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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?

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    $\begingroup$ because it is the average sum of all slopes from each point at $(x_0,y_0)$ till $(x_0+Δx,y_0+Δy)$ $\endgroup$ – Manoj Pandey May 12 '13 at 14:10
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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.

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  • $\begingroup$ Do you mean that when we sum the slopes of each point $R$ on the curve which lies between $P$ and $Q$ and divide it by the number of slopes, then we shall get the same quantity if we divide $\frac {\Delta y}{\Delta x}$? $\endgroup$ – Samama Fahim May 12 '13 at 14:22
  • $\begingroup$ That's what the average slope approximates. We would need integration to determine what the "sum" of all those slopes amounts, divided by the number of all those points evaluates to. The average slope given above approximates this by dividing the total change in $y$ by the total change in $x$, over the interval between $P$ and $Q$. $\endgroup$ – amWhy May 12 '13 at 14:26
  • $\begingroup$ @amWhy: Looks like they got it! :-) +1 $\endgroup$ – Amzoti May 13 '13 at 0:29
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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.

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