To determine the average rate of change of the function [tex]\( f(x) = \ln(3x) + 2.5 \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 7 \)[/tex], we'll follow these steps:
1. Calculate [tex]\( f(a) \)[/tex] and [tex]\( f(b) \)[/tex] where [tex]\( a = 2 \)[/tex] and [tex]\( b = 7 \)[/tex].
[tex]\[
f(a) = f(2) = \ln(3 \cdot 2) + 2.5 = \ln(6) + 2.5
\][/tex]
[tex]\[
f(b) = f(7) = \ln(3 \cdot 7) + 2.5 = \ln(21) + 2.5
\][/tex]
2. Compute the values of [tex]\( \ln(6) \)[/tex] and [tex]\( \ln(21) \)[/tex].
[tex]\[
\ln(6) \approx 1.79176
\][/tex]
[tex]\[
\ln(21) \approx 3.04452
\][/tex]
3. Substitute these values back into [tex]\( f(a) \)[/tex] and [tex]\( f(b) \)[/tex]:
[tex]\[
f(a) = \ln(6) + 2.5 \approx 1.79176 + 2.5 = 4.29176
\][/tex]
[tex]\[
f(b) = \ln(21) + 2.5 \approx 3.04452 + 2.5 = 5.54452
\][/tex]
4. Find the difference [tex]\( f(b) - f(a) \)[/tex]:
[tex]\[
f(b) - f(a) = 5.54452 - 4.29176 = 1.25276
\][/tex]
5. Calculate the difference [tex]\( b - a \)[/tex]:
[tex]\[
b - a = 7 - 2 = 5
\][/tex]
6. Determine the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a} = \frac{1.25276}{5} \approx 0.250552
\][/tex]
7. Round the result to the nearest hundredth.
[tex]\[
\text{Average rate of change} \approx 0.25
\][/tex]
So, the average rate of change of [tex]\( f(x) \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 7 \)[/tex] is approximately [tex]\( \boxed{0.25} \)[/tex].
