A car’s velocity (in m/s) is positive when the car moves east and negative when it…
MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A car’s velocity v(t) (in m/s) is positive when the car moves east and negative when it moves west. Which statement best explains why dividing ∫010v(t)dt by 10 yields the car’s average velocity but may not equal the car’s average speed over the same interval?
(a) Dividing by 10 differentiates the integral, so the result is the car’s average acceleration, not its velocity or speed.
(b) The integral accumulates signed displacement, so motions in opposite directions can cancel; dividing by 10 therefore gives the average velocity, whereas average speed requires the integrand’s magnitude.
(c) Average speed would require averaging v(t)2 to eliminate sign, so 101∫010v(t)dt cannot represent the average speed.
(d) Because ∫010v(t)dt equals the total distance traveled, dividing by 10 gives the average speed rather than the average velocity.