let $f(x)=ln(3x)+2.5$. what is the average rate of change of $f(x)$ from 2 to 7? round your answer to the…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 2$, $b = 7$, and $f(x)=\ln(3x)+2.5$.

Step2: Calculate $f(7)$

$f(7)=\ln(3\times7)+2.5=\ln(21)+2.5$. Using a calculator, $\ln(21)\approx3.04452$, so $f(7)\approx3.04452 + 2.5=5.54452$.

Step3: Calculate $f(2)$

$f(2)=\ln(3\times2)+2.5=\ln(6)+2.5$. Using a calculator, $\ln(6)\approx1.79176$, so $f(2)\approx1.79176+2.5 = 4.29176$.

Step4: Calculate the average rate of change

$\frac{f(7)-f(2)}{7 - 2}=\frac{(5.54452)-(4.29176)}{5}=\frac{1.25276}{5}=0.250552\approx0.25$.

Answer:

C. 0.25