Share JSXGraph: example "Extended mean value theorem"

JSXGraph
Share JSXGraph: example "Extended mean value theorem"
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Extended mean value theorem

The*'extended mean value theorem* (also called *Cauchy's mean value theorem*) is usually formulated as: Let $$ f, g: [a,b] \to \mathbb{R}$$ be continuous functions that are differentiable in the open interval $(a,b)$. If $g'(x)\neq 0$ for all $x\in(a,b)$, then there exists a value $\xi \in (a,b)$ such that $$ \frac{f'(\xi)}{g'(\xi)} = \frac{f(b)-f(a)}{g(b)-g(a)}. $$ __Remark:__ It seems to be easier to state the extended mean value theorem in the following form: Let $$f, g: [a,b] \to \mathbb{R}$$ be continuous functions that are differentiable in the open interval $(a,b)$. Then there exists a value $\xi \in (a,b)$ such that $$ f'(\xi)\cdot (g(b)-g(a)) = g'(\xi) \cdot (f(b)-f(a)). $$ This second formulation avoids the need that $g'(x)\neq 0$ for all $x\in(a,b)$ and is therefore much easier to handle numerically. The proof is similar, just use the function $$ h(x) = f(x)\cdot(g(b)-g(a)) - (g(x)-g(a))\cdot(f(b)-f(a)) $$ and apply *Rolle's theorem*. __Visualization:__ The extended mean value theorem says that given the curve $$C: [a,b]\to\mathbb{R}, \quad t \mapsto (f(t), g(t))$$ with the above prerequisites for $f$ and $g$, there exists a $\xi$ such that the tangent to the curve in the point $C(\xi)$ is parallel to the line through $C(a)$ and $C(b)$.
// Define the id of your board in BOARDID

const board = JXG.JSXGraph.initBoard(BOARDID, {
    boundingbox: [-5, 10, 7, -6],
    axis: true
});

// Some initial points
var p = [];
p.push(board.create('point', [0, -2], {size: 2, name: 'C(a)'}));
p.push(board.create('point', [-1.5, 5], {size: 2, name: ''}));
p.push(board.create('point', [1, 4], {size: 2, name: ''}));
p.push(board.create('point', [3, 3], {size: 2, name: 'C(b)'}));

// Lagrange interpolation through the points
var fg = JXG.Math.Numerics.Neville(p);
var graph = board.create('curve', fg, {strokeWidth: 3, strokeOpacity: 0.5});

// Line 
var line = board.create('line', [p[0], p[3]], {strokeColor: '#ff0000', dash: 1});

// Derivatives of the curve
var df = JXG.Math.Numerics.D(fg[0]);
var dg = JXG.Math.Numerics.D(fg[1]);

// Usually, the extended mean value theorem is formulated as
// df(t) / dg(t) == (p[3].X() - p[0].X()) / (p[3].Y() - p[0].Y())
// We can avoid division by zero with the following formulation:
var quot = (t) => df(t) * (p[3].Y() - p[0].Y()) - dg(t) * (p[3].X() - p[0].X());

// Construct the point C(ξ)
var r = board.create('glider', [
    () => fg[0](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)),
    () => fg[1](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)),
    graph
], {name: 'C(ξ)', size: 4, fixed: true, color: 'blue'});

// Tangent to C through C(ξ)
board.create('tangent', [r], {strokeColor: '#ff0000'});