{"id":156486,"date":"2022-03-26T01:48:39","date_gmt":"2022-03-25T20:18:39","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/rolles-theorem-and-lagranges-mean-value-theorem-representations-and-faqs\/"},"modified":"2022-12-22T21:41:19","modified_gmt":"2022-12-22T16:11:19","slug":"rolles-theorem-and-lagranges-mean-value-theorem-representations-and-faqs","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/rolles-theorem-and-lagranges-mean-value-theorem\/","title":{"rendered":"Rolle\u2019s Theorem and Lagrange\u2019s Mean Value Theorem &#8211; Representations, and FAQs"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/rolles-theorem-and-lagranges-mean-value-theorem\/#Rolles_Theorem_and_Lagranges_Mean_Value_Theorem_in_Detail\" title=\"Rolle\u2019s Theorem and Lagrange\u2019s Mean Value Theorem in Detail\">Rolle\u2019s Theorem and Lagrange\u2019s Mean Value Theorem in Detail<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/rolles-theorem-and-lagranges-mean-value-theorem\/#Lagranges_Mean_Value_Theorem_Proof\" title=\"Lagrange&#8217;s Mean Value Theorem Proof\">Lagrange&#8217;s Mean Value Theorem Proof<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/rolles-theorem-and-lagranges-mean-value-theorem\/#Geometrical_Interpretation_of_Lagranges_Mean_Value_Theorem\" title=\"Geometrical Interpretation of Lagrange\u2019s Mean Value Theorem\">Geometrical Interpretation of Lagrange\u2019s Mean Value Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/rolles-theorem-and-lagranges-mean-value-theorem\/#Rolles_theorem\" title=\"Rolle\u2019s theorem\">Rolle\u2019s theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/rolles-theorem-and-lagranges-mean-value-theorem\/#Geometrical_interpretation_of_Rolles_mean_value_theorem\" title=\"Geometrical interpretation of Rolle&#8217;s mean value theorem\">Geometrical interpretation of Rolle&#8217;s mean value theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/rolles-theorem-and-lagranges-mean-value-theorem\/#Solved_Example\" title=\"Solved Example\">Solved Example<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Rolles_Theorem_and_Lagranges_Mean_Value_Theorem_in_Detail\"><\/span>Rolle\u2019s Theorem and Lagrange\u2019s Mean Value Theorem in Detail<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Rolle\u2019s theorem states that a function has a maximum or a minimum at an interior point if and only if the function is differentiable at that point.<\/p>\n<p>Lagrange\u2019s mean value theorem states that, for a function that is differentiable on an interval, there is a point in the interval where the derivative is equal to the average of the function\u2019s derivatives at the two endpoints of the interval.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156485 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/rolles-theorem-and-lagranges-mean-value-theorem-representations-and-faqs.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/rolles-theorem-and-lagranges-mean-value-theorem-representations-and-faqs.jpg?v=1648239514 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/rolles-theorem-and-lagranges-mean-value-theorem-representations-and-faqs-300x212.jpg?v=1648239514 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Lagranges_Mean_Value_Theorem_Proof\"><\/span>Lagrange&#8217;s Mean Value Theorem Proof<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Lagrange mean value theorem states that if is a function that is continuous on the closed interval and differentiable on the open interval then there exists a point within the open interval where the derivative is equal to the average value of the function on the closed interval.<\/p>\n<p>To prove the Lagrange mean value theorem, we will use the mean value theorem for derivatives. The mean value theorem for derivatives states that if is a function that is continuous on the closed interval and differentiable on the open interval then there exists a point within the open interval where the derivative is equal to the average value of the function on the closed interval.<\/p>\n<p>We will use the mean value theorem for derivatives to show that there exists a point within the open interval where the derivative is equal to the average value of the function on the closed interval.<\/p>\n<p>We will start by assuming that there exists a point within the open interval where the derivative is equal to the average value of the function on the closed interval. We will then use the mean value theorem for derivatives to show that the derivative is also equal to the average value of the function on the open interval.<\/p>\n<p>We will start by assuming that there exists a point within the open interval where the derivative is equal to the average value of the function on the closed interval. We will then use the mean value theorem for derivatives to show that the derivative is also equal to the average value of the function on the open interval.<\/p>\n<p>We will start by assuming that<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Geometrical_Interpretation_of_Lagranges_Mean_Value_Theorem\"><\/span>Geometrical Interpretation of Lagrange\u2019s Mean Value Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Lagrange\u2019s mean value theorem states that for a given function and a given interval, there exists a point in the interval at which the function takes on its average value.<\/p>\n<p>The theorem can be illustrated geometrically by considering a line segment that connects two points on the graph of the function. The theorem states that there is a point on the line segment where the function takes on its average value.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Rolles_theorem\"><\/span>Rolle\u2019s theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A function is continuous on a closed interval if and only if it is continuous at each point in the interval.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Geometrical_interpretation_of_Rolles_mean_value_theorem\"><\/span>Geometrical interpretation of Rolle&#8217;s mean value theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Rolle&#8217;s mean value theorem states that if a function is continuous on a certain interval and differentiable on the interior of that interval, then there is a point in the interval where the function takes on its average value.<\/p>\n<p>This theorem can be interpreted geometrically as follows: imagine a function represented by a curve drawn on a coordinate plane. The curve will be continuous on a certain interval, and differentiable on the interior of that interval. There will be a point on the curve where the function takes on its average value. This point will be the midpoint of the interval.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Solved_Example\"><\/span>Solved Example<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Question:<\/p>\n<p>A bus travelling at a constant speed of 72 km\/h takes 1 hour to travel 108 km. Find the average speed of the bus.<\/p>\n<p>The average speed of the bus is 36 km\/h.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Rolle\u2019s Theorem and Lagrange\u2019s Mean Value Theorem in Detail Rolle\u2019s theorem states that a function has a maximum or a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Rolle\u2019s Theorem and Lagrange\u2019s Mean Value Theorem - Representations, and FAQs","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about Rolle's theorem and Lagrange's mean value theorem topic of maths in details explained by subject experts on infinitylearn.com. 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