Consider a twice-differentiable position function that models the motion of a…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Consider a twice-differentiable position function p that models the motion of a particle along a straight track. Suppose p′(c)=0 and p′′(c)<0 for some t=c in the domain of p.
(a) Explain, in full sentences, what the conditions p′(c)=0 and p′′(c)<0 each tell you about the motion of the particle at t=c.
(b) Using the meaning of velocity and acceleration, justify why p(c) must be a local maximum of the position function.
(c) Describe how your reasoning in part (b) would change if instead p′′(c)>0. What motion interpretation would then follow?
(d) Give a verbal description of a real-world situation that could be modeled by a position function satisfying p′(c)=0 and p′′(c)<0 at some instant t=c. Indicate the physical significance of c in your scenario.