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How Derivatives Affect The Shape of a Graph

How Derivatives Affect The Shape of a Graph. Section 4.3. Definition of concavity . Let be differentiable on an open interval: i . The graph of is concave upward if is increasing on the interval.

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How Derivatives Affect The Shape of a Graph

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  1. How Derivatives Affect The Shape of a Graph Section 4.3

  2. Definition of concavity • Let be differentiable on an open interval: i. The graph of is concave upward if is increasing on the interval. ii. The graph of is concave downward if is decreasing on the interval.

  3. Test for concavity • Let be a function whose second derivative exists on an open interval: i. If for all x in the interval, then the graph of is concave upward. ii. If for all x in the interval, then the graph of is concave downward.

  4. Definition of point of inflection • Let be a function whose graph has a tangent line at . The point is called the pointof inflection if the concavity of changes from upward to downward (or vice-versa) at that point. • If is a point of inflection of the graph of , then either .

  5. Second derivative test • Let be a function such that and the second derivative of exists on an open interval containing : i. If , then is a relativeminimum. ii. If , then is a relativemaximum. iii. If , then the test FAILS!!

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