To find the estimated average rates of change of the functions \( p(x) \) and \( q(x) \) over the interval \([0.1, 8.9]\), we need to calculate the average rate of change for both functions over this interval.
The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is given by the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
For p(x):
From the table, we have:
- \( p(0) = -5 \)
- \( p(9) = -2 \)
Since the interval we are considering is \([0.1, 8.9]\) and we have no exact values at these points but we can use the values near them from the table.
Using the closest values we have for \( p \):
- \( a = 0 \)
- \( b = 9 \)
Therefore, we calculate:
\[ \text{Average Rate of Change of } p(x) = \frac{p(9) - p(0)}{9 - 0} = \frac{-2 - (-5)}{9 - 0} = \frac{3}{9} = \frac{1}{3} \]
For q(x):
From the table, we have:
- \( q(1) = 0 \)
- \( q(9) = 10 \)
Again, the closest values we can use are:
- \( a = 1 \)
- \( b = 9 \)
Calculating the average rate of change:
\[ \text{Average Rate of Change of } q(x) = \frac{q(9) - q(1)}{9 - 1} = \frac{10 - 0}{9 - 1} = \frac{10}{8} = \frac{5}{4} = 1.25 \]
Conclusion
- The average rate of change for \( p(x) \) is \( \frac{1}{3} \).
- The average rate of change for \( q(x) \) is \( 1.25 \).
Now let's compare these values:
\(\frac{5}{4} > \frac{1}{3}\), which means:
The estimated average rate of change of \( q(x) \) is greater than the estimated average rate of change of \( p(x) \) over \([0.1, 8.9]\).
Thus, the correct response is:
The estimated average rate of change of q(x) is greater than the estimated average rate of change of p(x) over [0.1, 8.9].
1 answer