Math Insight

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.