More Calculus of a Single Variable by Peter R. Mercer

Author:Peter R. Mercer
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

With h as the independent variable, we apply L’Hospital’s Rule (Theorem 5.​13) to get

Now if f ′′ were continuous, we could apply L’Hospital’s Rule again to get

But since we only assumed that f ′′ exists we must take more care. Instead, we write

and this just as we wanted to show. ◇

Remark 8.5.

The quantity

is the average of the right-hand and left-hand derivatives of f at x. (See Exercise 4.​10.) This average is called the Schwarz derivative, or the symmetric derivative. It may exist even when f ′ (x) does not: Consider at x = 0. For Rolle – type theorems and Mean Value – type theorems for the symmetric derivative, see [2]. For Flett’s Mean Value Theorem (Theorem 7.​4) as regards the symmetric derivative, see [39]. See also Exercise 8.17. ∘

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