The Amazing Power of the Derivative in

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Presentation transcript:

The Amazing Power of the Derivative in Velocity Functions Calculus - Introduce topic Michelle Martin Period 4 AP #26

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. 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. - Read problem to be solved. D. Find the total distance traveled by the particle from t = 0 to .

Solving the Problem • Numerically • Graphically • Analytically - Restate slide.

Writing the Velocity Function Part A Writing the Velocity Function Given: v(0)=2 Knowledge: Solution: - Introduce the given information. - Explain the graph of acceleration. - Tie in the knowledge of how to get a velocity function from the acceleration function. - Explain solution step-by-step.

Writing the Position Function Part B Writing the Position Function Given: x(0)=5 Knowledge: Solution: - Introduce given information and explain graph of velocity. - Position function is the antiderivative of the velocity function. - Explain solution step-by-step.

Studying Direction of Movement Part C Studying Direction of Movement Knowledge: Solution: - State the velocity function. - Explain analytical method to find critical values. - Explain sign study. v(t) + No critical values The particle is always moving to the right.

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

Finding Total Distance Traveled Part Deux Calculus Overload! - Explain solving the integral. - Explain graph of integral of velocity function. Distance = 4.1416 units

The End