tag:blogger.com,1999:blog-86890611864773431452024-03-13T05:16:14.926-07:00Maths au lycéeAnonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.comBlogger121125tag:blogger.com,1999:blog-8689061186477343145.post-58261854370668174682011-11-18T08:54:00.001-08:002011-11-18T08:54:23.911-08:00Texte dynamique<iframe src="http://www.screenr.com/embed/i6Ls" width="650" height="396" frameborder="0"></iframe>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-55535936765993276472011-03-19T07:52:00.001-07:002012-02-24T07:38:46.116-08:00Trapèze avec Casyopée<span id="internal-source-marker_0.7489439041337691" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">On se donne un triangle $OAB$ rectangle en $O$, isocèle et de cotés 6. </span><br />
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">$C$ est un point du segment $[OA ]$ et $D$ est le point du segment $[AB]$ tel que $OCDB$ soit un trapèze.<br class="kix-line-break" />En utilisant <a href="http://www.inclassablesmathematiques.fr/tag/casyop%C3%A9e">Casyopée</a>, modélisez la situation et déterminez la position du point $C$ à partir de laquelle l’aire du trapèze $OCDB$ sera égale à la moitié de l’aire du triangle $OAB$ (pour les premières) ou égale à 10 pour les secondes.</span><br />
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;"><br />
</span><br />
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;"></span><img height="426px;" src="https://docs.google.com/drawings/image?id=sdgEh3tP0h_KzUR5sRSO_kw&w=506&h=426&rev=21&ac=1" width="506px;" /><br />
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;"></span><br />
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">Démontrez tous les résultats, en posant que $OC=x$.</span><br />
<br />
<br />
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">Voir les copies des fenêtres: <a href="https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B8LWwvTf2MtJNDIxODBmODktZGM4Zi00NGNhLTgzMGUtYTQ5NmM1NzlhZWMy&hl=fr">ICI </a></span><br />
<div><br />
</div>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-82968428930673967292010-07-27T02:24:00.001-07:002010-07-27T02:24:32.041-07:00Recherche de racines<script id="WolframAlphaScripta7d8ae4569120b5bec12e7b6e9648b86" src="http://www.wolframalpha.com/widget/widget.jsp?id=a7d8ae4569120b5bec12e7b6e9648b86" type="text/javascript">
</script>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-51048390909318328202010-07-25T10:03:00.001-07:002010-07-25T10:13:01.965-07:00Jeu complexe<p align='center'><code><iframe scrolling='no' src='http://classtools.net/widgets/quiz_7/9LIRq.htm?710?530' width='718' height='550' frameborder=0></iframe></code><p align='center'><a href='http://classtools.net/widgets/quiz_7/9LIRq.htm'>Click here for full screen version</a></p>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-17114554629909800622010-06-06T01:17:00.000-07:002010-06-06T02:18:05.703-07:00Chaines de Markov<b>Deux iles voisines A et B, isolées du monde, échangent une partie de leur population entre elles.</b><br />
<br />
<br />
On note $\;a_n\;$ et $\;b_n\;$ les populations respectives des iles A et B, à l'issue de $\;n\;$ cycles d'échanges, $\; p_n=\begin{bmatrix}a_n\\ b_n \end{bmatrix}\;$ et $\;A\;$ la matrice de passage d'un état à son successeur: $\;p_{n+1}=A \times p_n \;$.<br />
<br />
Le vecteur population initiale est $\; p_0=\begin{bmatrix}a_0\\ b_0 \end{bmatrix}\;$<br />
<br />
<br />
Exprimons le vecteur colonne $ \;p_n\;$ de la population des iles après $\;n\;$ cycles d'échanges en fonction du vecteur population initiale $ \;p_0\;$<br />
<br />
$p_1=A \times p_0 $<br />
<br />
$p_2=A \times p_1 $<br />
$p_2=A \times A \times p_0 $<br />
$p_2=A^2 \times p_0 $<br />
<br />
$p_3=A \times p_2 $<br />
$p_3=A \times A^2 \times p_0 $<br />
$p_3=A^3 \times p_0 $<br />
...<br />
$p_n=A^n \times p_0 $<br />
<br />
Si par exemple, les deux iles conservent 80 % de leur population et en exportent 20% sur l'autre ile à chaque cycle, la matrice $\;A\;$ se définit comme suit :<br />
<br />
$ A= \begin{bmatrix}0.8 &0.2 \\ 0.2 & 0.8 \end{bmatrix} $<br />
<br />
<div style="color: #38761d;"><br />
<applet archive="wrs_net_fr.jar" code="WirisApplet_net_fr.class" codebase="http://www.wiris.net/demo/wiris/wiris-codebase/" height="300px" width="100%"><br />
<param name="version" value="2.0" /><param name="toolbar" value="floating" /><param name="requestFirstEvaluation" value="true" /><param name="command" value="false" /><param name="commands" value="false" /><param name="interface" value="false" /><param name="XMLinitialText" value="<session lang="fr" version="2.0"><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd></mtr></mtable></mfenced><mo>*</mo><mfenced><mtable><mtr><mtd><mn>2000</mn></mtd></mtr><mtr><mtd><mn>1000</mn></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1800.</mn></mtd></mtr><mtr><mtd><mn>1200.</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>5</mn><mo>*</mo><mfenced><mtable><mtr><mtd><mn>2000</mn></mtd></mtr><mtr><mtd><mn>1000</mn></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1538.9</mn></mtd></mtr><mtr><mtd><mn>1461.1</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>10</mn><mo>*</mo><mfenced><mtable><mtr><mtd><mn>2000</mn></mtd></mtr><mtr><mtd><mn>1000</mn></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1503.</mn></mtd></mtr><mtr><mtd><mn>1497.</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>20</mn><mo>*</mo><mfenced><mtable><mtr><mtd><mn>2000</mn></mtd></mtr><mtr><mtd><mn>1000</mn></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1500.</mn></mtd></mtr><mtr><mtd><mn>1500.</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>" /><br />
<br />
<br />
<br />
<br />
<param name="Level" value="primary" /></applet></div><br />
Quelque soit la répartition initiale de la population sur chaque ile, il semble qu'un tel échange qui se répète, finisse par une stabilisation des populations, et l'égalisation de leur effectif sur chaque ile.<br />
<br />
Prenons un autre exemple:<br />
<br />
$ A= \begin{bmatrix}0.75 &0.15 \\ 0.25 & 0.85 \end{bmatrix} $<br />
<br />
Ici, l'ile A conserve 75% de sa population et en envoie 25% sur l'ile B alors que B en conserve 85% et en envoie 15% sur A.<br />
<br />
On peut regarder ce qui se passe à l'issue de 20 cycles avec différentes populations initiales :<br />
<br />
<applet archive="wrs_net_fr.jar" code="WirisApplet_net_fr.class" codebase="http://www.wiris.net/demo/wiris/wiris-codebase/" height="250px" width="100%"><br />
<param name="version" value="2.0" /><param name="toolbar" value="floating" /><param name="command" value="false" /><param name="commands" value="false" /><param name="interface" value="false" /><param name="XMLinitialText" value="<session lang="fr" version="2.0"><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>15</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>85</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>20</mn><mo>*</mo><mfenced><mtable><mtr><mtd><mn>1000</mn></mtd></mtr><mtr><mtd><mn>2000</mn></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1125.</mn></mtd></mtr><mtr><mtd><mn>1875.</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>15</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>85</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>20</mn><mo>*</mo><mfenced><mtable><mtr><mtd><mn>2000</mn></mtd></mtr><mtr><mtd><mn>1000</mn></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1125.</mn></mtd></mtr><mtr><mtd><mn>1875.</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>15</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>85</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>20</mn><mo>*</mo><mfenced><mtable><mtr><mtd><mn>3000</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1125.1</mn></mtd></mtr><mtr><mtd><mn>1874.9</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>" /><br />
<br />
<br />
<param name="Level" value="primary" /></applet>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-71224932953518792212010-06-05T08:07:00.000-07:002010-06-05T08:12:28.387-07:00Le théorème de MorleyUne illustration dynamique du très beau théorème de Morley qui s'énonce comme suit:<br />
<br />
<div style="color: #38761d; text-align: center;"><span style="font-size: large;"><b>Les intersections des trissectrices des angles d'un triangle forment un triangle équilatéral</b></span></div><br />
Pour les démonstrations, c'est un peu <a href="http://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Morley">plus sportif</a>...<br />
<br />
<br />
<div align="CENTER"><br />
<br />
<applet archive="http://www.geogebra.org/webstart/3.2/geogebra.jar" code="geogebra.GeoGebraApplet" codebase="http://www.geogebra.org/webstart/3.2/unsigned/" height="600" name="ggbApplet" width="400"><br />
<param name="filename" value="http://flora.allain.perso.sfr.fr/Geogebra/morley.ggb" /><br />
<br />
<br />
<br />
<br />
<param name="java_arguments" value="-Xmx1000m" /><br />
<br />
<br />
<br />
<br />
<param name="framePossible" value="true" /><br />
<br />
<br />
<br />
<br />
<param name="showResetIcon" value="true" /><br />
<br />
<br />
<br />
<br />
<param name="showAnimationButton" value="true" /><br />
<br />
<br />
<br />
<br />
<param name="enableRightClick" value="true" /><br />
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<br />
<param name="enableLabelDrags" value="true" /><br />
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<br />
<br />
<param name="showMenuBar" value="false" /><br />
<br />
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<br />
<br />
<param name="showToolBar" value="false" /><br />
<br />
<br />
<br />
<br />
<param name="showToolBarHelp" value="false" /><br />
<br />
<br />
<br />
<br />
<param name="showAlgebraInput" value="false" /><br />
<br />
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (<a href="http://java.sun.com/getjava">Click here to install Java now</a>)</applet></div>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-13609600559448746382010-06-01T09:30:00.000-07:002010-06-01T09:30:52.285-07:00Formules de trigonométrie<div class="prezi-player"><style type="text/css" media="screen">.prezi-player { width: 600px; } .prezi-player-links { text-align: center; }</style><object id="prezi_54uvhgsl6htz" name="prezi_54uvhgsl6htz" classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" width="600" height="400"><param name="movie" value="http://prezi.com/bin/preziloader.swf"/><param name="allowfullscreen" value="true"/><param name="allowscriptaccess" value="always"/><param name="bgcolor" value="#ffffff"/><param name="flashvars" value="prezi_id=54uvhgsl6htz&lock_to_path=0&color=ffffff&autoplay=no"/><embed id="preziEmbed_54uvhgsl6htz" name="preziEmbed_54uvhgsl6htz" src="http://prezi.com/bin/preziloader.swf" type="application/x-shockwave-flash" allowfullscreen="true" allowscriptaccess="always" width="600" height="400" bgcolor="#ffffff" flashvars="prezi_id=54uvhgsl6htz&lock_to_path=0&color=ffffff&autoplay=no"></embed></object><div class="prezi-player-links"><p><a title="Trigonométrie" href="http://prezi.com/54uvhgsl6htz/">Formules de trigonométrie</a> on <a href="http://prezi.com">Prezi</a></p></div></div>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-82763307028614108032010-05-27T10:42:00.001-07:002010-06-08T11:10:37.740-07:00Second degré GeoGebra et Javascript<div style="text-align: center;">$ \huge{{\color{blue}ax^2+bx+c=0 }}$</div><br />
<script language="JavaScript">
function init()
{
document.discriminant.a.value="";
document.discriminant.b.value="";
document.discriminant.c.value="";
document.discriminant.resultat.value="";
}
function discri()
{
a=document.discriminant.a.value;
b=document.discriminant.b.value;
c=document.discriminant.c.value;
a=parseFloat(a);
b=parseFloat(b);
c=parseFloat(c);
d=b*b-4*a*c;
if(d>0)
{x1=(-b+Math.sqrt(d))/(2*a);
x2=(-b-Math.sqrt(d))/(2*a);
document.discriminant.resultat.value="Delta= "+d+" et: x1="+x1+" x2="+x2;
}
if(d==0)
{x0=-b/2*a;
document.discriminant.resultat.value="Delta=0 et: x0="+x0;
}
if(d<0){document.discriminant.resultat.value="Delta= "+d+" et il n'y pas de solution";}
}
</script>
<table width="100%" border="1" cellspacing="1" cellpadding="1">
<tr>
<td colspan="5"> <center>$f(x)=ax^2+bx+c \:$ et $\: \Delta=b^2-4ac$</center> </td>
</tr>
<td></td> <td><center> $ \Delta > 0 $</center></td><td><center>$\Delta = 0$</center></td><td><center>$\Delta < 0$</center></td>
<tr> <td><center> Equation $f(x)=0$ </center></td><td><center> 2 solutions $x_1$ et $x_2$</center></td><td><center>une solution $x_0$</center></td> <td><center>pas de solution</center></td>
</tr>
<tr> <td><center> Factorisation de $f(x)$ </center></td><td><center> $a(x-x_1)(x-x_2)$</center></td><td><center>$a(x-x_0)^2$</center></td> <td><center>pas de factorisation</center></td>
</tr>
<tr>
<td colspan=5><center>
<applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.geogebra.org/webstart/3.2/unsigned/" archive="http://www.geogebra.org/webstart/3.2/geogebra.jar" width="600" height="400">
<param name="filename" value="http://flora.allain.perso.sfr.fr/Geogebra/seconddegre.ggb"/>
<param name="java_arguments" value="-Xmx1000m">
<param name="framePossible" value="true"/>
<param name="showResetIcon" value="true"/>
<param name="showAnimationButton" value="true"/>
<param name="enableRightClick" value="true"/>
<param name="enableLabelDrags" value="true"/>
<param name="showMenuBar" value="false"/>
<param name="showToolBar" value="false"/>
<param name="showToolBarHelp" value="false"/>
<param name="showAlgebraInput" value="false"/>
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (<a href="http://java.sun.com/getjava">Click here to install Java now</a>)
</param></applet>
</center>
</td></tr>
</table><form action="" method="post" name="discriminant">
<table width="100%"border="1" cellpadding="1" cellspacing="1">
<tbody>
<tr><td><b>a : </b>
<input name="a" size="5" />
</td></tr>
<tr><td><b>
b : </b><input name="b" size="5" />
</td></tr>
<tr><td><b>
c : </b>
<input name="c" size="5" />
<input name="bouton1" onclick="discri()" type="button" value="Lancer le calcul" />
</td></tr>
<tr><td><textarea cols="63" name="resultat" rows="3" wrap="virtual"></textarea>
<input onclick="init()" type="button" value="Recommencer" />
</td></tr>
</tbody></table></form><b><span class="Apple-style-span" style="font-size: x-large;"><a href="http://docs.google.com/View?id=df845nmp_79cv57gvdj">CODE JavaScript et HTML</a></span></b>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-24285303968955314522010-05-18T12:52:00.001-07:002010-05-18T12:53:14.577-07:00Script java "puissances de 2"<script language="JavaScript">
function puissancesDe2(combien)
{
var p=1;
for(var i=0;i<combien;i++)
{
p*=2;
document.write(p + "<br>");
}
}
</SCRIPT><br />
<br />
<br />
voici les premières puissances de 2 :<br />
<br />
<script language="JavaScript">puissancesDe2(30);</SCRIPT>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-70512989309923702012010-05-16T03:03:00.000-07:002010-05-16T04:06:39.656-07:00Patron de cube<center><br />
<script src="http://jmath3d.aspirine.org/download/deployJMath3D.js"></script><br />
<script>
var parametres = { model:"http://flora.allain.perso.sfr.fr/patroncube.g3", couleurfond:"#e0e0ff" };
deployJMath3D.runApplet(600, 400, parametres);
</script><br />
</center>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-82680770781593367302010-05-16T02:10:00.000-07:002010-05-16T04:08:19.442-07:00Ballon de foot<center><br />
<script src="http://jmath3d.aspirine.org/download/deployJMath3D.js"></script><br />
<script>
var parametres = { model:"http://jmath3d.aspirine.org/galerie/divers/icosaedre_tronque.obj", couleurfond:"#e0e0ff" };
deployJMath3D.runApplet(500, 400, parametres);
</script><br />
</center>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-32765876806361224532010-05-11T05:53:00.000-07:002010-05-11T06:11:52.327-07:00Lois de probabilité continues<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/_NxB_F1GNmYg/S-lROY7xAdI/AAAAAAAABCA/7Mh0WGMPKV4/s1600/gauss.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/_NxB_F1GNmYg/S-lROY7xAdI/AAAAAAAABCA/7Mh0WGMPKV4/s320/gauss.jpg" /></a></div><br />
<div style="text-align: center;"><span class="Apple-style-span" style="font-size: x-large;"><span class="Apple-style-span" style="color: #38761d;"><a href="http://www.univ-orleans.fr/irem/ecureuil/loi%20continue.htm">Densité de probabilité, loi continue, loi uniforme,la loi exponentielle</a></span></span></div><div style="text-align: center;"><span class="Apple-style-span" style="font-size: x-large;"><br />
</span><br />
<div style="text-align: left;"><span class="Apple-style-span">Les fichiers GeoGebra:</span></div><div style="text-align: left;"><a href="http://flora.allain.perso.sfr.fr/Geogebra/TS/loicontinue0.ggb">Loi continue</a></div><div style="text-align: left;"><a href="http://flora.allain.perso.sfr.fr/Geogebra/TS/loi_uniforme.ggb">Loi uniforme</a></div><div style="text-align: left;"><a href="http://flora.allain.perso.sfr.fr/Geogebra/TS/loi_expo.ggb">Loi exponentielle</a></div><div style="text-align: left;"><span class="Apple-style-span" style="font-size: x-large;"><br />
</span></div></div><div style="text-align: center;"><span class="Apple-style-span" style="font-size: x-large;"><br />
</span></div><div style="text-align: center;"><iframe height="780" src="http://docs.google.com/viewer?url=http%3A%2F%2Fpagesperso-orange.fr%2Fgilles.costantini%2FTS%2Fproba%2FTSProba2_yl.PDF&embedded=true" style="border: none;" width="600"></iframe></div>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-5720318801942394042010-05-06T05:14:00.000-07:002010-05-06T05:31:57.853-07:00Loi binomiale<div class="separator" style="clear: both; text-align: center;"><img border="0" src="http://2.bp.blogspot.com/_NxB_F1GNmYg/S-K0i5sG2dI/AAAAAAAABB0/kLndwvTiaA4/s320/binomiale.jpg" /></div><br />
<div style="text-align: center;">Pour afficher la loi binomiale $\textsl{B}(n,p)$, cliquez<span style="font-size: large;"><b><span style="color: #38761d;"> <a href="http://www.stat.tamu.edu/%7Ewest/applets/binomialdemo.html">ICI</a></span></b></span><br />
<br />
<br />
</div>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-90327862272867160952010-05-03T09:40:00.000-07:002010-05-29T02:07:33.379-07:00Coefficients binomiaux et triangle de Pascal<applet archive="http://flora.allain.perso.sfr.fr/Java/PascalDreieck.jar, http://flora.allain.perso.sfr.fr/Java/M14French.jar" code="PascalDreieck.class" codebase="../m14_jar" height="380" width="680"><br />
</applet>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-58309436344160102702010-05-03T04:15:00.000-07:002010-05-14T15:01:11.015-07:00Quatre amis se retrouvent dans quatre salles de cinéma...<div style="text-align: justify;">Chaque ami choisit indépendamment des autres une salle de cinéma. L'univers de cette expérience aléatoire est composé de $4 \times 4 \times 4 \times 4=4^4=256$ quadruplets correspondant chacun aux choix respectifs des quatre amis.</div><div style="text-align: justify;"><br />
</div><div style="text-align: justify;"><b><span style="color: #38761d;">Soit $A$ l'évènement: Ils se retrouvent dans des salles différentes.</span></b></div><div style="text-align: justify;"><br />
</div><div style="text-align: justify;">En utilisant la méthode des cases de choix, on obtient 4 choix pour le premier ami, puis 3 pour le deuxième qui ne eput se trouver dans la même salle, puis 2 pour le troisièmes et enfin 1 pour le dernier.<br />
<br />
Il y a donc $4 \times 3 \times 2 \times 1 = 24$ cas favorables.<br />
<br />
Nous sommes en situation d'équiprobabilité, puisque les choix se font indépendamment les uns des autres, la probabilité d'un évènement s'obtient donc comme le quotient des cas favorables sur la totalité des cas possibles.</div><div style="text-align: justify;"><br />
</div><div style="text-align: justify;">$$p(A)=\frac{24}{256} = \frac{3}{32}$$</div><div style="text-align: justify;"><br />
</div><div style="text-align: justify;"><b><span style="color: #38761d;">Soit $B$ l'évènement : Au moins deux se retrouvent dans la même salle.</span></b><br />
<br />
On remarque que $$B= \overline{A}$$.<br />
Ainsi:<br />
$$p(B)=1-p(A)= \frac{29}{32}$$<br />
<br />
On peut faire un dénombrement direct, mais cela n'est pas très adapté compte tenu de la simplicité du pasage par l'évènement contraire et le fait que l'évenment $B$ recouvre la quasi-totalité des issues possibles.<br />
<br />
Il faut considérer 3 cas:<br />
<br />
<b>Deux amis exactement se rencontrent dans la même salle. </b><br />
Et là il faut considérer deux sous-cas suivants que les deux amis restants se rencontrent ou non dans une même autre salle.<br />
<br />
<i>Deux amis se rencontrent dans une même salle et les deux autres dans deux salles différentes:</i><br />
On fixe la salle et les amis soit : $1\times 1 \times 3 \times 2 $.<br />
On multiplie par le nombre de salle possibles soit 4 et le nombre de positions (qui correspondent aux amis) de 2 éléments parmi 4 soit 6. On trouve donc <b>144</b> cas favorables.<br />
<br />
<i>Deux amis se rencontrent dans une même salle et les deux autre dans une même autre salle</i>.<br />
On fixe la salle et les amis soit $1 \times 1 \times 3 \times 1=3$ car le dernier ami doit aller dans la salle du troisième. Il n'a pas le choix, tout comme le deuxième.<br />
On fait maintenant varier les salles, il y en a 4 possibles et les positions . Il faut cependant faire attention car il n'y en a plus 6 mais 3 car il existe des symétries. Il ne faut pas compter deux fois les issues.<br />
On trouve donc <b>36</b> cas favorables.<br />
<br />
<b>Trois amis se retrouvent dans la même salle.</b><br />
Avec le même procédé que précédemment on trouve : $1\times 1 \times 1 \times 3$ multtiplié par le nombre de salles, soit 4, puis le nombre de possibilité de faire le choix de trois élément parmi 4, d'ailleurs identique aux choix possibles d'un élément parmi 4, soit 4. On a donc <b>48 </b>issues favorables.<br />
<br />
<b>Quatre amis se retrouvent dans la même salle.</b> On trouve aisément qu'il n'y a que <b>4 </b>issues favorables.<br />
<br />
On trouve donc :<br />
<br />
<div style="text-align: center;">$$p(B)= \frac{144+36+48+4}{256}=\frac{232}{256}=\frac{29}{32}$$</div><br />
</div>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-78216501225807277072010-05-02T09:03:00.000-07:002010-05-14T15:01:43.808-07:00Détermination de coefficients et recherche d'asymptotes$f$ est la fonction définie sur $]1;+\infty[$ par:<br />
<br />
$$f(x)=\frac{x^2+x+3}{x-1}$$<br />
<br />
<div style="color: #38761d;">Déterminons les réels $a,b,c$ tels que pour tout $x \in ]1;+\infty[ $</div><br />
$$f(x)=ax+b + \frac{c}{x-1}$$<br />
<br />
Pour cela on met l'expression de $f$ contenant les coefficients littéraux au même dénominateur et on identifie les coefficients par comparaison avec la forme développée, ordonnée et réduite du numérateur.<br />
<br />
$$f(x)=\frac{ax^2+(-a+b)x-b+c}{x-1}$$<br />
<br />
En identifiant les coefficients des "$x^2$", des "$x$" et la constante:<br />
<br />
<applet archive="wrs_net_fr.jar" code="WirisApplet_net_fr.class" codebase="http://www.wiris.net/demo/wiris/wiris-codebase/" height="100px" width="100%"><br />
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</applet><br />
<br />
On a donc :<br />
<br />
$$f(x)=x+2 + \frac{1}{x-1}$$<br />
<br />
<div style="color: #38761d;">Utilisons cette forme pour étudier les limites en $1$ et en $+\infty$<br />
<br />
<br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=%5C%5C%5Cleft.%5Cbegin%7Bmatrix%7D%20%5Clim_%7Bx%5Crightarrow%201%5E@plus;%7D%28x@plus;2%29=3%5C%5C%20%5Clim_%7Bx%5Crightarrow%201%5E@plus;%7D%5Cfrac%7B1%7D%7Bx-1%7D=@plus;%5Cinfty%20%5Cend%7Bmatrix%7D%5Cright%5C%7D%5Clim_%7Bx%5Crightarrow%201%5E@plus;%7Df%28x%29=@plus;%20%5Cinfty%5C%5C%20%5C%5C%20%5Cleft.%5Cbegin%7Bmatrix%7D%20%5Clim_%7Bx%5Crightarrow%20@plus;%5Cinfty%7D%28x@plus;2%29=@plus;%5Cinfty%5C%5C%20%5Clim_%7Bx%5Crightarrow%20@plus;%5Cinfty%7D%5Cfrac%7B1%7D%7Bx-1%7D=0%20%5Cend%7Bmatrix%7D%5Cright%5C%7D%5Clim_%7Bx%5Crightarrow%20@plus;%5Cinfty%7Df%28x%29=@plus;%20%5Cinfty%5C%5C" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5C%5C%5Cleft.%5Cbegin%7Bmatrix%7D%20%5Clim_%7Bx%5Crightarrow%201%5E+%7D%28x+2%29=3%5C%5C%20%5Clim_%7Bx%5Crightarrow%201%5E+%7D%5Cfrac%7B1%7D%7Bx-1%7D=+%5Cinfty%20%5Cend%7Bmatrix%7D%5Cright%5C%7D%5Clim_%7Bx%5Crightarrow%201%5E+%7Df%28x%29=+%20%5Cinfty%5C%5C%20%5C%5C%20%5Cleft.%5Cbegin%7Bmatrix%7D%20%5Clim_%7Bx%5Crightarrow%20+%5Cinfty%7D%28x+2%29=+%5Cinfty%5C%5C%20%5Clim_%7Bx%5Crightarrow%20+%5Cinfty%7D%5Cfrac%7B1%7D%7Bx-1%7D=0%20%5Cend%7Bmatrix%7D%5Cright%5C%7D%5Clim_%7Bx%5Crightarrow%20+%5Cinfty%7Df%28x%29=+%20%5Cinfty%5C%5C" title="\\\left.\begin{matrix} \lim_{x\rightarrow 1^+}(x+2)=3\\ \lim_{x\rightarrow 1^+}\frac{1}{x-1}=+\infty \end{matrix}\right\}\lim_{x\rightarrow 1^+}f(x)=+ \infty\\ \\ \left.\begin{matrix} \lim_{x\rightarrow +\infty}(x+2)=+\infty\\ \lim_{x\rightarrow +\infty}\frac{1}{x-1}=0 \end{matrix}\right\}\lim_{x\rightarrow +\infty}f(x)=+ \infty\\" /></a></div><br />
La fonction $f$ est de la forme $$f(x)=x+2+\phi(x)$$ <br />
avec $$\phi(x)=\frac{1}{x-1}$$<br />
et $$\lim_{x\rightarrow +\infty}\phi(x)=\lim_{x\rightarrow +\infty}\frac{1}{x-1}=0$$<br />
<br />
<b>Ainsi la droite$\Delta$ d'équation $y=x+2$ est asymptote à la courbe représentative de $f$ en $+\infty$.</b><br />
<br />
<b>La courbe possède aussi une asymptote verticale d'équation $x=1$<br />
puisque $$\lim_{x\rightarrow 1^+}f(x)=+\infty$$</b><br />
<br />
<div style="color: #38761d;">Résolvons l'inéquation $$\left |f(x)-(x+2) \right | \leq 0,1 $$</div><br />
<br />
Soit $$\left | \frac{1}{x-1} \right | \leq 0,1 $$<br />
<br />
ce qui est équivalent à$$ 0< \frac{1}{x-1} \leq 0,1 $$ puisque $x-1$ est strictement positif.
et donc à $x$>$11$
Ceci implique que pour $x$>$11$, l'écart entre la courbe représentative de $f$ et l'asymptote $\Delta$ est inférieur à 0,1.
<div class="separator" style="clear: both; text-align: center;"><br />
<a href="http://4.bp.blogspot.com/_NxB_F1GNmYg/S92wsF8AQzI/AAAAAAAABBU/aT4fOinG6u4/s1600/ImageSQN1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="287" src="http://4.bp.blogspot.com/_NxB_F1GNmYg/S92wsF8AQzI/AAAAAAAABBU/aT4fOinG6u4/s400/ImageSQN1.png" width="400" /></a></div>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-75461558182539877422010-04-30T10:12:00.000-07:002010-05-14T15:02:10.947-07:00Matrices inverses<div style="text-align: center;"><a href="http://www.codecogs.com/eqnedit.php?latex=%5C120dpi%20A=%5Cbegin%7Bpmatrix%7D%204%20&-6%20&5%20%5C%5C%203&-5%20&5%20%5C%5C%200%20&-1%20&%202%20%5Cend%7Bpmatrix%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5C120dpi%20A=%5Cbegin%7Bpmatrix%7D%204%20&-6%20&5%20%5C%5C%203&-5%20&5%20%5C%5C%200%20&-1%20&%202%20%5Cend%7Bpmatrix%7D" title="\120dpi A=\begin{pmatrix} 4 &-6 &5 \\ 3&-5 &5 \\ 0 &-1 & 2 \end{pmatrix}" /></a></div><applet archive="wrs_net_fr.jar" code="WirisApplet_net_fr.class" codebase="http://www.wiris.net/demo/wiris/wiris-codebase/" height="250px" width="100%"><br />
<param name="version" value="2.0" /><param name="toolbar" value="floating" /><param name="command" value="false" /><param name="commands" value="false" /><param name="interface" value="false" /><param name="XMLinitialText" value="<session lang="fr" version="2.0"><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>6</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>2</mn></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>6</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>3</mn></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>7</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd><mtd><mn>8</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>6</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>4</mn></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command></group></session>" /></applet><br />
<br />
$A^4=A^2\times A^2=I_3$<br />
En posant:<br />
$B=A^2$<br />
On a :<br />
$B \times B=I_3$<br />
Ainsi:<br />
$B=B^{-1}$Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-67677875214491845112010-04-30T09:55:00.000-07:002010-05-14T15:03:25.365-07:00Matrices inverses<div style="text-align: center;"><a href="http://www.codecogs.com/eqnedit.php?latex=%5C200dpi%20A=%5Cbegin%7Bpmatrix%7D%20-2%20&-1%20%5C%5C%205%20&%202%20%5Cend%7Bpmatrix%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5C120dpi%20A=%5Cbegin%7Bpmatrix%7D%20-2%20&-1%20%5C%5C%205%20&%202%20%5Cend%7Bpmatrix%7D" title="\120dpi A=\begin{pmatrix} -2 &-1 \\ 5 & 2 \end{pmatrix}" /></a></div><applet archive="wrs_net_fr.jar" code="WirisApplet_net_fr.class" codebase="http://www.wiris.net/demo/wiris/wiris-codebase/" height="300px" width="100%"><br />
<param name="version" value="2.0" /><param name="toolbar" value="floating" /><param name="requestFocus" value="true" /><param name="command" value="false" /><param name="commands" value="false" /><br />
<br />
<br />
<br />
<param name="interface" value="false" /><param name="XMLinitialText" value="<session lang="fr" version="2.0"><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>2</mn></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>3</mn></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>4</mn></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>3</mn><mo>*</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>2</mn><mo>*</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>2</mn></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>" /><br />
</applet><br />
<br />
$A^4=I_2$ <br />
ainsi <br />
$A \times A^3=I_2$ <br />
et donc <br />
$A^{-1}=A^3$<br />
<br />
En posant $B=A^2$<br />
<br />
De $A^4=I_2$<br />
Il vient $A^2\times A^2=I_2$<br />
Et donc $B \times B=I_2$<br />
D'où $B^{-1}=B$Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-79104722731598516732010-04-30T09:37:00.000-07:002010-05-14T15:04:07.051-07:00Inverse d'une matriceSoit $A$ une matrice d'ordre n. B est l'inverse de $A$ si et seulement si $A \times B=I_n$<br />
Exemple :<br />
<applet archive="wrs_net_fr.jar" code="WirisApplet_net_fr.class" codebase="http://www.wiris.net/demo/wiris/wiris-codebase/" height="300px" width="100%"><br />
<param name="version" value="2.0" /><param name="toolbar" value="floating" /><param name="requestFirstEvaluation" value="true" /><param name="requestFocus" value="true" /><param name="command" value="false" /><br />
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<param name="commands" value="false" /><param name="interface" value="false" /><param name="XMLinitialText" value="<session lang="fr" version="2.0"><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>5</mn></mtd><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd><mtd><mn>8</mn></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><mn>11</mn><mn>3</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>13</mn><mn>6</mn></mfrac></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>5</mn></mtd><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd><mtd><mn>8</mn></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mfenced><mo>*</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><mn>11</mn><mn>3</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>13</mn><mn>6</mn></mfrac></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>" /><br />
</applet>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-4456790235882408142010-04-29T07:02:00.000-07:002010-07-09T00:19:48.022-07:00Modélisation de croissances de population<a href="http://www.scribd.com/doc/30698376/introduction-modelisation" style="display: block; font: 14px Helvetica,Arial,Sans-serif; margin: 12px auto 6px; text-decoration: underline;" title="View introduction modélisation on Scribd">introduction modélisation</a> <object data="http://d1.scribdassets.com/ScribdViewer.swf" height="600" id="doc_782144639184523" name="doc_782144639184523" style="outline: medium none;" type="application/x-shockwave-flash" width="100%"> <param name="movie" value="http://d1.scribdassets.com/ScribdViewer.swf"><param name="wmode" value="opaque"><param name="bgcolor" value="#ffffff"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><param name="FlashVars" value="document_id=30698376&access_key=key-14xkda3s79wbxe09heby&page=1&viewMode=list"><embed id="doc_782144639184523" name="doc_782144639184523" src="http://d1.scribdassets.com/ScribdViewer.swf?document_id=30698376&access_key=key-14xkda3s79wbxe09heby&page=1&viewMode=list" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="600" width="100%" wmode="opaque" bgcolor="#ffffff"></embed> </object><br />
<a href="http://www.scribd.com/doc/30698407/quelques-modeles-d" style="display: block; font: 14px Helvetica,Arial,Sans-serif; margin: 12px auto 6px; text-decoration: underline;" title="View quelques modèles d on Scribd">quelques modèles d</a> <object data="http://d1.scribdassets.com/ScribdViewer.swf" height="600" id="doc_884350309099725" name="doc_884350309099725" style="outline: medium none;" type="application/x-shockwave-flash" width="100%"> <param name="movie" value="http://d1.scribdassets.com/ScribdViewer.swf"><param name="wmode" value="opaque"><param name="bgcolor" value="#ffffff"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><param name="FlashVars" value="document_id=30698407&access_key=key-1a98ez7rosaabtqfgz5v&page=1&viewMode=list"><embed id="doc_884350309099725" name="doc_884350309099725" src="http://d1.scribdassets.com/ScribdViewer.swf?document_id=30698407&access_key=key-1a98ez7rosaabtqfgz5v&page=1&viewMode=list" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="600" width="100%" wmode="opaque" bgcolor="#ffffff"></embed> </object><br />
<br />
<div style="color: #38761d;"><a href="http://www.scribd.com/full/30698391?access_key=key-2ljnsjxch5qfa1h6439t"><span class="Apple-style-span" style="color: red;"><span class="Apple-style-span" style="font-size: x-large;"><b>Le sujet élève</b></span></span></a></div><div style="color: #38761d;"><br />
</div><div style="color: #38761d;"><br />
</div><div align="CENTER"><br />
<a href="http://www.scribd.com/doc/30698391/seance-modele" style="display: block; font: 14px Helvetica,Arial,Sans-serif; margin: 12px auto 6px; text-decoration: underline;" title="View séance modèle on Scribd">séance modèle</a> <object data="http://d1.scribdassets.com/ScribdViewer.swf" height="500" id="doc_66863166842672" name="doc_66863166842672" rel="media:document" resource="http://d1.scribdassets.com/ScribdViewer.swf?document_id=30698391&access_key=key-2ljnsjxch5qfa1h6439t&page=1&viewMode=list" style="outline: medium none;" type="application/x-shockwave-flash" width="100%" xmlns:dc="http://purl.org/dc/terms/" xmlns:media="http://search.yahoo.com/searchmonkey/media/"> <param name="movie" value="http://d1.scribdassets.com/ScribdViewer.swf"><param name="wmode" value="opaque"><param name="bgcolor" value="#ffffff"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><param name="FlashVars" value="document_id=30698391&access_key=key-2ljnsjxch5qfa1h6439t&page=1&viewMode=list"><embed id="doc_66863166842672" name="doc_66863166842672" src="http://d1.scribdassets.com/ScribdViewer.swf?document_id=30698391&access_key=key-2ljnsjxch5qfa1h6439t&page=1&viewMode=list" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="500" width="100%" wmode="opaque" bgcolor="#ffffff"></embed> </object> <br />
<br />
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<br />
<applet archive="http://www.geogebra.org/webstart/3.2/geogebra.jar" code="geogebra.GeoGebraApplet" codebase="http://www.geogebra.org/webstart/3.2/unsigned/" height="500" name="ggbApplet" width="800"><br />
<param name="filename" value="http://flora.allain.perso.sfr.fr/Geogebra/TS/Malthus_Verhulst.ggb" /><br />
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<param name="java_arguments" value="-Xmx1000m" /><br />
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<param name="framePossible" value="true" /><br />
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<param name="showResetIcon" value="true" /><br />
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<param name="showAnimationButton" value="true" /><br />
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<param name="enableRightClick" value="true" /><br />
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<param name="enableLabelDrags" value="true" /><br />
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<param name="showMenuBar" value="false" /><br />
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<param name="showToolBar" value="false" /><br />
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<param name="showToolBarHelp" value="false" /><br />
<br />
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<param name="showAlgebraInput" value="false" /><br />
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (<a href="http://java.sun.com/getjava">Click here to install Java now</a>)<br />
</applet></div><br />
<br />
<br />
<div align="CENTER"><br />
</div><br />
<a href="http://mathsaulycee.blogspot.com/2010/04/chaos-et-carmetal.html#links"><span class="Apple-style-span" style="color: red;"><span class="Apple-style-span" style="font-size: x-large;"><b>La suite logistique (Animation CaRMetal)</b></span></span></a><br />
<br />
<br />
<br />
<div style="color: #bf9000;">-</div>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-17517387004163632242010-04-29T03:54:00.000-07:002010-05-14T15:02:54.174-07:00Calcul matriciel<div style="text-align: left;">Soit la matrice:</div><br />
<div style="text-align: center;"><a href="http://www.codecogs.com/eqnedit.php?latex=A=%5Cbegin%7Bbmatrix%7D%200%20&%20-1%5C%5C%201&%200%20%5Cend%7Bbmatrix%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?A=%5Cbegin%7Bbmatrix%7D%200%20&%20-1%5C%5C%201&%200%20%5Cend%7Bbmatrix%7D" title="A=\begin{bmatrix} 0 & -1\\ 1& 0 \end{bmatrix}" /></a></div><br />
<br />
<applet archive="wrs_net_fr.jar" code="WirisApplet_net_fr.class" codebase="http://www.wiris.net/demo/wiris/wiris-codebase/" height="400px" width="100%"><br />
<param name="version" value="2.0" /><param name="command" value="false" /><param name="commands" value="false" /><param name="interface" value="false" /><param name="XMLinitialText" value="<session lang="fr" version="2.0"><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>2</mn></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>3</mn></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>&Hat;</mo><mn>4</mn></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>" /><br />
</applet><br />
<br />
<br />
<br />
<b>Exprimez $A^2,A^3,A^4,A^5$ à l'aide de $A$ et de $I_2$.</b><br />
<br />
$A^2=-I_2$<br />
$A^3=-A$<br />
$A^4=I_2$<br />
<br />
<b><br />
Quelle expression conjecturez-vous pour $A^n$, pour tout$n$de $\mathbb{N}^*$?</b><br />
<br />
<br />
Soit $p$ un entier naturel non nul:<br />
<br />
$A^{4p+1}=A$<br />
$A^{4p+2}=-I_2$<br />
$A^{4p+3}=-A$<br />
$A^{4p}=I_2$Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-68266440541650119372010-04-29T01:57:00.000-07:002010-04-29T01:58:41.944-07:00wiris<applet archive="wrs_net_fr.jar" code="WirisApplet_net_fr.class" codebase="http://www.wiris.net/demo/wiris/wiris-codebase/" height="100%" width="100%"><param name="_cx" value="17304" /><param name="_cy" value="7250" /></applet>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-70321580183013982122010-04-26T09:53:00.000-07:002010-04-26T09:57:06.081-07:00Comportement asymptotique<center><br />
<applet archive="http://www.geogebra.org/webstart/3.2/geogebra.jar" code="geogebra.GeoGebraApplet" codebase="./" height="704" name="ggbApplet" width="600"> <br />
<br />
<br />
<br />
<br />
<br />
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<br />
<param name="filename" value="http://www.geogebra.org/en/upload/index.php?action=downloadfile&filename=comportement_asymptotique.ggb&directory=AAFrancais/OLeguay&" /><param name="java_arguments" value="-Xmx1000m" /><param name="framePossible" value="true" /><param name="showResetIcon" value="true" /><param name="showAnimationButton" value="true" /><param name="enableRightClick" value="true" /><param name="enableLabelDrags" value="false" /><param name="showMenuBar" value="false" /><param name="showToolBar" value="false" /><param name="showToolBarHelp" value="false" /><param name="showAlgebraInput" value="false" />Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (<a href="http://java.sun.com/getjava">Click here to install Java now</a>)<br />
</applet> <br />
</center><br />
<div style="color: #b45f06; text-align: center;"><b><span style="font-size: large;">Utilisez les curseurs pour modifier l'expression de $f$</span></b><br />
<br />
<br />
<br />
</div>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-25848891902644283962010-04-24T07:44:00.001-07:002010-04-24T07:44:40.565-07:00Les sujets 2010 du bac Pondichery<div style="text-align: center;"><a href="http://pedagogie.ac-toulouse.fr/lyc-francais-pondichery/espaceprofs/sujetbac/Bacpondy2010/index.html" linkindex="16"><span style="font-size: x-large;">ICI</span></a></div>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0tag:blogger.com,1999:blog-8689061186477343145.post-74782165683928130382010-04-21T06:21:00.000-07:002010-06-01T03:22:11.285-07:00Chaos, second degré et CaRMetal<div style="text-align: center;"><br />
</div><br />
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<br />
<br />
<center><br />
<br />
<applet align="center" archive="http://flora.allain.perso.sfr.fr/CarMetal/CaRMetal.jar" code="rene.zirkel.ZirkelApplet.class" height="600" width="500"><param name="_cx" value="13229" /><param name="_cy" value="15875" /></applet> </center><br />
<br />
<b><span style="font-size: large;">Déplacez $u_0$ en utilisant le clic droit. Zoomez avec la molette de la souris. Modifiez la valeur de a.</span></b><br />
<br />
<div style="text-align: center;"><br />
</div>Anonymoushttp://www.blogger.com/profile/10215812536572995002noreply@blogger.com0