To find the average velocity of the object over a given interval $[a, b]$, we use the formula:

$v_{\text{avg}} = \frac{s(b) - s(a)}{b - a}$

The position function is given as: $s(t) = -4t^2 + 20t$

We calculate each part step by step:

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(a) Interval $[1, 7]$

1. Compute $s(7)$ and $s(1)$: $s(7) = -4(7)^2 + 20(7) = -4(49) + 140 = -196 + 140 = -56$ $s(1) = -4(1)^2 + 20(1) = -4(1) + 20 = -4 + 20 = 16$

2. Substitute into the average velocity formula: $v_{\text{avg}} = \frac{s(7) - s(1)}{7 - 1} = \frac{-56 - 16}{6} = \frac{-72}{6} = -12$

Thus, the average velocity over $[1, 7]$ is: $\boxed{-12}$

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(b) Interval $[1, 6]$

1. Compute $s(6)$ and $s(1)$: $s(6) = -4(6)^2 + 20(6) = -4(36) + 120 = -144 + 120 = -24$ $s(1) = 16 \quad \text{(already computed above).}$

2. Substitute into the average velocity formula: $v_{\text{avg}} = \frac{s(6) - s(1)}{6 - 1} = \frac{-24 - 16}{5} = \frac{-40}{5} = -8$

Thus, the average velocity over $[1, 6]$ is: $\boxed{-8}$

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(c) Interval $[1, 5]$

1. Compute $s(5)$ and $s(1)$: $s(5) = -4(5)^2 + 20(5) = -4(25) + 100 = -100 + 100 = 0$ $s(1) = 16 \quad \text{(already computed above).}$

2. Substitute into the average velocity formula: $v_{\text{avg}} = \frac{s(5) - s(1)}{5 - 1} = \frac{0 - 16}{4} = \frac{-16}{4} = -4$

Thus, the average velocity over $[1, 5]$ is: $\boxed{-4}$

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(d) Interval $[1, 1 + h]$

1. Compute $s(1 + h)$ and $s(1)$: $s(1 + h) = -4(1 + h)^2 + 20(1 + h)$ Expand: $s(1 + h) = -4(1 + 2h + h^2) + 20(1 + h) = -4 - 8h - 4h^2 + 20 + 20h$ Simplify: $s(1 + h) = 16 + 12h - 4h^2$ $s(1) = 16 \quad \text{(already computed above).}$

2. Substitute into the average velocity formula: $v_{\text{avg}} = \frac{s(1 + h) - s(1)}{(1 + h) - 1}$ Simplify: $v_{\text{avg}} = \frac{(16 + 12h - 4h^2) - 16}{h} = \frac{12h - 4h^2}{h}$ Factor and simplify: $v_{\text{avg}} = 12 - 4h$

Thus, the average velocity over $[1, 1 + h]$ is: $\boxed{12 - 4h}$

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Summary of Results:

• (a) $-12$
• (b) $-8$
• (c) $-4$
• (d) $12 - 4h$

Let me know if you'd like more details or have any further questions! Here are some related questions for exploration:

1. What is the instantaneous velocity of the object at $t = 1$?
2. How does the average velocity behave as $h \to 0$ in part (d)?
3. Can you determine when the object changes direction based on the position function?
4. What is the displacement of the object over the interval $[1, 7]$?
5. How would the results change if the position function had a constant term?

Tip: The average velocity formula closely resembles the slope of the secant line; interpreting it geometrically helps build intuition.