1. **State the problem:** We are given velocity graphs of two particles and asked to determine intervals when each particle is speeding up or slowing down. 2. **Recall the rule:** A particle is speeding up when velocity and acceleration have the same sign (both positive or both negative). 3. **Recall the rule:** A particle is slowing down when velocity and acceleration have opposite signs. 4. **Analyze (a):** - Velocity $v(t)$ starts at 0, rises to a positive peak before $t=1$, then falls, crossing zero just after $t=1$, and becomes negative. - Acceleration $a(t)$ is the derivative of velocity, so it is positive when velocity is increasing and negative when velocity is decreasing. 5. **Find intervals for (a):** - For $0 < t < 1$, velocity is increasing (positive acceleration), and velocity is positive, so speeding up on $(0,1)$. - For $t > 1$, velocity is negative and decreasing (velocity negative, acceleration negative), so speeding up on $(1,\infty)$. - For $t$ just after 0 to 1, velocity positive but acceleration negative (velocity decreasing), so slowing down on $(1,\text{just after }1)$ is not correct; actually velocity decreases after peak but before crossing zero, so slowing down on $(1,\text{just after }1)$ is empty. - Actually, velocity decreases from peak to zero crossing, so velocity positive but acceleration negative means slowing down on $(\text{peak time},1)$. 6. **Summarize (a):** - Speeding up: $(0, t_{peak}) \cup (1, \infty)$ - Slowing down: $(t_{peak}, 1)$ 7. **Analyze (b):** - Velocity starts positive, peaks before $t=1$, then drops below zero after $t=1$. - Acceleration positive when velocity increasing, negative when velocity decreasing. 8. **Find intervals for (b):** - Speeding up when velocity and acceleration same sign. - From $0$ to peak: velocity positive and increasing, speeding up. - From peak to $1$: velocity positive but decreasing, slowing down. - After $1$: velocity negative and decreasing, speeding up. 9. **Final answers:** - (a) Speeding up: $(0, t_{peak}) \cup (1, \infty)$ - (a) Slowing down: $(t_{peak}, 1)$ - (b) Speeding up: $(0, t_{peak}) \cup (1, \infty)$ - (b) Slowing down: $(t_{peak}, 1)$ Since exact $t_{peak}$ is not given, approximate as $t_{peak} \approx 0.7$ for (a) and (b). **Answer:** (a) Speeding up: $(0,0.7) \cup (1,\infty)$ (a) Slowing down: $(0.7,1)$ (b) Speeding up: $(0,0.7) \cup (1,\infty)$ (b) Slowing down: $(0.7,1)$