
Figure:
The
tangent
plane
to
z
=
f
(
x
,
y
)
at
a
p
oint
320
Question
4.5.1
What
Is
a
T
angent
Plane?
Equation
If
the
graph
z
=
f
(
x
,
y
)
has
a
tangent
plane
at
(
x
0
,
y
0
),
then
it
has
the
equation:
z
−
z
0
=
f
x
(
x
0
,
y
0
)(
x
−
x
0
)
+
f
y
(
x
0
,
y
0
)(
y
−
y
0
)
.
Rema
rks
1
This
is
the
p
oint-slop
e
form
of
the
equation
of
a
plane.
f
x
(
x
0
,
y
0
)
and
f
y
(
x
0
,
y
0
)
a
re
the
slop
es.
2
x
0
and
y
0
a
re
numb
ers,
so
f
x
(
x
0
,
y
0
)
and
f
y
(
x
0
,
y
0
)
are
numb
ers.
The
va
riables
in
this
equation
a
re
x
,
y
and
z
.
321
Question
4.5.1
What
Is
a
T
angent
Plane?
The
cross
sections
of
the
tangent
plane
give
the
equation
of
the
tangent
lines
w
e
lea
rned
in
single
va
riable
calculus.
y
=
y
0
x
=
x
0
z
−
z
0
=
f
x
(
x
0
,
y
0
)(
x
−
x
0
)
+
0
z
−
z
0
=
0
+
f
y
(
x
0
,
y
0
)(
y
−
y
0
)
322
Example
4.5.2
W
riting
the
Equation
of
a
T
angent
Plane
Give
an
equation
of
the
tangent
plane
to
f
(
x
,
y
)
=
√
xe
y
at
(4
,
0)
323
Question
4.5.3
Ho
w
Do
We
Rewrite
a
T
angent
Plane
as
a
Function?
Definition
If
w
e
write
z
as
a
function
L
(
x
,
y
),
w
e
obtain
the
linea
rization
of
f
at
(
x
0
,
y
0
).
L
(
x
,
y
)
=
f
(
x
0
,
y
0
)
+
f
x
(
x
0
,
y
0
)(
x
−
x
0
)
+
f
y
(
x
0
,
y
0
)(
y
−
y
0
)
If
the
graph
z
=
f
(
x
,
y
)
has
a
tangent
plane,
then
L
(
x
,
y
)
app
ro
ximates
the
values
of
f
nea
r
(
x
0
,
y
0
).
Notice
f
(
x
0
,
y
0
)
just
calculates
the
value
of
z
0
.
This
formula
is
equivalent
to
the
tangent
plane
equation
after
w
e
solve
fo
r
z
b
y
adding
z
0
to
b
oth
sides.
324
Example
4.5.4
App
roximating
a
F
unction
Use
a
linea
rization
to
app
ro
ximate
the
value
of
√
4
.
02
e
0
.
05
.
325
Question
4.5.5
Ho
w
Do
es
Differential
Notation
W
o
rk
in
More
Va
riables?
The
differential
dz
measures
the
change
in
the
linea
rization
of
f
(
x
,
y
)
given
pa
rticula
r
changes
in
the
inputs:
dx
and
dy
.
It
is
a
useful
sho
rthand
when
one
is
estimating
the
erro
r
in
an
initial
computation.
Definition
F
o
r
z
=
f
(
x
,
y
),
the
differential
o
r
total
differential
dz
is
a
function
of
a
p
oint
(
x
0
,
y
0
)
and
t
w
o
indep
endent
va
riables
dx
and
dy
.
dz
=
f
x
(
x
0
,
y
0
)
dx
+
f
y
(
x
0
,
y
0
)
dy
=
∂
z
∂
x
dx
+
∂
z
∂
y
dy
Rema
rk
The
differential
fo
rmula
is
just
the
tangent
plane
fo
rmula
with
dz
=
z
−
z
0
dx
=
x
−
x
0
dy
=
y
−
y
0
.
326
Question
4.5.5
How
Do
es
Differential
Notation
Wo
rk
in
More
Va
riables?
An
old
trigonometry
application
is
to
measure
the
height
of
a
p
ole
by
standing
at
some
distance.
We
then
measure
the
angle
θ
of
incline
to
the
top,
as
w
ell
as
the
distance
b
to
the
base.
The
height
is
h
=
b
tan
θ
.
a
If
the
distance
to
the
base
is
13
m
and
the
angle
of
incline
is
π
6
,
what
is
the
height
of
the
p
ole?
b
Human
measurement
is
never
p
erfect.
If
our
measurement
of
b
is
off
b
y
at
most
0
.
1
m
and
our
measurement
of
θ
is
off
by
at
most
π
120
,
use
a
differential
to
app
ro
ximate
the
maximum
p
ossible
erro
r
in
our
h
.
327
Section
4.5
Summa
ry
Questions
Q1
What
do
y
ou
need
to
compute
in
o
rder
to
write
the
equation
of
a
tangent
plane
to
z
=
f
(
x
,
y
)
at
(
x
0
,
y
0
,
z
0
)?
Q2
F
o
r
what
kinds
of
functions
a
re
linea
r
app
roximations
useful?
Q3
Ho
w
a
re
the
tangent
plane
and
the
linea
rization
related?
Q4
Ho
w
is
the
differential
defined
fo
r
a
t
wo
va
riable
function?
What
do
es
each
va
riable
in
the
fo
rmula
mean?
328
Section
4.5
Q10
Let
g
(
x
,
y
)
=
3
x
2
+4
x
−
2
e
(
y
3
)
.
W
rite
the
equation
of
the
tangent
plane
to
z
=
g
(
x
,
y
)
at
(0
,
1).
329
Section
4.5
Q16
Sho
w
ho
w
to
use
an
app
ropriate
linea
rization
to
appro
ximate
1
5
.
12
sin
31
π
30
.
a
What
function
f
(
x
,
y
)
would
y
ou
linearize
to
make
this
app
ro
ximation?
b
What
(
x
0
,
y
0
)
w
ould
y
ou
use
to
write
y
our
linea
rization?
c
What
x
and
y
would
you
plug
into
L
(
x
,
y
)
to
appro
ximate
1
5
.
12
sin
31
π
30
?
330
Section
4.5
Q21
Bo
ris
is
measuring
the
a
rea
of
a
rectangula
r
field,
so
he
can
decide
ho
w
much
grass
seed
to
buy
.
Acco
rding
to
his
measurements,
the
field
is
30
m
b
y
50
m
,
giving
an
area
of
1500
m
2
.
If
w
e
accept
that
each
of
his
measurements
has
an
erro
r
no
la
rger
than
0
.
2
m
,
use
a
differential
to
app
ro
ximate
the
maximum
e
rro
r
in
his
area
computation.
331
Section
4.5
Q22
Supp
ose
I
decide
to
invest
$10
,
000
exp
ecting
a
6%
annual
rate
of
return
fo
r
12
y
ears,
after
which
I’ll
use
it
to
purchase
a
house.
The
fo
rmula
fo
r
comp
ound
interest
P
=
P
0
e
rt
indicates
that
when
I
w
ant
to
buy
a
house,
I
will
have
P
=
10
,
000
e
0
.
72
.
I
accept
that
my
exp
ected
rate
of
return
might
have
an
error
of
up
to
dr
=
2%.
Also,
I
ma
y
decide
to
buy
a
house
up
to
dt
=
3
yea
rs
b
efore
or
after
I
exp
ected.
a
W
rite
the
fo
rmula
for
the
differential
dP
at
(
r
0
,
t
0
)
=
(0
.
06
,
12).
b
Given
my
assumptions,
what
is
the
maximum
estimated
erro
r
dP
in
my
initial
calculation?
c
What
is
the
actual
maximum
erro
r
in
P
?
332
Section
4.5
Q24
Let
f
(
x
,
y
)
b
e
a
function.
What
differential
and
what
inputs
into
that
differential
w
ould
y
ou
use
to
app
roximate
f
(5
.
5
,
3
.
2)
−
f
(4
.
7
,
3
.
8).
333
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