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Miss Battaglia AP Calculus AB/BC

3.4 Concavity and the Secondary Derivative Test Objective: Determine on intervals on which a function is concave up or down; find any points of inflection; apply the Second Derivative Test. Miss Battaglia AP Calculus AB/BC. Definition of Concavity.

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Miss Battaglia AP Calculus AB/BC

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  1. 3.4 Concavity and the Secondary Derivative TestObjective: Determine on intervals on which a function is concave up or down; find any points of inflection; apply the Second Derivative Test. Miss Battaglia AP Calculus AB/BC

  2. Definition of Concavity • Let f be differentiable on an open interval I. The graph of f is concave upward on I if f’ is increasing on the interval and concave downward on I if f’ is decreasing on the interval.

  3. Thm 3.7 Test for Concavity Let f be a function whose second derivative exists on an open interval I. • If f’’(x)>0 for all x in I, then the graph of f is concave upward on I. • If f’’(x)<0 for all x in I, then the graph of f is concave downward on I.

  4. Determining Concavity Determine the open intervals on which the graph of is concave upward or downward.

  5. Determining Concavity Determine the open intervals on which the graph of is concave upward or downward.

  6. Definition of Concavity • Let f be a function that is continuous on an open interval and let c be a point in the interval. If the graph of f has a tangent line at this point (c,f(c)), then this point is a point of inflection of the graph of f if the concavity of f changes from upward to downward at the point.

  7. Theorem 3.8 Points of Inflection • If (c,f(c)) is a point of inflection of the graph of f, then either f’’(c)=0 or f’’(c) does not exist at x=c.

  8. Finding Points of Inflection Determine the points of inflection and discuss the concavity of the graph f(x)=x4 - 4x3

  9. Thm 3.9 The Second Derivative Test(oh no not another test!) Let f be a function such that f’(c)=0 and the second derivative of f exists on an open interval containing c. • If f’’(c)>0, then f has a relative min at (c,f(c)) • If f’’(c)<0, then f has a relative max at (c,f(c)) If f’’(c)=0, the test fails. That is, may have a relative max, a relative min, or neither. In such cases, you can use the First Derivative Test.

  10. Using the Second Derivative Test Find the relative extrema for f(x)=-3x5+5x3

  11. Classwork/Homework • Read 3.4 Page 195 #19, 25 (AB), 27 (BC), 29, 31, 34, 63, 65, 67, 91-94

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