MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 added 326 characters in body

I think your criticisms of calculus textbooks are on the mark.

In my view, if you are teaching students how to use calculus and not how to prove every assertion in the subject, there is no reason to state or use the Mean Value Theorem. If you must state it, the visual "proof" is best. Some claim it is needed for deriving the error term for a Taylor polynomial, but in my view the integral representation of the error is far more useful and easily derived from the Fundamental Theorem of Calculus.

(Wandering off topic here) The standard calculus textbook is, in my opinion, a confused logically inconsistent combination of rigorous reasoning and take-it-on-faith assertions. For example, a lot of textbooks devote a lot of attention to showing how to define $e$ and the exponential function rigorously (to the point that some textbooks find it necessary to define the logarithm first!). On the other hand, I have never seen any textbook worry about how to define degrees and radians rigorously.

To elaborate a little, many uses of the MVT:

There exists $c \in [a, b]$ such that $f(b) - f(a) = f'(c)(b-a)$

can be achieved just as well by the fundamental theorem of calculus:

$f(b) - f(a) = A(b-a),$

where

$A = \frac{1}{b-a}\int_a^b f'(t)\,dt$

is the average derivative of $f$ on the interval $[a,b]$.

Post Made Community Wiki by S. Carnahan
1

I think your criticisms of calculus textbooks are on the mark.

In my view, if you are teaching students how to use calculus and not how to prove every assertion in the subject, there is no reason to state or use the Mean Value Theorem. If you must state it, the visual "proof" is best. Some claim it is needed for deriving the error term for a Taylor polynomial, but in my view the integral representation of the error is far more useful and easily derived from the Fundamental Theorem of Calculus.

(Wandering off topic here) The standard calculus textbook is, in my opinion, a confused logically inconsistent combination of rigorous reasoning and take-it-on-faith assertions. For example, a lot of textbooks devote a lot of attention to showing how to define $e$ and the exponential function rigorously (to the point that some textbooks find it necessary to define the logarithm first!). On the other hand, I have never seen any textbook worry about how to define degrees and radians rigorously.