### Historical and theoretical comments: Mean Value Theorem

For several reasons, the traditional way that *Taylor
polynomials* are taught gives the impression that the ideas are
inextricably linked with issues about *infinite series*. This is
not so, but every calculus book I know takes that approach.
The reasons for this systematic mistake are complicated. Anyway, we
will *not* make that mistake here, although we may
talk about infinite series later.

Instead of following the tradition, we will immediately talk about
Taylor polynomials, *without* first tiring ourselves over infinite
series, and *without* fooling anyone into thinking that Taylor
polynomials have the infinite series stuff as prerequisite!

The theoretical underpinning for these facts about Taylor
polynomials is *The Mean Value Theorem*, which itself depends upon
some fairly subtle properties of the real numbers. It asserts that,
*for a function $f$ differentiable on an interval $[a,b]$, there is a
point $c$ in the interior $(a,b)$ of this interval so that*
$$f'(c)={f(b)-f(a)\over b-a}$$

Note that the latter expression is the formula for the slope
of the ‘chord’ or ‘secant’ line connecting the two points $(a,f(a))$
and $(b,f(b))$ on the graph of $f$. And the $f'(c)$ can be interpreted
as the slope of the *tangent* line to the curve at the point
$(c,f(c))$.

In many traditional scenarios a person is expected to commit the statement of the Mean Value Theorem to memory. And be able to respond to issues like ‘Find a point $c$ in the interval $[0,1]$ satisfying the conclusion of the Mean Value Theorem for the function $f(x)=x^2$.’ This is pointless and we won't do it.

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