@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-BLP2HLSP-6
  skos:prefLabel "calcul intégral"@fr, "integral calculus"@en ;
  a skos:Concept ;
  skos:narrower psr:-MMM6SQWN-M .

psr: a skos:ConceptScheme .
psr:-SKTRS1V0-R
  skos:prefLabel "real analysis"@en, "analyse réelle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-MMM6SQWN-M .

psr:-MMM6SQWN-M
  skos:inScheme psr: ;
  skos:definition """In mathematics, the <b>mean value theorem</b> (or <b>Lagrange theorem</b>) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
<br/>More precisely, the theorem states that if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> is a continuous function on the closed interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle [a,b]}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">[</mo>
<br/>        <mi>a</mi>
<br/>        <mo>,</mo>
<br/>        <mi>b</mi>
<br/>        <mo stretchy="false">]</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle [a,b]}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="[a,b]"></span> and differentiable on the open interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle (a,b)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo>,</mo>
<br/>        <mi>b</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (a,b)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="(a,b)"></span>, then there exists a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle c}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>c</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle c}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="c"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle (a,b)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo>,</mo>
<br/>        <mi>b</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (a,b)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="(a,b)"></span> such that the tangent at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle c}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>c</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle c}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="c"></span> is parallel to the secant line through the endpoints <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\ig (}a,f(a){\\ig )}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo maxsize="1.2em" minsize="1.2em">(</mo>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mi>a</mi>
<br/>        <mo>,</mo>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo maxsize="1.2em" minsize="1.2em">)</mo>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\ig (}a,f(a){\\ig )}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb948f947e2e80c6b7c211a1aa51fac8c7de2887" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:8.711ex; height:3.176ex;" alt="{\\\\displaystyle {\\ig (}a,f(a){\\ig )}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\ig (}b,f(b){\\ig )}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo maxsize="1.2em" minsize="1.2em">(</mo>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mi>b</mi>
<br/>        <mo>,</mo>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>b</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo maxsize="1.2em" minsize="1.2em">)</mo>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\ig (}b,f(b){\\ig )}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf862745e08ecab99533c829fd31b2d17f88c5f2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:8.247ex; height:3.176ex;" alt="{\\\\displaystyle {\\ig (}b,f(b){\\ig )}}"></span>, that is,
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f'(c)={\\rac {f(b)-f(a)}{b-a}}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>c</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>f</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>b</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>f</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>a</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>b</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>a</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f'(c)={\\rac {f(b)-f(a)}{b-a}}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e68eccc69d29a2a10d669fdd0a7f038417277b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:20.646ex; height:6.009ex;" alt="{\\\\displaystyle f'(c)={\\rac {f(b)-f(a)}{b-a}}.}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Mean_value_theorem#Mean_value_theorems_for_definite_integrals">https://en.wikipedia.org/wiki/Mean_value_theorem#Mean_value_theorems_for_definite_integrals</a>)"""@en, """En analyse réelle, le théorème de la moyenne est un résultat classique concernant l'intégration des fonctions continues d'une variable réelle, selon lequel la moyenne d'une fonction continue sur un segment se réalise comme valeur de la fonction. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_la_moyenne">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_la_moyenne</a>)"""@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_la_moyenne>, <https://en.wikipedia.org/wiki/Mean_value_theorem#Mean_value_theorems_for_definite_integrals> ;
  skos:broader psr:-BLP2HLSP-6, psr:-SKTRS1V0-R ;
  skos:prefLabel "théorème de la moyenne"@fr, "mean value theorem"@en ;
  a skos:Concept .