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This article says the following:

To find the slope at the desired point, the choice of the second point needed to calculate the ratio represents a difficulty because, in general, the ratio will represent only an average slope between the points, rather than the actual slope at either point (see figure).

I have simplified this as follows:

To find the slope at the desired point we need a second point to calculate the ratio. The choice of the second point represents difficulty. Because, in general, the ratio will not represent the actual slope at either point. Rather, it will represent an average slope between the points.

What is the "average slope"? What is the "actual slope"? What is the difference between these two?

user366312
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2 Answers2

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If we have two points $(x_1,y_1)$ and $(x_2,y_2)$, then the average slope is referring to $$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$$ and the actual slope at the point $(x_1,y_1)$ is simply $$\left.\frac{dy}{dx}\right|_{x=x_1}$$ Take this graph as an example: enter image description here the slope of the red line segment represents the average slope between points $B$ and $C$; the slopes of the orange and the green lines represent the actual slope at the points $C$ and $B$ respectively.

5201314
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When $f:\>[a,b]\to{\mathbb R}$ is convex then $$f'(a)<{f(b)-f(a)\over b-a}<f'(b)\ .$$ In this way the difference quotient ${f(b)-f(a)\over b-a}$ can be seen as a sort of "mean" or "average" between $f'(a)$ and $f'(b)$. But more is true: There is actually a point $\xi\in\>]a,b[\>$ such that $${f(b)-f(a)\over b-a}=f'(\xi)\ .$$ The word average slope is also justified by the following equation: $${f(b)-f(a)\over b-a}={1\over b-a}\int_a^b f'(t)\>dt\ .$$ Here on the RHS the actual integral mean, or average, of the pointwise slopes between $a$ and $b$ is computed.

Christian Blatter
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