Mu Alpha Theta Calculus Review
1
Introduction
This is a very brief review of AP Calculus for the purposes of doing well at the Mu Alpha Theta
State competition. It is by no means a way to learn calculus, and it does not go over basic facts that
the reader is assumed to know. Most of the things covered here will probably be useful at Mu Alpha
Theta, so it is a good idea to go over it, especially if you do not remember your calculus from the
beginning of the year.
2
Limits
2.1
Undefined Limits
These are generally when you have something in the denominator that equals zero and something in
the numerator that is not zero, for example,
lim
x
→
4
x
2
+ 3
x

6
x

4
=
4
2
+ 3(4)

6
4

4
=
22
0
.
The limit is considered undefined or
∞
.
2.2
Removable Discontinuities
These are generally when both the numerator and the denominator are equal to zero and we can
”plug” the hole in the graph, for example,
lim
x
→
4
x
2
+ 3
x

28
x

4
= lim
x
→
4
(
x
+ 7)(
x

4)
x

4
= lim
x
→
4
(
x
+ 7) = 11.
Just because the denominator is zero does not mean the limit is undefined!
2.3
L’Hopital’s Rule
This states that if
f
(
x
) and
g
(
x
) are functions such that
f
(
c
)
g
(
c
)
is of the form
0
0
or
∞
∞
we have
lim
x
→
c
f
(
x
)
g
(
x
)
= lim
x
→
c
f
(
x
)
g
(
x
)
.
For example, if we wanted to evaluate lim
x
→
0
sin
x
x
, we know that both go to zero, so we can apply
L’Hopital’s Rule to get
lim
x
→
0
sin
x
x
= lim
x
→
0
cos
x
1
= 1.
Be cautious, though; always check that it satisfies one of the indeterminate forms before using
L’Hopital’s Rule.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2.4
Using the Logarithm
Sometimes, you may need to take the logarithm of a limit and then apply L’Hopital’s Rule to evaluate
it. For example, if we wanted to evaluate
lim
x
→
0
x
x
plugging it in gives 0
0
which is not defined. So take the logarithm of it to get ln (
x
x
) =
x
ln
x
and
now take the limit by using L’Hopital cleverly:
lim
x
→
0
x
ln
x
= lim
x
→
0
ln
x
1
/x
= lim
x
→
0
1
/x

1
/x
2
= lim
x
→
0

x
= 0.
But remember at the end to exponentiate; so our original limit would be
lim
x
→
0
x
x
=
e
lim
x
→
0
x
ln
x
=
e
0
= 1.
3
Differentiation
3.1
Definition
For a function to be differentiable at a point, it must also be continuous; the reverse statement is not
necessarily true, however. If the function is differentiable, the derivative of
f
(
x
) is defined as
f
(
x
) = lim
h
→
0
f
(
x
+
h
)

f
(
x
)
h
.
Beware of problems that look like limits but are really the definition of the derivative; using this
knowledge can simplify them greatly. For example,
lim
h
→
0
((3 +
h
)
2
+ (3 +
h
))

(3
2
+ 3)
h
is just the derivative of
f
(
x
) =
x
2
+
x
at
x
= 3.
3.2
Basic Things
3.2.1
Critical Points and Points of Inflection
The critical points of a function
f
(
x
) are the points at which
f
(
x
) = 0. There are several classifica
tions of critical points: local minimum/maximum, absolution minimum/maximum, or just none of
them. Know how to find them and determine what they are.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 sd
 Calculus, Derivative, lim, dx

Click to edit the document details