We are given the displacement equation $s(t) = 2 \sin(\pi t) + 3 \cos(\pi t)$, where $s$ is the displacement in centimeters, and $t$ is the time in seconds.

(a) Average Velocity

The average velocity $v_{avg}$ over a time interval $[t_1, t_2]$ is given by:

$v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$

We will compute this for the four intervals:

(i) [1, 2]

$v_{avg} = \frac{s(2) - s(1)}{2 - 1}$

(ii) [1, 1.1]

$v_{avg} = \frac{s(1.1) - s(1)}{1.1 - 1}$

(iii) [1, 1.01]

$v_{avg} = \frac{s(1.01) - s(1)}{1.01 - 1}$

(iv) [1, 1.001]

$v_{avg} = \frac{s(1.001) - s(1)}{1.001 - 1}$

(b) Instantaneous Velocity at $t = 1$

The instantaneous velocity is the derivative of the displacement function $s(t)$. First, we calculate the derivative $s'(t)$:

$s'(t) = \frac{d}{dt}\left(2 \sin(\pi t) + 3 \cos(\pi t)\right)$

Using the chain rule: $s'(t) = 2 \pi \cos(\pi t) - 3 \pi \sin(\pi t)$

To find the instantaneous velocity at $t = 1$, we substitute $t = 1$ into $s'(t)$:

$s'(1) = 2 \pi \cos(\pi \times 1) - 3 \pi \sin(\pi \times 1)$

Now, let's calculate the values. I will solve these step by step.### Results:

(a) Average velocity for each time period:

1. For $[1, 2]$: $v_{avg} = 6 \, \text{cm/s}$
2. For $[1, 1.1]$: $v_{avg} \approx -4.71 \, \text{cm/s}$
3. For $[1, 1.01]$: $v_{avg} \approx -6.13 \, \text{cm/s}$
4. For $[1, 1.001]$: $v_{avg} \approx -6.27 \, \text{cm/s}$

(b) Instantaneous velocity at $t = 1$:

The instantaneous velocity is approximately $v(1) \approx -6.28 \, \text{cm/s}$.

Do you need further details or clarifications on any part?
Here are some related questions:

1. How does the instantaneous velocity relate to the average velocity as the time interval shrinks?
2. Can you derive the derivative of the displacement function step by step?
3. What is the physical meaning of negative velocity in this case?
4. How would the results change if we used different trigonometric functions in the displacement formula?
5. What is the behavior of the velocity as $t \to \infty$?

Tip: Instantaneous velocity can be thought of as the limit of average velocity over smaller and smaller time intervals.