To find the average velocity of the particle during the given time periods, we need to use the formula for average velocity:

$\text{Average velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$

where $s(t)$ is the displacement at time $t$, and $[t_1, t_2]$ is the time interval.

The displacement function is given as:

$s(t) = 4\sin(\pi t) + 3\cos(\pi t)$

Let's calculate the average velocity for each of the specified intervals.

(i) Interval $[1, 2]$

1. Calculate $s(1)$ and $s(2)$: $s(1) = 4\sin(\pi \cdot 1) + 3\cos(\pi \cdot 1) = 4\sin(\pi) + 3\cos(\pi) = 4(0) + 3(-1) = -3$ $s(2) = 4\sin(\pi \cdot 2) + 3\cos(\pi \cdot 2) = 4\sin(2\pi) + 3\cos(2\pi) = 4(0) + 3(1) = 3$

2. Calculate the average velocity: $\text{Average velocity} = \frac{s(2) - s(1)}{2 - 1} = \frac{3 - (-3)}{1} = \frac{6}{1} = 6 \text{ cm/s}$

(ii) Interval $[1, 1.01]$

1. Calculate $s(1.01)$: $s(1.01) = 4\sin(\pi \cdot 1.01) + 3\cos(\pi \cdot 1.01)$ Using the approximations: $\sin(\pi \cdot 1.01) \approx \sin(\pi + 0.01\pi) = -\sin(0.01\pi) \approx -0.0314$ $\cos(\pi \cdot 1.01) \approx \cos(\pi + 0.01\pi) = -\cos(0.01\pi) \approx -0.9995$ $s(1.01) \approx 4(-0.0314) + 3(-0.9995) = -0.1256 - 2.9985 \approx -3.1241$

2. Calculate the average velocity: $\text{Average velocity} = \frac{s(1.01) - s(1)}{1.01 - 1} = \frac{-3.1241 - (-3)}{0.01} = \frac{-0.1241}{0.01} = -12.41 \text{ cm/s}$

(iii) Interval $[1, 1.001]$

1. Calculate $s(1.001)$: $s(1.001) = 4\sin(\pi \cdot 1.001) + 3\cos(\pi \cdot 1.001)$ Using the approximations: $\sin(\pi \cdot 1.001) \approx \sin(\pi + 0.001\pi) = -\sin(0.001\pi) \approx -0.00314$ $\cos(\pi \cdot 1.001) \approx \cos(\pi + 0.001\pi) = -\cos(0.001\pi) \approx -0.999995$ $s(1.001) \approx 4(-0.00314) + 3(-0.999995) = -0.01256 - 2.999985 \approx -3.012545$

2. Calculate the average velocity: $\text{Average velocity} = \frac{s(1.001) - s(1)}{1.001 - 1} = \frac{-3.012545 - (-3)}{0.001} = \frac{-0.012545}{0.001} = -12.55 \text{ cm/s}$

Final Results:

• (i) Interval $[1, 2]$: Average velocity = $6$ cm/s
• (ii) Interval $[1, 1.01]$: Average velocity = $-12.41$ cm/s
• (iii) Interval $[1, 1.001]$: Average velocity = $-12.55$ cm/s

Would you like any further details or explanations?

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Related Questions:

1. What is the instantaneous velocity at $t = 1$ seconds?
2. How does the average velocity change as the time interval decreases?
3. Can you determine the acceleration of the particle at $t = 1$ second?
4. What is the maximum displacement of the particle and at what time does it occur?
5. How would the displacement function change if the sine and cosine coefficients were swapped?

Tip: Average velocity over very short intervals can be used to approximate instantaneous velocity.