To solve this problem, we'll break it into two parts. The first part involves finding the average velocity over specific time intervals, and the second part requires estimating the instantaneous velocity at $t = 1$.

Part A: Average Velocity

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

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

The displacement function is given by:

$s(t) = 5\sin(\pi t) + 4\cos(\pi t)$

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

Interval $[1, 2]$

• $t_1 = 1$, $t_2 = 2$
• $s(1) = 5\sin(\pi \times 1) + 4\cos(\pi \times 1)$
• $s(2) = 5\sin(\pi \times 2) + 4\cos(\pi \times 2)$

Interval $[1, 1.1]$

• $t_1 = 1$, $t_2 = 1.1$
• $s(1.1) = 5\sin(\pi \times 1.1) + 4\cos(\pi \times 1.1)$

Interval $[1, 1.01]$

• $t_1 = 1$, $t_2 = 1.01$
• $s(1.01) = 5\sin(\pi \times 1.01) + 4\cos(\pi \times 1.01)$

Interval $[1, 1.001]$

• $t_1 = 1$, $t_2 = 1.001$
• $s(1.001) = 5\sin(\pi \times 1.001) + 4\cos(\pi \times 1.001)$

Part B: Instantaneous Velocity

The instantaneous velocity at $t = 1$ can be found by taking the derivative of $s(t)$ and evaluating it at $t = 1$.

$v(t) = \frac{ds}{dt} = \frac{d}{dt} \left( 5\sin(\pi t) + 4\cos(\pi t) \right)$

$v(t) = 5\pi \cos(\pi t) - 4\pi \sin(\pi t)$

Evaluating this at $t = 1$:

$v(1) = 5\pi \cos(\pi \times 1) - 4\pi \sin(\pi \times 1)$

I'll calculate these values now.### Part A: Average Velocity

The average velocity over the given time intervals is:

• $[1, 2]$: $v_{\text{avg}} = 8.00 \, \text{cm/s}$
• $[1, 1.1]$: $v_{\text{avg}} = -13.49 \, \text{cm/s}$
• $[1, 1.01]$: $v_{\text{avg}} = -15.51 \, \text{cm/s}$
• $[1, 1.001]$: $v_{\text{avg}} = -15.69 \, \text{cm/s}$

Part B: Instantaneous Velocity

The instantaneous velocity of the particle at $t = 1$ is approximately:

$v(1) \approx -15.71 \, \text{cm/s}$

This shows that the velocity is negative, indicating that the particle is moving in the opposite direction at $t = 1$.

Would you like more details on these calculations or have any further questions?

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Here are some related questions you might find interesting:

1. How does the instantaneous velocity relate to the average velocity as the interval decreases?
2. What does the negative sign of the velocity indicate about the direction of motion?
3. How would the displacement function change if we added a phase shift to the sine and cosine terms?
4. What would be the impact on velocity if the amplitude of the sine or cosine terms was altered?
5. How does the velocity behave for other values of $t$ around 1?

Tip: The instantaneous velocity at a point is the slope of the tangent to the displacement-time graph at that point.