Difference between revisions of "Extended mean value theorem"

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Line 3: Line 3:
 
var p = [];
 
var p = [];
  
board.suspendUpdate();
 
 
p[0] = board.create('point', [-2,-2], {size:2});
 
p[0] = board.create('point', [-2,-2], {size:2});
 
p[1] = board.create('point', [-1.5, 5], {size:2});
 
p[1] = board.create('point', [-1.5, 5], {size:2});
Line 19: Line 18:
 
var dg = JXG.Math.Numerics.D(fg[1]);
 
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 that formulation:
 
var quot = function(t) {
 
var quot = function(t) {
    //var slope = (p[3].Y() - p[0].Y()) / (p[3].X() - p[0].X());
 
 
     return df(t) * (p[3].Y() - p[0].Y()) - dg(t) * (p[3].X() - p[0].X());
 
     return df(t) * (p[3].Y() - p[0].Y()) - dg(t) * (p[3].X() - p[0].X());
 
};
 
};
 
board.create('functiongraph', [quot]);
 
  
 
var r = board.create('glider', [
 
var r = board.create('glider', [
                    function() { return fg[0](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
+
            function() { return fg[0](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
                    function() { return fg[1](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
+
            function() { return fg[1](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
                    graph], {name: '', size: 4, fixed:true});
+
            graph], {name: '', size: 4, fixed:true, color: 'blue'});
  
 
board.create('tangent', [r], {strokeColor:'#ff0000'});
 
board.create('tangent', [r], {strokeColor:'#ff0000'});
  
board.unsuspendUpdate();
 
 
</jsxgraph>
 
</jsxgraph>
  
 
===The underlying JavaScript code===
 
===The underlying JavaScript code===
 
<source lang="javascript">
 
<source lang="javascript">
 +
var board = JXG.JSXGraph.initBoard('box', {boundingbox: [-5, 10, 7, -6], axis:true});
 +
var p = [];
 +
 +
p[0] = board.create('point', [-2,-2], {size:2});
 +
p[1] = board.create('point', [-1.5, 5], {size:2});
 +
p[2] = board.create('point', [1,4], {size:2});
 +
p[3] = board.create('point', [3,-1], {size:2});
 +
 +
// Curve
 +
var fg = JXG.Math.Numerics.Neville(p);
 +
var graph = board.create('curve', fg, {strokeWidth:3, strokeOpacity:0.5});
 +
 +
// Secant
 +
line = board.create('line', [p[0], p[3]], {strokeColor:'#ff0000', dash:1});
 +
 +
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 that formulation:
 +
var quot = function(t) {
 +
    return df(t) * (p[3].Y() - p[0].Y()) - dg(t) * (p[3].X() - p[0].X());
 +
};
 +
 +
var r = board.create('glider', [
 +
            function() { return fg[0](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
 +
            function() { return fg[1](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
 +
            graph], {name: '', size: 4, fixed:true, color: 'blue'});
 +
 +
board.create('tangent', [r], {strokeColor:'#ff0000'});
 
</source>
 
</source>
  
 
[[Category:Examples]]
 
[[Category:Examples]]
 
[[Category:Calculus]]
 
[[Category:Calculus]]

Revision as of 17:10, 29 January 2019

The underlying JavaScript code

var board = JXG.JSXGraph.initBoard('box', {boundingbox: [-5, 10, 7, -6], axis:true});
var p = [];

p[0] = board.create('point', [-2,-2], {size:2});
p[1] = board.create('point', [-1.5, 5], {size:2});
p[2] = board.create('point', [1,4], {size:2});
p[3] = board.create('point', [3,-1], {size:2});

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

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

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 that formulation:
var quot = function(t) {
    return df(t) * (p[3].Y() - p[0].Y()) - dg(t) * (p[3].X() - p[0].X());
};

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

board.create('tangent', [r], {strokeColor:'#ff0000'});