Second Derivative Test

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Presentation transcript:

Second Derivative Test 4.3b Concavity And the Second Derivative Test State Standard – 9.0a Students find the intervals in which the graph of functions is concave up and concave down. – 9.0c Students can identify points of inflection of graphs of functions. Objective – To be able to use the 2nd derivative test to find concavity and points of inflection.

In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator of computer to draw complicated graphs, it is still important to understand the relationships between derivatives and graphs.

First derivative: is positive Curve is rising. is negative Curve is falling. is zero Constant - Possible local maximum or minimum. Second derivative: is positive Curve is concave up. is negative Curve is concave down. is zero Possible inflection point (where concavity changes).

Possible inflection point at . Example: We will use the 2nd derivative test to find the Concavity of the function. Possible inflection point at . negative Curve is concave down. positive Curve is concave up. inflection point at

Possible inflection point at . Example: We will use the 2nd derivative test to find the Concavity of the function. Possible inflection point at . negative Curve is concave down. positive inflection point at Curve is concave up.

Possible inflection point at Example: We will use the 2nd derivative test to find the Concavity of the function. Possible inflection point at positive Curve is concave up. negative Curve is concave down. inflection point at positive Curve is concave up.

Find the intervals of concavity and the inflection point for 4.3b Assignment Find the intervals of concavity and the inflection point for # 33 – 37 on pg. 305