Chapter 4.
Partial Differentiation
4.1.
Review on functions of one variable
Let
f
:
R
→
R
be a
realvalued function of one variable
x
∈
R
.
The
graph
consists of
the points in the Cartesian plane
R
2
whose coordinates are the inputoutput pairs for
f
. In set
notation, the graph is
{
(
x, f
(
x
))
T
∈
R
2
:
x
∈
R
}
.
Limits
•
Informally, a function
f
(
x
)
has a limit
l
at a point
a
if the value of
f
(
x
)
can be made as
close to
l
as desired, by making
x
close enough to
a
. We write
lim
x
→
a
f
(
x
) =
l
to mean that
l
is the limit of
f
(
x
)
as
x
→
a
.
•
When computing a limit, we are NOT concerned with what happens to
f
(
x
)
when
x
=
a
,
but only with what happens to
f
(
x
)
when
x
is sufficiently close to
a
. In fact, the limit
value does not depend on how the function is defined at
a
.
•
OneSided Limits
As
x
approaches
a
from the right
,
f
(
x
)
approaches
l

and we write
lim
x
→
a
+
f
(
x
) =
l

.
As
x
approaches
a
from the left
,
f
(
x
)
approaches
l
+
and we write
lim
x
→
a

f
(
x
) =
l
+
.
Limits of these forms are called
onesided limits
.
The limit is a
righthand limit
if the
approach is from the right. From the left, it is a
lefthand limit
. If both
lim
x
→
a

f
(
x
)
and
lim
x
→
a
+
f
(
x
)
exist, and
lim
x
→
a

f
(
x
) =
lim
x
→
a
+
f
(
x
)
,
then
lim
x
→
a
f
(
x
)
exists.
•
Limits at Infinity
There is also a concept for limits as
x
approaches infinity, that is, as
x
either grows without
bound positively or negatively. Example:
lim
x
→
+
∞
f
(
x
) = 1
Continuity
•
A function
f
(
x
)
is
continuous at
x
=
c
if and only if it meets the following three conditions.
1.
f
(
c
)
exists
2.
lim
x
→
c
f
(
x
)
exists
3.
lim
x
→
c
f
(
x
) =
f
(
c
)
•
The function
f
(
x
) = 1
/x
is continuous on its natural domain
R
\ {
0
}
. It has a point of
discontinuity at
x
= 0
, as it is not defined there. We also say that
f
(
x
)
has an
infinite
discontinuity
at
x
= 0
. Every polynomial function
P
(
x
) =
a
n
x
n
+
a
n

1
x
n

1
+
· · ·
+
a
0
is
continuous at every point
c
∈
R
, that is, continuous on
R
, since
lim
x
→
c
P
(
x
) =
P
(
c
)
for
any
c
∈
R
.
•
Intermediate Value Theorem for Continuous Functions
Geometrically, the Intermediate Value Theorem says that any horizontal line
y
=
k
crossing
the
y
axis between the numbers
f
(
a
)
and
f
(
b
)
will cross the curve
y
=
f
(
x
)
at least once
over the interval
[
a, b
]
. In particular, the theorem tells us that if
f
(
x
)
is continuous, then
any interval on which
f
changes sign contains a
root
of the function, that is, a solution
to the equation
f
(
x
) = 0
.
Differentiation
•
Difference quotient and Derivatives
The
difference quotient
of a function
f
(
x
)
at
x
=
a
with increment
h
is given by
f
(
a
+
h
)

f
(
a
)
h
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and the
derivative
of
f
(
x
)
at
x
=
a
is the limit
lim
h
→
0
f
(
a
+
h
)

f
(
a
)
h
and we denote this limit by
f
0
(
a
)
or
d
dx
f
(
x
)
x
=
a
.
•
Geometric interpretation of derivatives
The
tangent line
of the graph of
f
(
x
)
at the point
P
= (
a, f
(
a
))
is the line through
P
whose slope is the limit of the secant slopes as a nearby point
Q
approaches
P
. This limit
is the derivative
f
0
(
a
)
of
f
at
a
. The equation for the tangent line to
f
(
x
)
at the
(
a, f
(
a
))
is then given by
y
=
f
(
a
) +
f
0
(
a
)(
x

a
)
.
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