8.6

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1. The Cauchy Mean-Value Theorem

For a review of the mean-value theorem see Section 5.1 Theorem 5.1.

where D₂, C₂, G₂, and F₂ are the counterparts for point c of D, C, G, and F respectively.

Fig. 1.1

The property expressed by Eq. [1.1] is called the Cauchy mean-value theorem or the generalized mean-value theorem,
because from Eq. [1.1] we get:

Also note that Eq. [1.1] says that the ratio of the total changes of f and g on [a, b] is equal to the ratio of their
instantaneous rates of change at c.

Theorem 1.1 – The Cauchy (Or Generalized) Mean-Value Theorem

Proof
Let:

h(x) = ( f(b) – f(a))g(x) – (g(b) – g(a)) f(x).

Clearly h is continuous on [a, b] and differentiable on (a, b). Also:

h(a) = ( f(b) – f(a))g(a) – (g(b) – g(a)) f(a) = f(b)g(a) – f(a)g(b)

and:

h(b) = ( f(b) – f(a))g(b) – (g(b) – g(a)) f(b) = f(b)g(a) – f(a)g(b) = h(a).

So, by Rolle's theorem, there exists c in (a, b) such that h'(c) = 0. We have:

h'(x) = ( f(b) – f(a))g'(x) – (g(b) – g(a)) f '(x),

yielding:

h'(c) = ( f(b) – f(a))g'(c) – (g(b) – g(a)) f '(c).

Thus:

( f(b) – f(a))g'(c) = (g(b) – g(a)) f '(c).

EOP

The Cauchy mean-value theorem is also called the generalized mean-value theorem.

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In Section 1.1.3 we encountered limits of the indeterminate quotient form of type 0/0, and in Section 1.1.7 we

Remark that the right-hand side is the quotient f '/g' of the derivatives f ' and g', not the derivative of the quotient f/g,
which is ( f 'g – fg' )/g².

If:

Proof

The proofs for one-sided limits are contained in this proof. The proofs for limits at infinity are similar to this proof.
EOP

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Example 3.1

Find:

Solution

EOS

Applying More Than Once

Example 3.2

Find:

Solution

EOS

Arranging Into Forms For The Rule

Example 3.3

Evaluate:

where a is a positive constant.

Solution

EOS

Trying To Simplify Before Trying To Apply The Rule

In Example 3.3, the limit:

So before trying to apply the rule we should first try to simplify the limit's expression as much as possible. After
simplification it may or may not turn out that there's no more indeterminate form. If you just keep applying the rule
without trying to simplify first, the process may never end.

For Functions Of The Form ( f (x)) ^g⁽^x⁾