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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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