Mean Value Theorem/Examples/x^3/Formulation 2
Jump to navigation
Jump to search
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
- 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 $\text G$