Let's go step by step to solve the given problem.

The displacement function $s(t)$ of a particle is given by:

$s(t) = t^2 - 7t + 15$

(a) Find the average velocity over each time interval:

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

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

Let’s compute the average velocity for each sub-interval.

(i) For the interval $[3, 4]$:

• $s(3) = 3^2 - 7(3) + 15 = 9 - 21 + 15 = 3$
• $s(4) = 4^2 - 7(4) + 15 = 16 - 28 + 15 = 3$

Now calculate the average velocity:

$v_{avg} = \frac{s(4) - s(3)}{4 - 3} = \frac{3 - 3}{1} = 0 \, \text{m/s}$

(ii) For the interval $[3.5, 4]$:

• $s(3.5) = (3.5)^2 - 7(3.5) + 15 = 12.25 - 24.5 + 15 = 2.75$
• $s(4) = 3$ (as computed earlier)

Now calculate the average velocity:

$v_{avg} = \frac{s(4) - s(3.5)}{4 - 3.5} = \frac{3 - 2.75}{0.5} = 0.5 \, \text{m/s}$

(iii) For the interval $[4, 5]$:

• $s(4) = 3$ (as computed earlier)
• $s(5) = 5^2 - 7(5) + 15 = 25 - 35 + 15 = 5$

Now calculate the average velocity:

$v_{avg} = \frac{s(5) - s(4)}{5 - 4} = \frac{5 - 3}{1} = 2 \, \text{m/s}$

(iv) For the interval $[4, 4.5]$:

• $s(4) = 3$ (as computed earlier)
• $s(4.5) = (4.5)^2 - 7(4.5) + 15 = 20.25 - 31.5 + 15 = 3.75$

Now calculate the average velocity:

$v_{avg} = \frac{s(4.5) - s(4)}{4.5 - 4} = \frac{3.75 - 3}{0.5} = 1.5 \, \text{m/s}$

(b) Find the instantaneous velocity when $t = 4$.

The instantaneous velocity is the derivative of the displacement function $s(t)$, which gives the velocity function $v(t)$:

$v(t) = \frac{ds}{dt} = 2t - 7$

Now, plug in $t = 4$:

$v(4) = 2(4) - 7 = 8 - 7 = 1 \, \text{m/s}$

Summary of answers:

(a) Average velocities:

1. $[3, 4]$: $0 \, \text{m/s}$
2. $[3.5, 4]$: $0.5 \, \text{m/s}$
3. $[4, 5]$: $2 \, \text{m/s}$
4. $[4, 4.5]$: $1.5 \, \text{m/s}$

(b) Instantaneous velocity at $t = 4$: $1 \, \text{m/s}$

Let me know if you need further details on any part, or have any questions!

Related Questions:

1. What is the meaning of average velocity over an interval?
2. How do you differentiate a quadratic function?
3. What is the geometric interpretation of instantaneous velocity?
4. How does the velocity change over time for this particle?
5. How do you calculate displacement from a velocity function?

Tip: When working with displacement or velocity functions, remember that the instantaneous velocity is the derivative of the displacement, and acceleration is the derivative of the velocity!