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h Height t Time 1 sec2 sec3 sec4 sec Velocity = 0, Slope =0 Moving upward, Slope > 0 Moving downward, slope <0 Touch down Throwing a glass ball up into the air 5 sec6 sec m>0 m=0 m<0

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A Night Ride in a Roller Coaster m=0 Inflection Point Steepest Slope Use the light beam as the tangent line Change in Concavity Tangent line Goes Below Tangent line Goes Above

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Critical Points & Signs of f ’(x) f’(x)=0 f “ (x) = 0 Inflection Point + + + + + + + + + + + + + + + - - - - - - f ‘(x) > 0 Rising f ‘(x) < 0 Falling f’(x)=0

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f(x) f’(x),m 1 2 4 3 -3 -2 m=+4 m=+2 m=+1 m=0 m=-1 m=-1.5 m=-1 m=0 m=+1.5 m=+3 Visualizing the Derivative Locate the critical points: m = 0 ; Inflection point Positive Slope Negative Slope m = 0

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f(x) f’(x),m + + + 0 - - - 0 + + Visualizing the Derivative (another method) Locate the critical points: m = 0 ; Inflection point Positive Slope Negative Slope m = 0

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f(x)f(x) f “(x) f ’(x) + + + + + - -- - 0 0 f(x)f(x) f “(x) 0 - 0 - - - + + + + + + - - - - 0 0 0 + + - - - - 0 0 0 Inflection points

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AP Calculus Section 3.4 Concavity and the Second Derivative Test

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