Mean Value Theorem/Examples/x^3/Formulation 1

Example of Use of Mean Value Theorem

Let $f$ be the real function defined as:

$\map f x = x^3$

Let:

$a = 1$, $b = 2$

Then when $\xi = \sqrt {\dfrac 7 3}$:

$\map {f'} \xi = \dfrac {\map f b - \map f a} {b - a}$

Proof

We have:

\(\ds \map f a\)

\(=\)

\(\ds 1\)

\(\ds \map f b\)

\(=\)

\(\ds 8\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac {\map f b - \map f a} {b - a}\)

\(=\)

\(\ds \dfrac {8 - 1} {2 - 1}\)

\(\ds \leadsto \ \ \)

\(\ds \map {f'} \xi\)

\(=\)

\(\ds 7\)

Mean Value Theorem

\(\ds \)

\(=\)

\(\ds 3 \xi^2\)

Power Rule for Derivatives

\(\ds \leadsto \ \ \)

\(\ds \xi\)

\(=\)

\(\ds \sqrt {\dfrac 7 3}\)

$\blacksquare$

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.4$ Law of the Mean: Example $\text G$