To determine the estimated average rates of change for the functions \( p(x) = \sqrt{x} - 5 \) and \( q(x) = \frac{\sqrt{5x - 1}}{3} \) over the interval \([0.1, 8.9]\), we will first find the average rate of change for each function in that interval. The average rate of change of a function \( f(x) \) from \( a \) to \( b \) is given by the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
1. Calculating for \( p(x) \)
Let \( a = 0.1 \) and \( b = 8.9 \).
Evaluate \( p(0.1) \): \[ p(0.1) = \sqrt{0.1} - 5 \approx 0.316 - 5 \approx -4.684 \]
Evaluate \( p(8.9) \): \[ p(8.9) = \sqrt{8.9} - 5 \approx 2.983 - 5 \approx -2.017 \]
Now we calculate the average rate of change for \( p(x) \): \[ \text{Average Rate of Change of } p = \frac{p(8.9) - p(0.1)}{8.9 - 0.1} = \frac{-2.017 - (-4.684)}{8.8} = \frac{2.667}{8.8} \approx 0.303 \]
2. Calculating for \( q(x) \)
Now we compute for \( q(x) \):
Evaluate \( q(0.1) \): \[ q(0.1) = \frac{\sqrt{5(0.1) - 1}}{3} = \frac{\sqrt{0.5 - 1}}{3} \] Since \( 0.5 - 1 < 0 \), \( q(0.1) \) is not defined.
Instead, let's check \( x = 1 \) and \( x = 8 \):
Evaluate \( q(1) \): \[ q(1) = \frac{\sqrt{5(1) - 1}}{3} = \frac{\sqrt{4}}{3} = \frac{2}{3} \]
Evaluate \( q(8.9) \): \[ q(8.9) = \frac{\sqrt{5(8.9) - 1}}{3} = \frac{\sqrt{44.5 - 1}}{3} = \frac{\sqrt{43.5}}{3} \approx \frac{6.6}{3} \approx 2.2 \]
Now calculate the average rate of change for \( q(x) \): \[ \text{Average Rate of Change of } q = \frac{q(8.9) - q(1)}{8.9 - 1} = \frac{2.2 - \frac{2}{3}}{7.9} \approx \frac{2.2 - 0.667}{7.9} \approx \frac{1.533}{7.9} \approx 0.194 \]
Conclusion
After comparing the average rates of change:
- Average Rate of Change of \( p(x) \approx 0.303 \)
- Average Rate of Change of \( q(x) \approx 0.194 \)
Thus, the results indicate that:
The estimated average rate of change of \( q(x) \) is less than the estimated average rate of change of \( p(x) \) over \([0.1, 8.9]\).