Matt Wolf
|Learning Goals: |At the completion of the lesson, you will be able to… |
| |Define, identify, and differentiate between concave up and concave down intervals |
| |Apply the Second Derivative Test to identify intervals of concave up and concave down |
| |Define and identify coordinates of points of inflection |
Examples: Perform the first and second derivative tests on the following functions.
[pic] [ -3, 4]
[pic]
-----------------------
[pic]
Concavity
Let [pic] be a differentiable function. The graph of[pic] is…
• concave up (U-shaped ) on every interval for which
[pic]
• concave down ()"-shaped ) on every interval for which
[pic]
*A point of inflection (POI) is a point at which the graph of a function changes concavity
The Second Derivative Test
Suppose that a function [pic]is continuous on the interval [pic] and differentiable on the interval [pic]. The Second Derivative Test is used to identify the concavity of the graph of[pic].
1) Find all critical points of [pic]. The endpoints and critical points divide the graph of [pic] into intervals starting with the first endpoint (a) then the critical points ([pic]...) and ending with the final endpoint (b) as shown in the figure above
2) Arrange endpoints/critical points into intervals and create a table as shown in the examples
3) Pick a test value (v) in each interval and evaluate the second derivative at each
4) If[pic] is positive then the graph of[pic] is concave up on that interval
If[pic] is negative then the graph of[pic] is concave down on that interval
[pic]_______________ Critical Pts: x = ________
|Interval | |
|Test Value (v) | |
|[pic] | |
|CU/CD | |
[pic]_______________ Critical Pts: x = ________
|Interval | |
|Test Value (v) | |
|[pic] | |
|Inc/Dec | |
Abs. Max. @ _______ Rel. Max. @ _______ Abs. Min. @ _______ Rel. Min. @ _______ POI @ _______
[pic]_______________ Critical Pts: x = ________
|Interval | |
|Test Value (v) | |
|[pic] | |
|CU/CD | |
[pic]_______________ Critical Pts: x = ________
|Interval | |
|Test Value (v) | |
|[pic] | |
|Inc/Dec | |
Abs. Max. @ _______ Rel. Max. @ _______ Abs. Min. @ _______ Rel. Min. @ _______ POI @ _______
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