Thread: Average Rate of Change of a Function

1. Average Rate of Change of a Function

I will take Jhevon's advice and ask as many questions as I want. I am having trouble with a problem. The problem is:

What is the average rate of change of the function $f$ defined by $f(x)=100\cdot 2^x$ on the interval $[0,4]$?
I know the average value of a function to be $Avg(f(x))=\frac{1}{b-a}\int_a^bf(x)dx$ so I assume that the average value of the rate of change of the function is given by $Avg(f'(x)) = \frac{1}{b-a}\int_a^bf'(x)dx$ which is just $\frac{1}{b-a}f(x)\big|_a^b=25(2^4-2^0)=25(15)=375$. Is my reasoning correct?

2. One does not need an integral to find the average rate of change of a function.
The average rate of change of $f(x)$ on $\left[ {a,b} \right]$ is $\frac{{f(b) - f(a)}}{{b - a}}$.

3. Which is just what I did. So the average value of a function $\frac{1}{b-a}\int_a^bf(x)dx$?

What would the average value of a function's integral be in the closed interval $[a,b]$? $\frac{1}{b-a}\int_a^b\left(\int f(x)dx\right)dx$? Or does it make no sense to take the average value of an integral? If it does, are there any applications you know of? Thanks

4. Originally Posted by rualin
Or does it make no sense to take the average value of an integral? If it does, are there any applications you know of? Thanks
The point is: what you are doing is circular.