1
Review
•
Limit laws
•
Derivative rules
2
Antiderivatives
Recall when we were looking at the motion of a particle, we were given a function for its position at a given
time. We could figure out how fast it was moving by looking at the derivative. What if we had the velocity,
but wanted the position? Well, velocity is the derivative of the position, so if we could find a function that
has a derivative of the velocity we’re looking for that’d be the position.
Definition 1.
A function
F
is an antiderivative of
f
in an interval [
a, b
] if
F
(
x
) =
f
(
x
) for all
x
in [
a, b
].
Example 2.1.
f
(
x
) =
x
2
.
It’s not hard to see that an antiderivative is
F
(
x
) =
1
3
x
3
.
But what about
G
(
x
) =
1
3
x
3
+ 1?
A consequence of the mean value theorem is that if two functions have the same derivative on an interval,
then they differ by a constant. So if
G
and
F
are both antiderivatives of
f
, then
F
(
x
) =
G
(
x
) +
C
.
Theorem 1.
If
F
is an antiderivative of
f
on an interval [
a, b
], then the most general antiderivative of
f
on
I
is:
F
(
x
) +
C
Where
C
is an arbitrary constant.