1. **State the problem:** We have a particle's position function given by $$f(t) = t^2 - \sin(2t)$$ for time $$t$$ in the interval $$0 \leq t \leq 3$$ seconds. We want to analyze the particle's motion by finding: - Position at $$t=0$$ - Velocity (speed and direction) at $$t=0$$ - Acceleration at $$t=0$$ - Whether the particle is accelerating or decelerating at $$t=0$$ - Average velocity over the interval $$[0,3]$$ 2. **Formulas and rules:** - Position is given by $$f(t)$$. - Velocity $$v(t)$$ is the first derivative of position: $$v(t) = f'(t)$$. - Acceleration $$a(t)$$ is the derivative of velocity: $$a(t) = f''(t)$$. - Average velocity over $$[a,b]$$ is $$\frac{f(b)-f(a)}{b-a}$$. - Acceleration and velocity signs determine acceleration/deceleration: if they have the same sign, the particle accelerates; if opposite, it decelerates. 3. **Calculate position at $$t=0$$:** $$f(0) = 0^2 - \sin(0) = 0 - 0 = 0$$ 4. **Find velocity function $$v(t)$$:** $$f(t) = t^2 - \sin(2t)$$ Derivative: $$v(t) = \frac{d}{dt}(t^2) - \frac{d}{dt}(\sin(2t)) = 2t - 2\cos(2t)$$ 5. **Evaluate velocity at $$t=0$$:** $$v(0) = 2\cdot0 - 2\cos(0) = 0 - 2\cdot1 = -2$$ Velocity is $$-2$$, meaning the particle moves in the negative direction at $$t=0$$. 6. **Find acceleration function $$a(t)$$:** $$a(t) = \frac{d}{dt}v(t) = \frac{d}{dt}(2t - 2\cos(2t)) = 2 - 2\cdot(-2\sin(2t)) = 2 + 4\sin(2t)$$ 7. **Evaluate acceleration at $$t=0$$:** $$a(0) = 2 + 4\sin(0) = 2 + 0 = 2$$ Acceleration is $$2$$, positive. 8. **Determine acceleration or deceleration at $$t=0$$:** Velocity is negative ($$-2$$), acceleration is positive ($$2$$), signs differ, so the particle is decelerating at $$t=0$$. 9. **Calculate average velocity over $$[0,3]$$:** $$\text{Average velocity} = \frac{f(3) - f(0)}{3 - 0}$$ Calculate $$f(3)$$: $$f(3) = 3^2 - \sin(6) = 9 - \sin(6)$$ Using $$\sin(6) \approx -0.2794$$ (radians), $$f(3) \approx 9 - (-0.2794) = 9 + 0.2794 = 9.2794$$ So, $$\text{Average velocity} = \frac{9.2794 - 0}{3} = 3.0931$$ **Final answers:** - Position at $$t=0$$: $$0$$ - Velocity at $$t=0$$: $$-2$$ (moving backward) - Acceleration at $$t=0$$: $$2$$ (positive) - Particle is decelerating at $$t=0$$ - Average velocity over $$[0,3]$$: approximately $$3.093$$