Power Series for sine and cosine: Difference between revisions

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<html>
== Power Series for Sine ==
<link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" />
:<math>\sum_{k=0}^n (-1)^k\frac{1}{(2k+1)!}x^{2k+1}</math>
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/prototype.js"></script>
 
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraphcore.js"></script>
<jsxgraph box="jxgbox1" width="700" height="300">
<div id="jxgbox1" class="jxgbox" style="width:600px; height:300px;"></div>
board1 = JXG.JSXGraph.initBoard('jxgbox1', {axis:true, boundingbox: [-6, 3, 8, -3]});
<script language="JavaScript">
// Board
board1 = JXG.JSXGraph.initBoard('jxgbox1', {originX: 300, originY: 150, unitX: 50, unitY: 50});
board1.suspendUpdate();
board1.suspendUpdate();
// Axes and Properties
board1.create('functiongraph', [function(t){ return Math.sin(t); },-10, 10],{strokeColor: "#cccccc"});
board1.createElement('axis', [[1,0], [0,0]], {});
var s = board1.create('slider', [[0.75,-1.5],[5.75,-1.5],[0,0,10]], {name:'S',snapWidth:1});
board1.createElement('axis', [[0,1], [0,0]], {});
board1.create('functiongraph', [
//
board1.createElement('curve', [function(t){ return t; }, function(t){ return Math.sin(t); }, "t", -10, 10],{strokeColor: "#cccccc"});
//
var s = board1.createElement('slider', [0.75,-2.5,5,0,10,0], {name:'S'});
//
board1.createElement('curve', [
function(t){ return t; },
function(t) {
function(t) {
var val = 0;
var val = 0, i, sv = s.Value()+1;
for(var i = 0; i < Math.floor(s.X()) + 1; i++) {
for(i = 0; i < sv; i++) {
var f = 1;
   val = val + Math.pow(-1, i) * Math.pow(t, 2 * i + 1) / JXG.Math.factorial(2*i+1);
var n = 1;
while (n <= 2 * i + 1) { f = f * n++; }
   val = val + Math.pow(-1, i) * Math.pow(t, 2 * i + 1) / f;
}
}
return val;
return val;
},
},
"t", -10, 10], {strokeColor: "#bb0000"});
-10, 10], {strokeColor: "#bb0000"});
board1.unsuspendUpdate();
board1.unsuspendUpdate();
/*
</jsxgraph>
*/
 
</script>  
<source lang="javascript">
<h2>Power Series for Cosine</h2>
board1 = JXG.JSXGraph.initBoard('jxgbox1', {axis:true, boundingbox: [-6, 3, 8, -3]});
<div id="jxgbox2" class="jxgbox" style="width:600px; height:300px;"></div>
board1.suspendUpdate();
<script language="JavaScript">
board1.create('functiongraph', [function(t){ return Math.sin(t); },-10, 10],{strokeColor: "#cccccc"});
// Board
var s = board1.create('slider', [[0.75,-1.5],[5.75,-1.5],[0,0,10]], {name:'S',snapWidth:1});
board2 = JXG.JSXGraph.initBoard('jxgbox2', {originX: 300, originY: 150, unitX: 25, unitY: 25});
board1.create('functiongraph', [
  function(t) {
    var val = 0, i, sv = s.Value()+1;
    for(i = 0; i < sv; i++) {
      val = val + Math.pow(-1, i) * Math.pow(t, 2 * i + 1) / JXG.Math.factorial(2*i+1);
    }
    return val;
  }, -10, 10], {strokeColor: "#bb0000"});
board1.unsuspendUpdate();
</source>
 
 
== Power Series for Cosine ==
<jsxgraph box="jxgbox2" width="700" height="300">
board2 = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox: [-6, 3, 8, -3]});
board2.suspendUpdate();
board2.suspendUpdate();
// Axes and Properties
board2.create('functiongraph', [function(t){ return Math.cos(t); }, -10, 10],{strokeColor: "#cccccc"});
board2.createElement('axis', [[1,0], [0,0]], {});
var s2 = board2.create('slider', [[0.75,-1.5],[5.75,-1.5],[0,0,10]], {name:'T',snapWidth:1});
board2.createElement('axis', [[0,1], [0,0]], {});
board2.create('functiongraph', [
board2.snapToGrid = true;
//
board2.createElement('curve', [function(t){ return t; }, function(t){ return Math.cos(t); }, "t", -10, 10],{strokeColor: "#cccccc"});
//
var s2 = board2.createElement('slider', [2,-4,5,0,10,0], {name:'T'});
//
board2.createElement('curve', [
function(t){ return t; },
function(t) {
function(t) {
var val = 0;
var val = 0, i, sv = s2.Value()+1;
for(var i = 0; i < Math.floor(s2.X()) + 1; i++) {
for(i = 0; i < sv; i++) {
var f = 1;
   val = val + Math.pow(-1, i) * Math.pow(t, 2 * i) / JXG.Math.factorial(2*i);
var n = 1;
while (n <= 2 * i) { f = f * n++; }
   val = val + Math.pow(-1, i) * Math.pow(t, 2 * i) / f;
}
}
return val;
return val;
},
},
"t", -10, 10],{strokeColor: "#009900"});
-10, 10],{strokeColor: "#009900"});
board2.unsuspendUpdate();
</jsxgraph>
 
<source lang="javascript">
board2 = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox: [-6, 3, 8, -3]});
board2.suspendUpdate();
board2.create('functiongraph', [function(t){ return Math.cos(t); }, -10, 10],{strokeColor: "#cccccc"});
var s2 = board2.create('slider', [[0.75,-1.5],[5.75,-1.5],[0,0,10]], {name:'T',snapWidth:1});
board2.create('functiongraph', [
  function(t) {
    var val = 0, i, sv = Math.floor(s2.Value())+1;
    for(i = 0; i < sv; i++) {
      val = val + Math.pow(-1, i) * Math.pow(t, 2 * i) / JXG.Math.factorial(2*i);
    }
    return val;
  }, -10, 10],{strokeColor: "#009900"});
board2.unsuspendUpdate();
board2.unsuspendUpdate();
</script>
</source>
 
=== References ===
* [http://en.wikipedia.org/wiki/Trigonometric_functions http://en.wikipedia.org/wiki/Trigonometric_functions]
 
[[Category:Examples]]
[[Category:Calculus]]

Latest revision as of 15:42, 20 February 2013

Power Series for Sine

[math]\displaystyle{ \sum_{k=0}^n (-1)^k\frac{1}{(2k+1)!}x^{2k+1} }[/math]

board1 = JXG.JSXGraph.initBoard('jxgbox1', {axis:true, boundingbox: [-6, 3, 8, -3]});
board1.suspendUpdate();
board1.create('functiongraph', [function(t){ return Math.sin(t); },-10, 10],{strokeColor: "#cccccc"});
var s = board1.create('slider', [[0.75,-1.5],[5.75,-1.5],[0,0,10]], {name:'S',snapWidth:1});
board1.create('functiongraph', [
  function(t) {
    var val = 0, i, sv = s.Value()+1;
    for(i = 0; i < sv; i++) {
      val = val + Math.pow(-1, i) * Math.pow(t, 2 * i + 1) / JXG.Math.factorial(2*i+1);
    }
    return val;
  }, -10, 10], {strokeColor: "#bb0000"});
board1.unsuspendUpdate();


Power Series for Cosine

board2 = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox: [-6, 3, 8, -3]});
board2.suspendUpdate();
board2.create('functiongraph', [function(t){ return Math.cos(t); }, -10, 10],{strokeColor: "#cccccc"});
var s2 = board2.create('slider', [[0.75,-1.5],[5.75,-1.5],[0,0,10]], {name:'T',snapWidth:1});
board2.create('functiongraph', [
  function(t) {
    var val = 0, i, sv = Math.floor(s2.Value())+1;
    for(i = 0; i < sv; i++) {
      val = val + Math.pow(-1, i) * Math.pow(t, 2 * i) / JXG.Math.factorial(2*i);
    }
    return val;
  }, -10, 10],{strokeColor: "#009900"});						
board2.unsuspendUpdate();

References

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