1. **Problem statement:** For the position function $s = t^2 - 3t + 2$ on the interval $0 \leq t \leq 2$, find: a. The displacement and average velocity over the interval. c. When, if ever, the body changes direction during the interval. 2. **Displacement:** Displacement is the change in position from the start to the end of the interval. Calculate $s(0)$ and $s(2)$: $$s(0) = 0^2 - 3\cdot0 + 2 = 2$$ $$s(2) = 2^2 - 3\cdot2 + 2 = 4 - 6 + 2 = 0$$ Displacement $= s(2) - s(0) = 0 - 2 = -2$ meters. 3. **Average velocity:** Average velocity is displacement divided by time interval length. $$\text{Average velocity} = \frac{\text{displacement}}{\Delta t} = \frac{-2}{2 - 0} = -1 \text{ m/s}$$ 4. **Change of direction:** The body changes direction when velocity changes sign. Velocity is the derivative of position: $$v(t) = \frac{ds}{dt} = 2t - 3$$ Find when $v(t) = 0$: $$2t - 3 = 0 \Rightarrow t = \frac{3}{2} = 1.5$$ Check if $t=1.5$ is in the interval $[0,2]$: yes. Check velocity sign before and after $t=1.5$: - At $t=1$, $v(1) = 2(1) - 3 = -1$ (negative) - At $t=2$, $v(2) = 4 - 3 = 1$ (positive) Velocity changes from negative to positive, so the body changes direction at $t=1.5$ seconds. --- 5. **Problem statement:** For the position function $s = 6t - t^2$ on the interval $0 \leq t \leq 6$, find: a. The displacement and average velocity over the interval. c. When, if ever, the body changes direction during the interval. 6. **Displacement:** Calculate $s(0)$ and $s(6)$: $$s(0) = 6\cdot0 - 0^2 = 0$$ $$s(6) = 6\cdot6 - 6^2 = 36 - 36 = 0$$ Displacement $= s(6) - s(0) = 0 - 0 = 0$ meters. 7. **Average velocity:** $$\text{Average velocity} = \frac{0}{6 - 0} = 0 \text{ m/s}$$ 8. **Change of direction:** Velocity is derivative of position: $$v(t) = \frac{ds}{dt} = 6 - 2t$$ Find when $v(t) = 0$: $$6 - 2t = 0 \Rightarrow t = 3$$ Check if $t=3$ is in $[0,6]$: yes. Check velocity sign before and after $t=3$: - At $t=2$, $v(2) = 6 - 4 = 2$ (positive) - At $t=4$, $v(4) = 6 - 8 = -2$ (negative) Velocity changes from positive to negative, so the body changes direction at $t=3$ seconds. **Final answers:** - For $s = t^2 - 3t + 2$, displacement = $-2$ m, average velocity = $-1$ m/s, changes direction at $t=1.5$ s. - For $s = 6t - t^2$, displacement = $0$ m, average velocity = $0$ m/s, changes direction at $t=3$ s.