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Calculus: The Power of Derivatives in Velocity Functions

Discover the significance of derivatives in velocity functions and calculate total distance traveled by a moving particle along the x-axis using numerical, graphical, and analytical methods. Explore how to determine particle movement direction and find position and velocity expressions.

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Calculus: The Power of Derivatives in Velocity Functions

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  1. Calculus The Amazing Power of the Derivative in Velocity Functions Michelle Martin Period 4 AP #26

  2. A particle moves along the x-axis with acceleration given by for all . At , the velocity v(t) of the particle is 2 and the position x(t) is 5. D. Find the total distance traveled by the particle from t = 0 to . A. Write an expression for the velocity v(t) of the particle. B. Write an expression for the position x(t). C. For what values of t is the particle moving to the right? Justify your answer.

  3. Solving the Problem • Numerically • Graphically • Analytically

  4. Knowledge: Given: v(0)=2 Part A Writing the Velocity Function Solution:

  5. Given: Knowledge: x(0)=5 Part B Writing the Position Function Solution:

  6. Knowledge: + v(t) 0 The particle is always moving to the right. Part C Studying Direction of Movement Solution: No critical values

  7. Knowledge: Distance traveled = Area under velocity graph Part D Finding Total Distance Traveled What is the integral of "one over cabin" with respect to "cabin"?Answer: Natural log cabin + c = houseboat.

  8. Distance = Calculus Overload! Part D Finding Total Distance Traveled Part Deux Distance = 4.1416 units

  9. The End

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