Download increasing and decreasing functions and the first derivative test

INCREASING AND DECREASING FUNCTIONS AND THE FIRST DERIVATIVE TEST
Derivatives can be used to classify relative extrema as either relative minima, or relative maxima.
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The derivative is related to the slope of a function.
constant
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FIRST DERIVATIVE TEST
Let c be a critical number of a function f that is continuous on an open interval. If f is differentiable on the interval, except possibly at c, then f (c) can be classified as follows:
1. If f ' (x) changes from negative to positive at c, then f (c) is a relative minimum of f.
2. If f ' (x) changes from positive to negative at c, then f (c) is a relative maximum of f.
3. If f ' (x) does not change sign at c, then f (c) is neither a relative minimum nor a relative maximum of f.
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EX. #1: Find local extrema, intervals on which f is increasing/decreasing, sketch graph.
Step 1:
Step 2:
Critical numbers: Step 3: Find intervals for increasing/decreasing
Test #
Crit #
Sign f'(x)
Conclusion
Increasing:
Decreasing:
Extrema: Step 4: Graph
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EX. #2: Find local extrema of , x ­ sin x on the interval 2
(0, 2π); sketch the graph.
Note that f is continuous on the entire interval.
Step 1:
Step 2:
Critical numbers: Step 3: Find intervals for increasing/decreasing
Test #
Crit #
Sign f'(x)
Conclusion
Local Max @:
Local Min @:
Extrema: Step 4: Graph
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EX. #3: Find local extrema of , f(x) = ( x2 ­ 4)2/3 ; sketch the graph.
Note that f is continuous on the entire real line.
Step 1:
Step 2:
f'(x) DNE
Critical numbers: Step 3: Find intervals for increasing/decreasing
Test #
Crit #
Sign f'(x)
Conclusion
Local Max @:
Local Min @:
Extrema: Step 4: Graph
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Attachments
Grid.docx