Cauchy Mean Value Theorem/Geometrical Interpretation

From ProofWiki
Jump to navigation Jump to search

Geometrical Interpretation of Cauchy Mean Value Theorem

Consider two functions $\map f x$ and $\map g x$:

continuous on the closed interval $\closedint a b$
differentiable on $\openint a b$.

For every $x \in \closedint a b$, we consider the point $\tuple {\map f x, \map g x}$.

If we trace out the points $\tuple {\map f x, \map g x}$ over every $x \in \closedint a b$, we get a curve in two dimensions, as shown in the graph:

Cauchy's Mean Value Theorem.png

In the drawing, the slope of the red line is $\dfrac {\map g b - \map g a} {\map f b - \map f a}$.

This is because:

$\dfrac {\Delta y} {\Delta x} = \dfrac {\map g b - \map g a} {\map f b - \map f a}$

assuming that the vertical axis, which contains the value of $\map f x$, is the $y$-axis.

The slope of the green line is $\dfrac {\map {g'} c} {\map {f'} c}$.

This is because:

$\valueat {\dfrac {\d g} {\d f} } {x \mathop = c} = \valueat {\dfrac {\d g / \d x} {\d f / \d x} } {x \mathop = c} = \dfrac {\map {g'} c} {\map {f'} c}$

The drawing illustrates that for the value of $c$ chosen in the pictures, the slopes of the red line and green line are the same.

That is:

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