# Exam 3

**Created:**2014-04-10

**Last Modified:**2014-04-10

Absolute Maximum/Minimum

Highest/lowest point on the whole graph.

Local Maximum/Minimum

Highest/lowest point on a part of a graph.

Fermat's Theorem

If f has a local max/min at point c, and if f'(c) exists, then f'(c)=__0__

Critical Number

A number such that either f'(c)=0 or f'(c) does not exist.

Critical Number

A number 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 at the critical numbers

2) Find the values at the endpoints

3) The largest valuesnfrom steps 1 and 2 is the absolute max/min

Rolle's Theorem

If f satisfies the 3 hypothesis:

1) f is continuous on the closed interval [a, b]

2) f is differentiable on the open interval (a, b)

3) f (a) = f (b)

Then there is c that f (c)=0

Mean Value Theorem

If f satisfies the hypotheses:

1) f is continuous on the closed interval [a, b]

2) f is differentiable on the open interval (a, b)

Then there is a number c that

f (b)-f (a)=f'(c)(b-a)

f'(x)=0 theorem

If all x in an interval (a, b) satisfy f'(x)=0, then f is constant on (a, b)

Increasing/Decreasing Test

If f'(x) > 0 on an interval, then f is increasing on that interval

If f'(x) < 0 on an interval, then f is decreasing on that interval

First Derivative Test

1) If f' changes from + to - at c (critical number), then f has a local maximum at c

2) If f' changes from - to + at c, then f has a local minimum at c

3) if f' does not change at c, then there is no max or min

Concavity

If all tangents on an interval lie above the graph, it is concave downward

If all tangents on an inter lie below the graph, it is concave upward

Concavity Test

If f''(x)>0 for all x in I, then graph is concave upward

if f''(x)<0 for all x in I, then graph is concave downward

Inflection Point

Point where a graph changes concavity

Second Derivative Test

If f'(c)=0 and f''(c)> 0, then f has a local minimum at c

If f'(c)=0 and f''(c)< 0, then f has a local maximum at c

L'Hospital's Rule

Separately taking the derivatives of the numerator and denominator when the limit reaches an indeterminate form

0/0, INF/INF

Curve Sketching

A) Domain

B) Intercepts

C) Symmetry or Periodic

D) Asymptotes

E) Inc/Dec

F) Local Max or Min

G) Concavity and P.O.I.

H) Sketch Graph

Slant Asymptotes

Found by long division

For rational functions, if the numerator is one more degree than the denominator there is a slant asymptotes

First Derivative Test for Absolute Extreme Values

If f'(x)> 0 for all x <c and f'(c)<0 for all x> c, then f(c) is an absolute max

If f'(x)<0 for all x<c and f'(x)> 0 for all x> c, then f (c) is an absolute min

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