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Evan B.

Absolute Maximum/Minimum

Highest/lowest point on the whole graph.

Local Maximum/Minimum

Highest/lowest point on a part of a graph.

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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

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

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

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

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

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Concavity

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 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 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

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

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

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