Question

complete the following table.
time interval | average velocity
1, 2 | 56
1, 1.5 | 64
1, 1.1 | 70.4
1, 1.01 | 71.84
1, 1.001 | 71.984
(type exact answers. type integers or decimals.)
the value of the instantaneous velocity at t = 1 is
(round to the nearest integer as needed.)

Explanation:

Step1: Observe the trend of average velocities

As the time interval around \( t = 1 \) gets smaller (from \([1,2]\) to \([1,1.0001]\)), the average velocity values are approaching a limit. The values are 56, 64, 70.4, 71.84, 71.984. As the interval length decreases, the average velocity is getting closer to 72.

Step2: Determine the instantaneous velocity

Instantaneous velocity at \( t = 1 \) is the limit of the average velocities as the time interval around \( t = 1 \) approaches zero. From the given values, as the interval becomes very small (like \([1,1.0001]\)), the average velocity is 71.984, which is very close to 72. Rounding to the nearest integer, we get 72.

Answer:

72