Anti- Derivatives.

Question

fot t less than or = 0, let v(t)= 6-4t be the velocity for a particle moving on a straight line. find the average speed and average velocity during the first 5 seconds

 

Answer 1

Average velocity: (1/5) ∫₀⁵ (6-4t) dt = (1/5) [6t-2t²]₀⁵ = -4 m/s

 

Average speed: (1/5) ∫₀⁵ |6-4t| dt = (1/5) ( ∫₀^1.5 (6-4t) dt - ∫_1.5⁵ (6-4t) dt )

= (1/5) ( [6t-2t²]₀^1.5 - [6t-2t²]_1.5⁵ ) = 5.8 m/s

 

Note: average speed is based on distance traveled, which is always positive, hence the absolute value bars. Since the velocity changes direction at t = 1.5 s, the integral needs to be split.