The displacement (in meters) of a particle moving in a straight line is given by $$ s(t)=t^{2}-8 t+18 $$, where $$ t $$ is measured in seconds. (i) Find the average velocity over the time interval [3,4]. Average Velocity = $$ \square $$ meters per second. (ii) Find the average velocity over the time interval $$ [3,5,4] $$. Average Velocity = $$ \square $$ meters per second. (iii) Find the average velocity over the time interval $$ [4,5] $$. Average Velocity = $$ \square $$ meters per second. (iv) Find the average velocity over the time interval $$ [4,4,5] $$. Average Velocity = meters per second. Note: lou can earm partial creait on this problem.

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Let's analyze the displacement function and compute the average velocity for each specified time interval.

Given the displacement function:
$$ s(t) = t^{2} - 8t + 18 $$

The **average velocity** over a time interval $$[a, b]$$ is calculated using the formula:
$$ \text{Average Velocity} = \frac{s(b) - s(a)}{b - a} $$

Let's compute each part step by step.

---

### (i) Average Velocity over the Time Interval $$[3, 4]$$

1. **Calculate $$ s(3) $$ and $$ s(4) $$:**
   $$
   s(3) = 3^{2} - 8(3) + 18 = 9 - 24 + 18 = 3 \, \text{meters}
   $$
   $$
   s(4) = 4^{2} - 8(4) + 18 = 16 - 32 + 18 = 2 \, \text{meters}
   $$

2. **Compute the average velocity:**
   $$
   \text{Average Velocity} = \frac{s(4) - s(3)}{4 - 3} = \frac{2 - 3}{1} = -1 \, \text{meters per second}
   $$

**Answer:**
$$
\text{Average Velocity} = -1 \, \text{meters per second}
$$

---

### (ii) Average Velocity over the Time Interval $$[3, 5, 4]$$

**Interpretation:**
The interval $$[3, 5, 4]$$ suggests a two-segment journey:
1. From $$ t = 3 $$ to $$ t = 5 $$
2. From $$ t = 5 $$ to $$ t = 4 $$

**Steps:**

1. **Calculate $$ s(3) $$, $$ s(5) $$, and $$ s(4) $$:**
   $$
   s(3) = 3^{2} - 8(3) + 18 = 3 \, \text{meters}
   $$
   $$
   s(5) = 5^{2} - 8(5) + 18 = 25 - 40 + 18 = 3 \, \text{meters}
   $$
   $$
   s(4) = 2 \, \text{meters} \quad (\text{from part i})
   $$

2. **Compute total displacement:**
   $$
   \text{From } 3 \text{ to } 5: s(5) - s(3) = 3 - 3 = 0 \, \text{meters}
   $$
   $$
   \text{From } 5 \text{ to } 4: s(4) - s(5) = 2 - 3 = -1 \, \text{meters}
   $$
   $$
   \text{Total Displacement} = 0 + (-1) = -1 \, \text{meters}
   $$

3. **Compute total time:**
   $$
   \text{From } 3 \text{ to } 5: 2 \, \text{seconds}
   $$
   $$
   \text{From } 5 \text{ to } 4: -1 \, \text{seconds} \quad (\text{but time is absolute, so } 1 \, \text{second})
   $$
   $$
   \text{Total Time} = 2 + 1 = 3 \, \text{seconds}
   $$

4. **Compute the average velocity:**
   $$
   \text{Average Velocity} = \frac{-1}{3} \approx -0.333 \, \text{meters per second}
   $$

**Answer:**
$$
\text{Average Velocity} \approx -0.333 \, \text{meters per second}
$$

---

### (iii) Average Velocity over the Time Interval $$[4, 5]$$

1. **Calculate $$ s(4) $$ and $$ s(5) $$:**
   $$
   s(4) = 2 \, \text{meters} \quad (\text{from part i})
   $$
   $$
   s(5) = 3 \, \text{meters} \quad (\text{from part ii})
   $$

2. **Compute the average velocity:**
   $$
   \text{Average Velocity} = \frac{s(5) - s(4)}{5 - 4} = \frac{3 - 2}{1} = 1 \, \text{meter per second}
   $$

**Answer:**
$$
\text{Average Velocity} = 1 \, \text{meter per second}
$$

---

### (iv) Average Velocity over the Time Interval $$[4, 4, 5]$$

**Interpretation:**
The interval $$[4, 4, 5]$$ suggests a two-segment journey:
1. From $$ t = 4 $$ to $$ t = 4 $$
2. From $$ t = 4 $$ to $$ t = 5 $$

**Steps:**

1. **Calculate $$ s(4) $$ and $$ s(5) $$:**
   $$
   s(4) = 2 \, \text{meters} \quad (\text{from part i})
   $$
   $$
   s(5) = 3 \, \text{meters} \quad (\text{from part ii})
   $$

2. **Compute total displacement:**
   $$
   \text{From } 4 \text{ to } 4: s(4) - s(4) = 0 \, \text{meters}
   $$
   $$
   \text{From } 4 \text{ to } 5: s(5) - s(4) = 3 - 2 = 1 \, \text{meter}
   $$
   $$
   \text{Total Displacement} = 0 + 1 = 1 \, \text{meter}
   $$

3. **Compute total time:**
   $$
   \text{From } 4 \text{ to } 4: 0 \, \text{seconds}
   $$
   $$
   \text{From } 4 \text{ to } 5: 1 \, \text{second}
   $$
   $$
   \text{Total Time} = 0 + 1 = 1 \, \text{second}
   $$

4. **Compute the average velocity:**
   $$
   \text{Average Velocity} = \frac{1}{1} = 1 \, \text{meter per second}
   $$

**Answer:**
$$
\text{Average Velocity} = 1 \, \text{meter per second}
$$

---

### Summary of Results

1. **(i)** Average Velocity over $$[3, 4]$$: $$-1 \, \text{m/s}$$
2. **(ii)** Average Velocity over $$[3, 5, 4]$$: $$-0.333 \, \text{m/s}$$
3. **(iii)** Average Velocity over $$[4, 5]$$: $$1 \, \text{m/s}$$
4. **(iv)** Average Velocity over $$[4, 4, 5]$$: $$1 \, \text{m/s}$$

**Note:** If the interpretation of the intervals $$[3, 5, 4]$$ and $$[4, 4, 5]$$ differs from the intended meaning, please provide additional clarification.
**Solutions:** 1. **(i) Average Velocity over [3, 4]:** $$-1$$ meters per second 2. **(ii) Average Velocity over [3, 5, 4]:** $$-0.333$$ meters per second 3. **(iii) Average Velocity over [4, 5]:** $$1$$ meter per second 4. **(iv) Average Velocity over [4, 4, 5]:** $$1$$ meter per second

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