Consider the curve $\displaystyle \sqrt{x - 2}$
Find the average slop of the curve from:
x = 2 to x = 2.5
What am I supposed to do here exactly?
It is just asking you to find the slope between the points
$\displaystyle (2,f(2))$ and $\displaystyle (2.5,f(2.5))$
This gives
$\displaystyle (2,0)$ and $\displaystyle (2.5,\sqrt{.5})$
Now we know that the slope between two points is
$\displaystyle \displaystyle m=\frac{y_2-y_1}{x_2-x_1}$
I started wondering if the "average of the slope", that is, the average of the slope function over an interval, wasn't what was wanted rather than just the slope between two points on the graph. It turns out they are the same!
The "average" value of a function, f(x), between x= a and x= b, is given by $\displaystyle \frac{\int_a^b f(x)dx}{b-a}$. Here you are asked to find the average value of the slope at each point which is just f'(x)- that is, you want $\displaystyle \frac{\int_a^b f'(x)dx}{b- a}$.
But by the "Fundamental Theorem of Calculus", $\displaystyle \int_a^b f'(x)dx= f(b)- f(a)$ so that is really just $\displaystyle \frac{f(b)- f(a)}{b- a}$ as TheEmptySet said!