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MOTION Derivatives Continued

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MOTION

Derivatives Continued

Derivatives of Trig Functions• sin’x => cos xcos’x => - sin x• tan’x => sec2x cot’x => - csc2x• sec’x => sec x tan x csc’x => - csc x cot x

Remember what you already know!

Page 3: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

Example of Harmonic Motion

• (jumping up and down)• (bouncing a ball)

• Example of weight on a spring.

• motion s = 5 cos t• velocity s’ = -5 sin t• acceleration s’’ = - 5 cos t

Page 4: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

Time for an experiment.• You need: a motion

detector, TI-83 Plus, bouncing ball program, and a ball.

• Bounce the ball and collect the data using the program.

• Graph the “Time vs. Distance” graph.

• Run the regression equation for a sin graph.

Page 5: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

Answer these questions:

1)In 5 seconds how many times did the ball hit the ground?

Height

time

Page 6: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

2) What is the height of the ball at t=2?

Page 7: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

3)At what time is the ball at it’s highest point?

(maximum)

Page 8: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

4)Take the heights at each bounce and plot them in the stat plot, find the exponential equation for the heights.

Page 9: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

5)Multiply the exponential equation by the sinusoidal equation that you got earlier and you will have the dampened equation that represents your ball exactly.

Page 10: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

6)Using your new equation, find the first derivative and the second derivative.

Page 11: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

7)What does the second derivative represent?

Page 12: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

8)At what time was the balls velocity the greatest?

Page 13: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

9)At what time was the balls speed the greatest?

Page 14: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

10)At what time or times was the balls velocity equal to zero? What does this mean and what is happening to the ball at these times?

Page 15: MOTION Derivatives Continued Derivatives of Trig Functions sin’x =>cos xcos’x => - sin x tan’x => sec 2 xcot’x => - csc 2 x sec’x => sec x tan xcsc’x

11)What is the limit as t approaches infinity?

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