
Critcal Number
A critical number of a function F is a number "C" in the domain of F such that either F'(c) = 0 or F'(c) does not exist.

Closed interval method
 To find the absolute maximum and minimum values of a continuous function F on 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 of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.

Roll's Therom
 Let F be a function that satisfies the following three hypotheses:
 1) F is contiuous on the closed interval [a,b].
 2) F is differentiable on the open interval (a,b).
 3) F(a) = F(b)
 Then there is a number "c" in (a,b) such that F'(c) = 0

Guidlines for sketching a curve (ah)
 A. Domain
 B. Intercepts
 C. Symmetry (even/odd)
 D. Asymptotes
 E. Increasing/Decreasing
 F. Local min and max's values
 G. Concavity & points of inflection
 H. Sketch

Mean Value Therom
 Let F be a function that satisfies the following hypotheses:
 1) F is continuouse on the closed interval [a,b].
 2) F is differentiable on the open interval (a,b).
 Then there is a number c in (a,b) such that;
 F'(c) = (F(b)  F(a))/(ba)
 OR EQUIVALENTLY,
 F(b)  F(a) = F'(c)(ba)

Theorem 5
If F'(x) = 0 for all x in an interval (a,b), then F is consistant on (a,b).

Corollary
 If F'(x) = g'(x) for all x in an interval (a,b), then F  g is constant on (a,b); that is,
 F(x) = g(x) + c is a constant.

Increasing/decreasing test
 A) If F'(x) > 0 on an interval, then F is increasing on that interval.
 B) If F'(x) < 0 on an interval, then F is decreasing on that interval.

First derivitive test
 Suppose that "c" is a critical number of a continuous function F.
 A) If F' changes from positive to negitive at c, then F has a local maximum at c.
 B) If F' changes from negitive to positive at c, then F has a local minimum at c.
 C) If F' does not change sign at c (for example, if F' is positive on both sides of c or negative on both sides), then F has no local maximum or minimum at c.

Concavity
If the graph of F lies above all of its tangents on an interval I, then it is called concave upward on I. If the graph of F lies below all of its tangents on I, it is called concave downward on I.

Concavity test
 A) If F''(x) > 0 for all x in I, then the graph of F is concave upward on I.
 B) If F''(x) < 0 for all x in I, then the graph of F is concave downward on I.

Inflection point
A point P on a curve y = F(x) is called an INFLECTION POINT if F is continuous there and the curve changes from concave upward to concave downward or vis versa at P.

Secound Derivitive test
 Suppose F'' is continuous near C.
 A) If F'(c) = 0 and F''(c) > 0, then F has a local minimum at C.
 B) If F'(c) = 0 and F''(c) < 0,
 Then F has a local maximum at c.

L'Hospital's Rule
 Suppose F and g are differentaible and g'(x) =/ 0 on an open interval I that contains a (except possibly at a). Suppose that
 lim F(x) = 0. And lim g(x) = 0
 x>a. x>a
 OR THAT
 lim F(x) = + inf.
 x>a. AND
 Lim g(x) =+inf
 x>a
 (In other words, we have an indeterminate form of type 0/0 or inf/inf) Then
 Lim (F(x)/g(x))=Lim(F'(x)/g'(x))
 x>a. x>a
if the limit on the right side exists (or is inf or  inf).

1st Derivitive Test for Absalute Extreme Values
 Suppose that c is a critical number of a continuous function F defined on an interval.
 A) If F'(x) > 0 for all x < c and F'(x) < 0 for all x > c, then F(c) is the absolute maximum value of F.
 B) If F'(x) < 0 for all x < c and F'(x) > 0 for all x > c, then F(c) is the absolute minimum value of F.

