concavity and second derivative rizzi – calc bc. warm up given derivative graph below, find a....
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CONCAVIT
Y AND S
ECOND
DERIVAT
IVE
RI Z
ZI
– CA
L C B
C
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WARM UP
Given derivative graph below, find a. intervals where the original function is increasing b. intervals where the original function is decreasing c. x-coordinates of the local maximums and minimums
of the function
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WHAT IS CONCAVITY?
Concavity is another physical interpretation of a function
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SECOND DERIVATIVE = CONCAVITY
The second derivative tells us intervals where the function is concave up and concave down.
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INFLECTION POINTS
Inflection points are the points where the graph changes concavity
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INTERVALS OF CONCAVE UP AND DOWN
Find the intervals of concavity for the function
Step 1: Find the second derivative, f”(x) = 0
Step 2: Determine the x-coordinates of the points of inflection
Step 3: Test the concavity for each interval in f”(x)
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AP PROBLEM
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SECOND DERIVATIVE TEST FOR EXTREMA
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TRY IT – SECOND DERIVATIVE TEST
Find the relative extrema of the function.
Step 1: Find the critical numbers of the function, where f’(x) = 0
Step 2: Find the second derivative and test each x-value to see the concavity at each point.
Step 3: Plug x-values into original to find coordinates
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COMPARISON OF 1ST AND 2ND DERIVATIVE
=0 Intervals Extrema Test
1st Derivative Critical Points(m=0)
Increasing/Decreasing
Use critical points and intervals of increasing/decreasing
2nd Derivative Inflection Points(concavity changes)
Concave Up/Down
Use critical points and concavity