AP Calculus AB: Average Value and Motion Applications — Drill 1

About This Drill

AP Calculus AB: Average Value and Motion Applications — Drill 1 is a Math practice drill covering Applications of Integration. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice applying the average value formula, the Mean Value Theorem for Integrals, and integration-based motion analysis including displacement versus total distance. These AP Calculus AB topics appear on both the multiple-choice and free-response sections.

Questions in This Drill

1. The average value of \( f(x) = 6\sqrt{x} \) on the interval \( [0, 4] \) is:
2. Let \( v(t) = t^2 \) on the interval \( [0, 3] \). The Mean Value Theorem for Integrals guarantees a value \( c \) in \( (0, 3) \) such that \( v(c) \) equals the average value of \( v \) on \( [0, 3] \). What is the value of \( c \)?
3. A particle moves along the x-axis with velocity \( v(t) \) feet per second. On the interval \( [0, 5] \), the velocity is negative for \( 0 < t < 3 \) and positive for \( 3 < t < 5 \). Which of the following expressions gives the total distance traveled by the particle over \( [0, 5] \)?
4. A particle moves along a straight line with velocity \( v(t) = t^2 - 6t + 8 \) feet per second for \( 0 \le t \le 5 \). What is the total distance traveled by the particle?
5. A particle moves along the x-axis with velocity \( v(t) = 2t + 1 \) feet per second. If the particle's position at \( t = 0 \) is \( x(0) = 3 \), what is the particle's position at \( t = 4 \)?