MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
On the interval 0≤t≤8, three twice-differentiable functions g, h, and k are defined by
g(t)=3t2−24t+36h(t)=t3−12t2+36t+5k(t)=6t−24
Exactly one of these functions models the height s(t), in meters, of a projectile above ground at time t seconds. The other two are its first and second derivatives with respect to time.
(1) Part 1: Determine which function is s, which is s′ (velocity), and which is s′′ (acceleration). Justify your choices analytically.
(2) Part 2: Using the function you identified as s, determine the time at which the projectile reaches its maximum height and compute that maximum height.
(3) Part 3: Determine the interval(s) on which the projectile’s speed is increasing. Show all reasoning based on the relationships among s, s′, and s′′.
(4) Part 4: A safety regulation requires that the magnitude of the acceleration remain below 15m/s2 for the entire flight. Does the motion modeled above satisfy this regulation? Explain fully.