Average slope of a curve If $(x_0,y_0)$ and $(x_0 + \Delta x, y_0 + \Delta y)$ are two points on the curve, then the "average slope" of the curve between these two points is defined as the ration of change in $y$ to the change in $x$, 

average slope= $\frac {\Delta y}{\Delta x}$

Why do we call it the average slope?
 A: Note that $\Delta x$ and $\Delta y$ can be thought of as the change in $x$ and the change in $y$, respectively, between the two points: $P = (x_0, y_0)$ and $Q = (x_0+Δx,y_0+Δy)$.
So dividing $\frac {\Delta y}{\Delta x}$ gives us the average slope when summing the slopes of each point $R$ on the curve, which lies between $P$ and $Q$. It is the value of the change in $y$ over that interval, with respect to the change in $x$ over that same interval.
A: It is literally an average slope.
Let $f$ be a continuous function on an interval $[a,b]$. Then, we define its average over $[a,b]$ to be the number
$$
f_{\operatorname{avg}} := {1 \over b-a} \int_a^b f(t) dt
$$
Let $y = F(x)$ be a curve, which I assume to be at least $C^1$. Fix the starting point $x_0$ and the ending point $x_1=x_0+\Delta x$. Then, the slope at a point $z$ is the derivative $F'(z)$. Thus, the "average slope'' should mean
\begin{equation*}
(F')_{\operatorname{avg}} = {1 \over x_1 - x_0}\int_{x_0}^{x_1} F'(t) dt = {F(x_1) - F(x_0) \over x_1-x_0}.
\end{equation*}
The final equality is the so-called Fundamental Theorem of Calculus.
