Section 3 - Ms. Monaco's Math...
Transcript of Section 3 - Ms. Monaco's Math...
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Section 3.4
Concavity and the Second Derivative Test
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Concavity
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Definition of Concavity: f is concave upward when f’ is increasing and f is concave downward when f’ is decreasing
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Test for Concavity
• Let f be a function whose second derivative exits on an open interval I.
1. If for all x in I, then the graph of f is concave upward in I.
2. If for all x in I, then the graph of f is concave downward in I.
0)( xf
0)( xf
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Definition of a Point of Inflection
• A point of inflection is an ordered pair where a graph changes concavity.
• THEOREM:
If (c, f(c)) is a point of inflection of the graph of f, then either or is not differentiable at
x = c.
0)( xf
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Draw a function with an inflection point
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• Ex1: a) Determine the open intervals on which the graph is concave upward or downward and state any points of inflection
3
6)(
2
xxf
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• Ex1: b) Determine the open intervals on which the graph is concave upward or downward and state any points of inflection
34 4)( xxxf
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The Second Derivative Test
• This test is an alternative way to find the relative maximums and minimums of a function
• If , the test fails and you must use the fist derivative test
0)( xf
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Ex2: Find the relative extrema for
using the second derivative test.
35 53)( xxxf
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Homework
• p. 195 # 1-3 odd, 13-49 EOO, 61-65 odd