The problem in the image is asking to find the average velocity of an object over different intervals. The position of the object is given by the function:

$s(t) = -14t^2 + 42t$

Step 1: Formula for Average Velocity

The average velocity $v_{\text{avg}}$ over a time interval $[a, b]$ is given by:

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

Step 2: Solve for the interval $[1, 7]$

• First, compute the position $s(1)$ and $s(7)$.

$s(t) = -14t^2 + 42t$

• $s(1) = -14(1)^2 + 42(1) = -14 + 42 = 28$
• $s(7) = -14(7)^2 + 42(7) = -14(49) + 42(7) = -686 + 294 = -392$

Step 3: Find the average velocity

$v_{\text{avg}} = \frac{s(7) - s(1)}{7 - 1} = \frac{-392 - 28}{6} = \frac{-420}{6} = -70$

Final Answer:

The average velocity of the object over the interval $[1, 7]$ is $-70$.

Would you like more details or have any questions?

Here are five related questions you might want to explore:

1. How do you compute the average velocity over the interval $[1, 6]$?
2. How do you compute the average velocity over the interval $[1, 5]$?
3. What is the physical significance of a negative average velocity?
4. How would you compute the instantaneous velocity at $t = 3$?
5. How would the average velocity change if the position function was modified?

Tip: The average velocity gives a broad sense of how fast something is moving over an interval, but instantaneous velocity gives more precise, moment-by-moment information.