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Miss Battaglia AP Calculus AB/BC

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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.

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Let f be a function whose second derivative exists on an open interval I. 1. If f’’(x)>0 for all x in I, then the graph of f is concave upward on I. 2. If f’’(x)<0 for all x in I, then the graph of f is concave downward on I.

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

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

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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.

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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.

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Determine the points of inflection and discuss the concavity of the graph f(x)=x 4 - 4x 3

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Let f be a function such that f’(c)=0 and the second derivative of f exists on an open interval containing c. 1. If f’’(c)>0, then f has a relative min at (c,f(c)) 2. 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.

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Find the relative extrema for f(x)=-3x 5 +5x 3

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Read 3.4 Page 195 #19, 25 (AB), 27 (BC), 29, 31, 34, 63, 65, 67, 91-94

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AP Calculus Section 3.4 Concavity and the Second Derivative Test

AP Calculus Section 3.4 Concavity and the Second Derivative Test

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