Mean Value Theorem/Examples/x^3/Formulation 2

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Example of Use of Mean Value Theorem

Let $f$ be the real function defined as:

$\map f x = x^3$

Let:

$c = 2$, $h = -1$

Then when $\theta = 2 - \sqrt {\dfrac 7 3}$:

$\map {f'} {c + \theta h} = \dfrac {\map f {c + h} - \map f c} h$


Proof

We have:

\(\ds \map f {c + h}\) \(=\) \(\ds \map f 1\)
\(\ds \) \(=\) \(\ds 1\)
\(\ds \map f c\) \(=\) \(\ds 8\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {\map f {c + h} - \map f c} h\) \(=\) \(\ds \dfrac {1 - 8} {-1}\)
\(\ds \leadsto \ \ \) \(\ds \map {f'} {c + \theta h}\) \(=\) \(\ds 7\) Mean Value Theorem
\(\ds \) \(=\) \(\ds 3 \paren {c + \theta h}^2\) Power Rule for Derivatives
\(\ds \leadsto \ \ \) \(\ds 2 - \theta\) \(=\) \(\ds \sqrt {\dfrac 7 3}\)
\(\ds \leadsto \ \ \) \(\ds \theta\) \(=\) \(\ds 2 - \sqrt {\dfrac 7 3}\)

and we see that:

$1 < 2 - \sqrt {\dfrac 7 3} < 2$

$\blacksquare$


Sources