Derniers messages sur Zeste de Savoirhttps://zestedesavoir.com/forums/2021-08-30T11:34:36+02:00Les derniers messages parus sur le forum de Zeste de Savoir.Limite et dérivabilité, message #2371202021-08-30T11:34:36+02:00Holosmos/@Holosmoshttps://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237120<p>D’ailleurs, même si on dépasse le cadre du sujet ici, ces quantités <em>conjuguées</em> ont le même sens algébrique que la conjugaison complexe.</p>
<p>Si <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span></span></span></span></span> est corps avec <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false">[</mo><msqrt><mi>a</mi></msqrt><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">K[\sqrt{a}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.05028em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">[</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">a</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.23972em;"></span><span class="mord">−</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">a</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span></span></span></span></span> peut être <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mn>2</mn></msqrt></mrow><annotation encoding="application/x-tex">\sqrt 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.13278em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;">2</span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span></span></span></span></span> Ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mrow><mo>−</mo><mn>1</mn></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.17444499999999996em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8655550000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-2.825555em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.17444499999999996em;"><span></span></span></span></span></span></span></span></span></span>, ce qui compte vraiment c’est l’équation <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mtext>ˆ</mtext><mn>2</mn><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">xˆ2 =a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mord">ˆ</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></p>
<p>Maintenant, pour tout <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">x\in K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><msqrt><mi>a</mi></msqrt><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msqrt><mi>a</mi></msqrt><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x+\sqrt a)(x - \sqrt a)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.05028em;vertical-align:-0.25em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="padding-left:0.833em;">a</span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.05028em;vertical-align:-0.25em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="padding-left:0.833em;">a</span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> est invariant par la conjugaison de Galois et appartient donc à <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span></span></span></span></span>. Le calcul montre que c’est <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mtext>ˆ</mtext><mn>2</mn><mo>−</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">xˆ2 -a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">x</span><span class="mord">ˆ</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span>.</p>
<p>Mais ce qui est pas mal ici, c’est que le fait d’être exprimable dans <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span></span></span></span></span> (ce qui signifie que le nombre est devenu plus simple) est systématique par invariance du groupe de Galois.</p>Limite et dérivabilité, message #2371192021-08-30T11:21:19+02:00cvanaret/@cvanarethttps://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237119<figure><blockquote>
<p>Merci Renault, je m’en suis rendu compte pile après.</p>
<p>À titre personnel est-ce que tu as le coup d’œil facile sur ce genre de chose ? J’ai quand même mis des heures à buter là-dessus.</p>
</blockquote><figcaption><a href="https://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237105">Ge0</a></figcaption></figure>
<p>Ca s’appelle la quantité conjuguée : <a href="https://fr.wikipedia.org/wiki/Quantit%C3%A9_conjugu%C3%A9e">https://fr.wikipedia.org/wiki/Quantit%C3%A9_conjugu%C3%A9e</a></p>
<p>Lorsque les racines passent du numérateur au dénominateur (ou inversement) avec un changement de signe entre les deux, c’est bon signe <img src="/static/smileys/svg/clin.svg" alt=";)" class="smiley"></p>Limite et dérivabilité, message #2371152021-08-29T21:00:01+02:00Holosmos/@Holosmoshttps://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237115<p>Tu dérives à gauche et à droite l’égalité <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\sqrt{x^2}=x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.08494599999999997em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9550540000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.915054em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.08494599999999997em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span></span> par la règle de composition </p>Limite et dérivabilité, message #2371122021-08-29T19:27:45+02:00Ge0/@Ge0https://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237112<p>Merci à tous pour vos réponses.</p>
<figure><blockquote>
<p>Tu as un exercice parallèle où tu peux calculer la dérivée de sqrt(x) comme dérivée de la réciproque de x2. Si tu ne l’as jamais fait, c’est l’occasion</p>
</blockquote><figcaption><a href="https://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237110">Holosmos</a></figcaption></figure>
<p>Tu peux me fournir plus de détails s’il te plaît ?</p>Limite et dérivabilité, message #2371102021-08-29T18:13:06+02:00Holosmos/@Holosmoshttps://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237110<p>Tu as un exercice parallèle où tu peux calculer la dérivée de sqrt(x) comme dérivée de la réciproque de x2. Si tu ne l’as jamais fait, c’est l’occasion</p>Limite et dérivabilité, message #2371092021-08-29T17:21:29+02:00Lucas-84/@Lucas-84https://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237109<figure><blockquote>
<p>Aussi, si vous savez comment mettre mettre <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x\to0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span> en-dessous de la <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟶</mo></mrow><annotation encoding="application/x-tex">\longrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟶</span></span></span></span></span>, je suis preneur.</p>
</blockquote><figcaption><a href="https://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237100">Ge0</a></figcaption></figure>
<p>Perso avec mathjax j’utilise <code>\underset{x\to 0}{\longrightarrow}</code>. Pour ce qui est de cette histoire de se débarrasser des racines au dénominateur, c’est à ranger avec les astuces calculatoires plus utilisées après le lycée. C’est pas grave de ne pas "voir" ce genre de choses au début, mais c’est bien d’arriver à comprendre d’où ça vient (en se rendant compte que la formule du corrigé est essentiellement équivalente à <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msqrt><mi>x</mi></msqrt><mo>−</mo><msqrt><mi>a</mi></msqrt><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msqrt><mi>x</mi></msqrt><mo>+</mo><msqrt><mi>a</mi></msqrt><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>−</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">(\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a})=x-a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.05028em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.05028em;vertical-align:-0.25em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">a</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
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c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.05028em;vertical-align:-0.25em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">a</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
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c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span>).</p>Limite et dérivabilité, message #2371082021-08-29T16:51:55+02:00Renault/@Renaulthttps://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237108<blockquote>
<p>À titre personnel est-ce que tu as le coup d’œil facile sur ce genre de chose ? J’ai quand même mis des heures à buter là-dessus.</p>
</blockquote>
<p>Ça dépend du nombre de cafés avant. <img src="/static/smileys/svg/heureux.svg" alt=":D" class="smiley"></p>
<p>Disons que comme c’est un cas assez classique en lycée / prépa, cela est devenu un réflexe de regarder de ce côté en priorité pour gagner du temps. Mais ça m’arrive de louper des choses évidentes pendant longtemps.</p>Limite et dérivabilité, message #2371052021-08-29T16:10:25+02:00Ge0/@Ge0https://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237105<p>Merci Renault, je m’en suis rendu compte pile après.</p>
<p>À titre personnel est-ce que tu as le coup d’œil facile sur ce genre de chose ? J’ai quand même mis des heures à buter là-dessus.</p>Limite et dérivabilité, message #2371042021-08-29T16:09:06+02:00Renault/@Renaulthttps://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237104<p>Pense aux identités remarquables du second degré : <a href="https://fr.wikipedia.org/wiki/Identit%C3%A9_remarquable">https://fr.wikipedia.org/wiki/Identit%C3%A9_remarquable</a> <img src="/static/smileys/svg/clin.svg" alt=";)" class="smiley"></p>Limite et dérivabilité, message #2371002021-08-29T16:02:21+02:00Ge0/@Ge0https://zestedesavoir.com/forums/sujet/15650/limite-et-derivabilite/?page=1#p237100<p>Salut à toutes et à tous,</p>
<p>Dans un de mes livres, je suis tombé sur un exemple où il est question d’un exercice de démonstration de dérivabilité d’une fonction :</p>
<blockquote>
<p>Soit <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span></span></span></span></span> la fonction définie sur <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[0, +\infty[</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mopen">[</span></span></span></span></span> par <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mi>x</mi></msqrt></mrow><annotation encoding="application/x-tex">f(x) = \sqrt{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.23972em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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c5.3,-9.3,12,-14,20,-14
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span></span></span></span></span>.</p>
<ul>
<li>Si <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a > 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span>, montrer que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span></span></span></span></span> est dérivable en <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span>.</li>
<li>La fonction <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span></span></span></span></span> est-elle dérivable en <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span> ?</li>
</ul>
</blockquote>
<p>Étant incapable à mon niveau d’aboutir à une pareille démonstration, je suis allé voir le corrigé :</p>
<blockquote>
<p>La fonction <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>↦</mo><msqrt><mi>x</mi></msqrt></mrow><annotation encoding="application/x-tex">f : x \mapsto \sqrt{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">↦</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.23972em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span></span></span></span></span> est définie sur <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[0, +\infty[</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mopen">[</span></span></span></span></span> par <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mi>x</mi></msqrt></mrow><annotation encoding="application/x-tex">f(x) = \sqrt{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.23972em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span></span></span></span></span>.</p>
<ul>
<li>Si <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a > 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span>, pour tout <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo mathvariant="normal">≠</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x \neq a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mrel"></span></span><span class="fix"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span>, on a :</li>
</ul>
<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><msqrt><mi>x</mi></msqrt><mo>−</mo><msqrt><mi>a</mi></msqrt></mrow><mrow><mi>x</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><msqrt><mi>a</mi></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\sqrt{x}-\sqrt{a}}{x - a} = \frac{1}{\sqrt{x}+\sqrt{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.24661em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.47728em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">a</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25144em;vertical-align:-0.9300000000000002em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.30972em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">a</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9300000000000002em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div>
<p>et donc <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span></span></span></span></span> dérivable en <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span> avec</p>
<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><mo><mi>lim</mi><mo></mo></mo><mrow><mi>x</mi><mo>→</mo><mi>a</mi></mrow></munder><mfrac><mrow><msqrt><mi>x</mi></msqrt><mo>−</mo><msqrt><mi>a</mi></msqrt></mrow><mrow><mi>x</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mi>a</mi></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">\lim_{x\to a} \frac{\sqrt{x}-\sqrt{a}}{x - a} = \frac{1}{2\sqrt{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.24661em;vertical-align:-0.7693300000000001em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="mrel mtight">→</span><span class="mord mathdefault mtight">a</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.47728em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">a</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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<ul>
<li>En <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span>, elle n’est pas dérivable car pour tout <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x > 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span> :</li>
</ul>
<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mi>x</mi></mfrac><mo>=</mo><mfrac><msqrt><mi>x</mi></msqrt><mi>x</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><msqrt><mi>x</mi></msqrt></mfrac><msub><mo>⟶</mo><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\frac{f(x)-f(0)}{x} = \frac{\sqrt{x}}{x} = \frac{1}{\sqrt{x}} \longrightarrow_{x\to0} +\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.16328em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.47728em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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</blockquote>
<p>(Je n’ai pas réussi à mettre <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x\to0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span> en-dessous de la <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟶</mo></mrow><annotation encoding="application/x-tex">\longrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟶</span></span></span></span></span> comme indiqué dans mon livre).</p>
<p>Ce que je ne comprends pas, c’est l’égalité suivante :</p>
<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><msqrt><mi>x</mi></msqrt><mo>−</mo><msqrt><mi>a</mi></msqrt></mrow><mrow><mi>x</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><msqrt><mi>a</mi></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\sqrt{x}-\sqrt{a}}{x - a} = \frac{1}{\sqrt{x}+\sqrt{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.24661em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.47728em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">a</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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<p>Je n’arrive pas à passer de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><msqrt><mi>x</mi></msqrt><mo>−</mo><msqrt><mi>a</mi></msqrt></mrow><mrow><mi>x</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\sqrt{x}-\sqrt{a}}{x - a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.441331em;vertical-align:-0.403331em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.038em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">a</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4738665em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8059050000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathdefault mtight">x</span></span></span><span style="top:-2.765905em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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<p>Aussi, si vous savez comment mettre mettre <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x\to0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span> en-dessous de la <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟶</mo></mrow><annotation encoding="application/x-tex">\longrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟶</span></span></span></span></span>, je suis preneur.</p>
<p>Merci d’avance.</p>
<hr>
<p>Edit : en fait c’est tout con, c’est une identité remarquable type <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext>²</mtext><mo>−</mo><mi>b</mi><mtext>²</mtext><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a² - b² = (a + b)(a - b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">a</span><span class="mord">²</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span><span class="mord">²</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span></span>…</p>
<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><msqrt><mi>x</mi></msqrt><mo>−</mo><msqrt><mi>a</mi></msqrt></mrow><mrow><mi>x</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msqrt><mi>x</mi></msqrt><mo>−</mo><msqrt><mi>a</mi></msqrt></mrow><mrow><mo stretchy="false">(</mo><msqrt><mi>x</mi></msqrt><mo>+</mo><msqrt><mi>a</mi></msqrt><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msqrt><mi>x</mi></msqrt><mo>−</mo><msqrt><mi>a</mi></msqrt><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><msqrt><mi>a</mi></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\sqrt{x}-\sqrt{a}}{x-a} = \frac{\sqrt{x}-\sqrt{a}}{(\sqrt{x}+\sqrt{a})(\sqrt{x}-\sqrt{a})} = \frac{1}{\sqrt{x}+\sqrt{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.24661em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.47728em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">a</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathdefault">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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<p>C’était pas intuitif de suite, quand même…</p>Cherche une équipe pour le TFJM, message #1732272018-02-03T18:20:10+01:00GouleFutée/@GouleFut%C3%A9ehttps://zestedesavoir.com/forums/sujet/10239/cherche-une-equipe-pour-le-tfjm/?page=1#p173227<p>Oui, j’ai posté un message sur la page Facebook, je n’ ai eu qu’une réponse d’une personne qui ne m’as jamais recontacté…</p>Cherche une équipe pour le TFJM, message #1731992018-02-03T09:38:44+01:00Holosmos/@Holosmoshttps://zestedesavoir.com/forums/sujet/10239/cherche-une-equipe-pour-le-tfjm/?page=1#p173199<p>Tu as essayé de contacter les organisateurs ?</p>Cherche une équipe pour le TFJM, message #1731982018-02-03T09:20:17+01:00GouleFutée/@GouleFut%C3%A9ehttps://zestedesavoir.com/forums/sujet/10239/cherche-une-equipe-pour-le-tfjm/?page=1#p173198<p>bonjour à tous, </p>
<p>j’ai découvert cet été le <a href="https://tfjm.org/">Tournoi Français des Jeunes mathématiciens & mathématiciennes</a> et j ’ai depuis très envie de participer. Malheureusement, je ne trouve pas d équipe et le seul espoir que j avais est tombé à l’eau. Si parmis vous, il y a des participants sur Paris ou ses environs qui cherchent un dernier membre pour participer, j’aimerais beaucoup rejoindre votre équipe. Merci beaucoup de vos réponses <img alt=":)" src="/static/smileys/smile.png"> .</p>À la recherche de la lumière bleue, message #1442902017-03-13T17:54:52+01:00Holosmos/@Holosmoshttps://zestedesavoir.com/forums/sujet/8148/a-la-recherche-de-la-lumiere-bleue/?page=1#p144290<p>Bonjour,</p>
<p>La bêta du contenu « À la recherche de la lumière bleue » a été désactivée.</p>À la recherche de la lumière bleue, message #1434942017-03-06T16:09:08+01:00Emel/@Emelhttps://zestedesavoir.com/forums/sujet/8148/a-la-recherche-de-la-lumiere-bleue/?page=1#p143494<p>Bêta mise à jour pour tenir compte des remarques faites.</p>
<p>Résumé des changements :</p>
<ul>
<li>ajout d’une image montrant que bleu+vert+rouge=blanc ;</li>
<li>ajout d’un lien vers le <a href="https://zestedesavoir.com/tutoriels/279/energie-solaire-du-panneau-photovoltaique-au-reseau-electrique/">tutoriel sur l’énergie solaire</a> pour ceux qui souhaitent en savoir plus sur le fonctionnement d’un semi-conducteur ; </li>
<li>correction de la dernière partie sur l’efficacité lumineuse.</li>
</ul>À la recherche de la lumière bleue, message #1434872017-03-06T14:40:54+01:00Emel/@Emelhttps://zestedesavoir.com/forums/sujet/8148/a-la-recherche-de-la-lumiere-bleue/?page=1#p143487<p>Merci pour vos commentaires, ils sont tous très appréciés. <img alt=":)" src="/static/smileys/smile.png"></p>
<p>Pour le principe du semi-conducteur, je l’explique déjà (très) vite fait. Si je donnais plus de détails, ce serait beaucoup plus long et je sortirais du cadre de l’article. Je peux éventuellement mettre en lien <a href="https://zestedesavoir.com/tutoriels/279/energie-solaire-du-panneau-photovoltaique-au-reseau-electrique/">le tuto sur l’énergie solaire</a> qui détaille un peu plus ce qu’est un semi-conducteur (dans le sens lumière -> électricité ; pour la diode électroluminescente, c’est dans le sens électricité -> lumière, mais le principe reste le même).</p>
<p>En ce qui concerne le "pourquoi il faut du bleu pour faire du blanc", c’est vrai que je dis simplement qu’il faut mélanger le rouge, le vert et le bleu. Idem, si je détaille trop, ce ne sera plus un bref article. Je peux ajouter une image des trois couleurs primaires qui se mélangent, peut-être que ce sera plus parlant.</p>
<p>Sur le "consomme quatre fois moins", tel que c’est écrit, c’est une erreur. <img alt=":-°" src="/static/smileys/siffle.png"> Merci de l’avoir signalée. En fait, la valeur de 300 lm/W correspond au maximum de ce qu’on peut atteindre aujourd’hui en recherche et développement. Les ampoules à LED qui sont en vente en supermarché (celles qui sont directement destinées à remplacer les ampoules à incandescence) ont une efficacité plus faible, autour de 65 lm/W : là on est bien quatre fois plus efficace (seulement, si j’ose dire). Je vais revoir ce passage.</p>À la recherche de la lumière bleue, message #1434722017-03-06T11:05:40+01:00Gwend@l/@Gwend%40lhttps://zestedesavoir.com/forums/sujet/8148/a-la-recherche-de-la-lumiere-bleue/?page=1#p143472<figure>
<blockquote>
<blockquote>
<p>Et notre grande gagnante est donc… la LED ! Et de très loin. Avec ces chiffres, on peut déduire qu’une LED consomme quatre fois moins d’électricité qu’une vieille ampoule. </p>
</blockquote>
<p>Comment on peut consommer 4 fois moins seulement en passant de 300ml/W à 16lm/W ? Il n’y a pas une petite erreur ou c’est moi qui suis trop noob en physique ?^^ Moi j’aurai envie de dire qu’elle consomme presque 20 fois moins non ?</p>
<p>Sinon je trouve l’article intéressant et facile à lire. Je me coucherai moins bête ce soir, merci à toi. <img alt=":)" src="/static/smileys/smile.png"></p>
</blockquote>
<figcaption><a href="https://zestedesavoir.com/forums/sujet/8148/a-la-recherche-de-la-lumiere-bleue/?page=1#p143456">Demandred</a></figcaption>
</figure>
<p>J’allais poser la même question. Du coup il doit y avoir un truc à expliquer là <img alt=":)" src="/static/smileys/smile.png"></p>
<p>Sinon j’ai beaucoup aimé la brève !
Le ton est sympa et le contenu original et bien expliquer.</p>
<p>En effet, pour être "plus complet", expliquer pourquoi il nous faut du bleu pour faire du blanc et ce qu’est un peu plus en détail un semi-conducteur pourrait être bien… mais là on s’oriente sur un article plus conséquent. Peut-être un simple renvoi vers des liens ? (J’ai découvert <a href="https://zestedesavoir.com/tutoriels/706/les-espaces-de-couleurs-rvb-et-tsv/">ce tuto</a> du coup <img alt=":P" src="/static/smileys/langue.png"> )</p>À la recherche de la lumière bleue, message #1434562017-03-05T23:54:39+01:00Demandred/@Demandredhttps://zestedesavoir.com/forums/sujet/8148/a-la-recherche-de-la-lumiere-bleue/?page=1#p143456<blockquote>
<p>Et notre grande gagnante est donc… la LED ! Et de très loin. Avec ces chiffres, on peut déduire qu’une LED consomme quatre fois moins d’électricité qu’une vieille ampoule. </p>
</blockquote>
<p>Comment on peut consommer 4 fois moins seulement en passant de 300ml/W à 16lm/W ? Il n’y a pas une petite erreur ou c’est moi qui suis trop noob en physique ?^^ Moi j’aurai envie de dire qu’elle consomme presque 20 fois moins non ?</p>
<p>Sinon je trouve l’article intéressant et facile à lire. Je me coucherai moins bête ce soir, merci à toi. <img alt=":)" src="/static/smileys/smile.png"></p>À la recherche de la lumière bleue, message #1434462017-03-05T21:20:41+01:00pierre_24/@pierre_24https://zestedesavoir.com/forums/sujet/8148/a-la-recherche-de-la-lumiere-bleue/?page=1#p143446<p>Je ne sais pas à quel point c’est compliqué à faire en quelques lignes, mais je crois que le principe d’un semi-conducteur devrait être expliqué <img alt=":)" src="/static/smileys/smile.png"></p>À la recherche de la lumière bleue, message #1434432017-03-05T19:56:01+01:00Emel/@Emelhttps://zestedesavoir.com/forums/sujet/8148/a-la-recherche-de-la-lumiere-bleue/?page=1#p143443<p>Tout le monde se secoue ! <img alt=":D" src="/static/smileys/heureux.png"></p>
<p>J’ai commencé (il y a une heure) la rédaction d’un article au doux nom
de « À la recherche de la lumière bleue » et j’ai pour objectif de proposer en validation
un texte aux petits oignons. Je fais donc appel à votre bonté sans
limites pour dénicher le moindre pépin, que ce soit à propos
du fond ou de la forme. Vous pourrez consulter la bêta à votre guise à
l’adresse suivante :</p>
<div align="center">
<p><a href="https://zestedesavoir.com/contenus/beta/1752/a-la-recherche-de-la-lumiere-bleue/">À présent, c’est à vous !</a> </p>
</div>
<p>Merci !</p>
<p>Voilà un article qui traînait depuis longtemps dans mes tiroirs numériques. Je l’ai passé en format "brève", et hop !
L’objectif est de présenter rapidement le prix Nobel de physique 2014 : il s’agit de vulgarisation.</p>