what is the average value of e^x on the interval 3,9? choose 1 answer: (a) (e^9 + e^3)/6 (b) (e^9 + e^3)/2…

Explanation:

Step1: Recall average - value formula

The average value of a function $y = f(x)$ on the interval $[a,b]$ is given by $\frac{1}{b - a}\int_{a}^{b}f(x)dx$. Here, $a = 3$, $b = 9$ and $f(x)=e^{x}$, so the average value is $\frac{1}{9 - 3}\int_{3}^{9}e^{x}dx$.

Step2: Integrate $e^{x}$

We know that $\int e^{x}dx=e^{x}+C$. So, $\int_{3}^{9}e^{x}dx=e^{x}\big|_{3}^{9}=e^{9}-e^{3}$.

Step3: Calculate the average value

The average value is $\frac{1}{6}(e^{9}-e^{3})=\frac{e^{9}-e^{3}}{6}$.

Answer:

C. $\frac{e^{9}-e^{3}}{6}$