Throwing a glass ball up into the air

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Presentation transcript:

Throwing a glass ball up into the air h Height Velocity = 0, Slope =0 m=0 Moving downward, slope <0 m>0 Moving upward, Slope > 0 m<0 Touch down t Time 1 sec 2 sec 3 sec 4 sec 5 sec 6 sec

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

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

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

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

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