Questions: Heather is driving a racecar. The table below gives the distance D(t) (in meters) she has driven Time t (seconds) Distance D(t) (meters) ---------------------------------------- 0 0 2 78.6 5 151.5 7 205.1 9 255.9 (a) Find the average rate of change for the distance driven from 0 seconds to 2 seconds. meters per second (b) Find the average rate of change for the distance driven from 5 seconds to 9 seconds. meters per second

Solution Steps

To find the average rate of change of a function over a given interval, we use the formula:

\[ \text{Average Rate of Change} = \frac{D(t_2) - D(t_1)}{t_2 - t_1} \]

(a) For the interval from 0 to 2 seconds, use the distances at these times.

(b) For the interval from 5 to 9 seconds, use the distances at these times.

To find the average rate of change of distance from \( t = 0 \) seconds to \( t = 2 \) seconds, we use the formula:

\[ \text{Average Rate of Change} = \frac{D(2) - D(0)}{2 - 0} \]

Substituting the values:

\[ \text{Average Rate of Change} = \frac{78.6 - 0}{2 - 0} = \frac{78.6}{2} = 39.3 \text{ m/s} \]

Next, we calculate the average rate of change of distance from \( t = 5 \) seconds to \( t = 9 \) seconds using the same formula:

\[ \text{Average Rate of Change} = \frac{D(9) - D(5)}{9 - 5} \]

Substituting the values:

\[ \text{Average Rate of Change} = \frac{255.9 - 151.5}{9 - 5} = \frac{104.4}{4} = 26.1 \text{ m/s} \]

Final Answer