Single Variable Calculus, Chapter 3, 3.7, Section 3.7, Problem 6

Below are the graphs of the position functions of two particles, where $t$ is measured in seconds. When is each particle speeding up? When is it slowing down? Explain.















Since the position of the particle is relevant, we must graph first the velocity and acceleration functions. And we know that the particle is speeding up when the velocity and acceleration have the same sign (either in positive or negative direction). On the other hand, the particle is slowing down when the velocity and acceleration have opposite sign.







Hence, the particle is speeding up at interval $1 < t < 2$ and $3 < t \leq 4$ while the particle is slowing down at interval $0 \leq t \leq 1$ and $2 \leq t \leq 3$.








Based from the graph, the particle is speeding up at intervals $1 \leq t \leq 2$ and $3 \leq t \leq 4$ while it shows down at interval $0 \leq t < 1$ and $2 < t < 3$