Chapter 4
Section
4.1 through 4.5
Absolute Max/Min:
Only one of an each.
-A
function f has an absolute max (min) at c if f(c) 
(
) f(x) for all x in domain.
Extreme Value Theorem:
If f is a continuous function on a closed interval [a, b], then the
function f has an absolute max valued f(c), and absolute min valued f (d) at
some numbers c, d, in [a, b].
[Insert graphs!]
Fermats
Theorem: If f has a local max/min at c, and f (c) exists
then f (c) = 0.
Critical Numbers: a value of a function in the
domain of f, such that f (c) = 0 or f (c) DNE.
Closed Interval Method: used to find absolute max/min,
when F is continuous with a closed interval [a, b].
1.
Find
the values of f at the critical numbers of f in [a, b]
2.
Find
the values of f at the endpoints of the interval
3.
The
largest value is the absolute max., smallest value is the absolute min.
Rolles Theorem: special case theorem. Let f be a function that satisfies the
following 3 hypotheses:
1.
f
is continuous on a closed interval [a,b]
2.
f
is differentiable on open interval (a,b)
3.
f(a)
= f(b)
Then there exists a number c in [a,
b) such that f (c) = 0
Mean Value Theorem:
1.
f
is continuous on a closed interval [a,b]
2.
f
is differentiable on open interval (a,b)
3.
f(a)
= f(b)
Then there exists a number c in [a,
b] such that f (c) = 
Increasing/Decreasing Test:
If f (x) > 0 on an interval,
then f(x) is increasing on that interval.
If f (x) < 0 on an interval,
then f(x) is decreasing on that interval
First Derivative Test:
Suppose
c is a critical number of a continuous function f
If f
changes from positive to negative at c, then there is a local max at c
If f
changes from negative to positive at c, then there is a local min at c
Concavity Test:
second derivative
If f
(x) is > 0 for all x in an interval then f is concave up on that interval
If f
(x) < 0 for all x in an interval then f is concave down on that interval
Horizontal Asymptotes:
Degree
numerator = degree denominator
Y=
leading coefficient in numerator/ leading coefficient in denominator
Degree
numerator < degree in denominator
Y
= 0
Degree
numerator > degree denominator
No
horizontal asymptote
Degree
in numerator= exactly one more than degree denominator
Y
=
D.I.S.A.C.
D.)
Domain
I.)
Intercepts
S.)
Symmetry
A.)
Asymptotes
C.)
Calculus!