File "graphs-and-derivatives.html"
Path: /StudyIB/mathsanalysis/page/658/graphs-and-derivativeshtml
File size: 64.26 KB
MIME-type: text/html
Charset: utf-8
<!DOCTYPE html><html lang="EN"><head> <script>(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-PZBWZ8J');</script> <title>Graphs and Derivatives</title><!-- Removed by WebCopy --><!--<base href="/">--><!-- Removed by WebCopy --><meta http-equiv="x-ua-compatible" content="IE=Edge"><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1"><link href="../../../css/bootstrap.flatly.min.css" rel="stylesheet" media="screen"><link rel="stylesheet" href="../../../fonts/awesome/css/font-awesome.min.css"><link href="../../../js/jquery-fancybox/jquery.fancybox.min.css" type="text/css" rel="stylesheet"><link rel="stylesheet" href="../../../css/style.min.css?v=202301301645"><meta name="robots" content="index, follow"><meta name="googlebot" content="noarchive"><meta prefix="og: http://ogp.me/ns#" property="og:title" name="og:title" content="StudyIB Maths: Analysis & Approaches: Graphs and Derivatives"> <meta prefix="og: http://ogp.me/ns#" property="og:image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta prefix="og: http://ogp.me/ns#" property="og:description" name="og:description" content="On the following page, we will look at graphs and derivatives. We will get some practice in sketching gradient functions and we will carefully consider stationary points (maximum, minimum and points of inflexion) as well as non-stationary points of inflexi"> <meta prefix="og: http://ogp.me/ns#" property="og:url" name="og:url" content="https://studyib.net/mathsanalysis/page/658/graphs-and-derivatives"> <meta prefix="og: http://ogp.me/ns#" property="og:site_name" name="og:site_name" content="StudyIB - Your IB learning companion"> <meta prefix="og: http://ogp.me/ns#" property="og:locale" name="og:locale" content="en"> <meta prefix="og: http://ogp.me/ns#" property="og:type" name="og:type" content="website"> <meta name="description" content="On the following page, we will look at graphs and derivatives. We will get some practice in sketching gradient functions and we will carefully consider stationary points (maximum, minimum and points of inflexion) as well as non-stationary points of inflexi"> <meta name="image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="keywords" content="Gradient function, Sketch, Graph, Stationary points, Maximum, Minimum, Point of inflexion, Inflection, IB, IBDP, InThinking, International Baccalaureate, Revision, Revision websites, Student Sites, Students, Learning, Guidance, Exam, Questions"> <meta itemprop="name" content="StudyIB Maths: Analysis & Approaches: Graphs and Derivatives"> <meta itemprop="description" content="On the following page, we will look at graphs and derivatives. We will get some practice in sketching gradient functions and we will carefully consider stationary points (maximum, minimum and points of inflexion) as well as non-stationary points of inflexi"> <meta itemprop="image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:title" content="StudyIB Maths: Analysis & Approaches: Graphs and Derivatives"> <meta name="twitter:description" content="On the following page, we will look at graphs and derivatives. We will get some practice in sketching gradient functions and we will carefully consider stationary points (maximum, minimum and points of inflexion) as well as non-stationary points of inflexi"> <meta name="twitter:image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="twitter:creator" content="@inthinker"><link rel="stylesheet" href="../../../css/snippets.min.css?v=202209111300"><link rel="stylesheet" href="../../../css/article.min.css?v=202211221000"><link rel="stylesheet" type="text/css" href="../../../js/slick-carousel/slick.min.css"><link rel="stylesheet" type="text/css" href="../../../js/slick-carousel/slick-theme.min.css"><style type="text/css">.filter-sl-only { display: none; }</style><script src="../../../ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script><script src="../../../js/ifvisible.min.js"></script><script>ifvisible.setIdleDuration(300);</script><link rel="apple-touch-icon-precomposed" sizes="57x57" href="../../../img/favicon/apple-touch-icon-57x57.png"> <link rel="apple-touch-icon-precomposed" sizes="114x114" href="../../../img/favicon/apple-touch-icon-114x114.png"> <link rel="apple-touch-icon-precomposed" sizes="72x72" href="../../../img/favicon/apple-touch-icon-72x72.png"> <link rel="apple-touch-icon-precomposed" sizes="144x144" href="../../../img/favicon/apple-touch-icon-144x144.png"> <link rel="apple-touch-icon-precomposed" sizes="60x60" href="../../../img/favicon/apple-touch-icon-60x60.png"> <link rel="apple-touch-icon-precomposed" sizes="120x120" href="../../../img/favicon/apple-touch-icon-120x120.png"> <link rel="apple-touch-icon-precomposed" sizes="76x76" href="../../../img/favicon/apple-touch-icon-76x76.png"> <link rel="apple-touch-icon-precomposed" sizes="152x152" href="../../../img/favicon/apple-touch-icon-152x152.png"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-196x196.png" sizes="196x196"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-96x96.png" sizes="96x96"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-32x32.png" sizes="32x32"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-16x16.png" sizes="16x16"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-128.png" sizes="128x128"> <meta name="application-name" content="StudyIB: Your IB Learning Companion"> <meta name="msapplication-TileColor" content="#A5BED5"> <meta name="msapplication-TileImage" content="/img/favicon/mstile-144x144.png"> <meta name="msapplication-square70x70logo" content="/img/favicon/mstile-70x70.png"> <meta name="msapplication-square150x150logo" content="/img/favicon/mstile-150x150.png"> <meta name="msapplication-wide310x150logo" content="/img/favicon/mstile-310x150.png"> <meta name="msapplication-square310x310logo" content="/img/favicon/mstile-310x310.png"><script>var stdHash = "6da675cbcb3d25f050b05557349abc79", stdTicket = "082b9c9c4ae3624d";</script><script src="../../../js/user/local-stats.min.js?v=202205311700"></script><link href="../../../css/subjects-frontpage.min.css?v=202302031000" rel="stylesheet"></head><body class="public mathsanalysis"> <noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-PZBWZ8J" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript> <div id="top-header"> <div class="wmap general hidden-sm hidden-xs subjects"> <div class="layout-wrapper"> <div class="container-fluid"> <a href=""> <img class="header-thinker" src="../../../img/header-thinker.svg"> </a> <h1> <a href="../../../mathsanalysis.html"> <span style="font-family: 'Helvetica Narrow','Arial Narrow',Tahoma,Arial,Helvetica,sans-serif;font-stretch: condensed;letter-spacing: -1px">IBDP Maths: Analysis & Approaches</span> </a> <a href="https://inthinking.net" title="inthinking.net" class="inthinking-logo pull-right"> <img src="../../../img/header-logo.svg" style="height: 80px; width: auto;"> </a> </h1> <p class="slogan">InThinking Revision Sites for students</p> <p class="author">Website by <strong>Richard Wade</strong></p> <p class="updated">Updated 3 February 2023</p> </div> </div> </div> <nav id="public-topnav" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <a class="navbar-brand hidden-md hidden-lg" href="../../../index.htm"> <img class="header-xs-thinker" src="../../../img/header-thinker.svg"> </a> <button type="button" class="collapsed navbar-toggle" data-toggle="collapse" data-target="#subject-navbar-collapse" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <i class="fa fa-fw fa-2x fa-user"></i> </button> <div class="brand hidden-lg hidden-md hidden-sm"> <a class="brand-xs" href="../../../mathsanalysis.html"> <strong class="title" style="white-space: nowrap;font-size: 18px;">Maths: A&A</strong> </a> </div> </div> <div class="collapse navbar-collapse" id="subject-navbar-collapse"> <ul class="nav navbar-bar navbar-userbox hidden-md hidden-lg"><li class="dropdown"><a href="#" class="dropdown-toggle" data-toggle="dropdown"><i class="fa fa-fw fa-lg fa-user-circle" style="margin-right: 5px;"></i><span style="font-size: 16px;"></span></a><ul class="dropdown-menu"><li class="dropdown-submenu"><a class="dropdown-toggle" href="#" data-toggle="dropdown"><i class="fa fa-fw fa-globe"></i> Subjects</a><ul class="dropdown-menu"><li></li><li class=""><a href="../../../biology.html">DP Biology</a></li><li class=""><a href="../../../chemistry.html">DP Chemistry</a></li><li class=""><a href="../../../englishalanglit.html">DP English A: Language & Literature</a></li><li class="active"><a href="../../../mathsanalysis.html"><i class="fa fa-caret-right"></i> DP Maths: Analysis & Approaches</a></li><li class=""><a href="../../../mathsapplications.html">DP Maths: Applications & Interpretations SL</a></li><li class=""><a href="../../../physics.html">DP Physics</a></li><li class=""><a href="../../../spanishb.html">DP Spanish B</a></li></ul></li><li class="menu-item"><a href="../../../user.html"><i class="fa fa-fw fa-dashboard"></i> Dashboard</a></li><li class="menu-item"><a href="../../../user/profile.html"><i class="fa fa-fw fa-cog"></i> My profile</a></li><li class="menu-item"><a href="../../../user/messages.html"><i class="fa fa-fw fa-envelope"></i> Messages</a></li><li class="menu-item hidden-md hidden-lg mt-xs-3"><a href="../../../index.htm?logout=1" class="btn btn-primary btn-xs-block"><i class="fa fa-fw fa-lg fa-power-off text-danger"></i>Log out</a></li></ul></li></ul> <ul class="nav navbar-bar"><li class=""><a class="home" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-home"></i> Home</a></li><li class=""><a class="topics" href="#"><i class="fa fa-fw fa-th-large"></i> Topics</a></li><li class=""><a class="favorites" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-star"></i> My favorites</a></li><li class=""><a class="qbank" href="../../test-your-knowledge.html"><i class="fa fa-fw fa-question"></i> Question bank</a></li><li class=""><a class="sitemap" href="../../sitemap.html"><i class="fa fa-fw fa-sitemap"></i> Sitemap</a></li><li class=""><a class="activity" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-calendar-check-o"></i> My activity</a></li></ul> <ul class="nav navbar-bar navbar-right"> <li> <form class="navbar-form hidden-md hidden-lg" role="search" method="get" action="mathsanalysis/search"> <div class="input-group"> <input class="form-control nav-search" name="glob" type="search" placeholder="Search..."> <span class="input-group-btn"> <button type="submit" class="btn bg-blue"> <i class="fa fa-search"> </i> </button> </span> </div> </form> <a href="#" class="toggle-menu-search hidden-sm hidden-xs" data-toggle="dropdown"> <i class="fa fa-lg fa-search"></i> </a> </li></ul> </div> </div></nav><nav id="nav-menu-search" class="shadow-bottom hidden-sm hidden-xs" style="display: none;"> <div class="container-fluid"> <form class="" role="search" method="get" action="mathsanalysis/search"> <input class="form-control nav-search" name="glob" type="search" placeholder="Search Maths: A&A..."> <button class="btn btn-sm btn-primary" type="submit"> Search </button> </form> </div></nav><div class="modal fade" tabindex="-1" role="dialog" id="modal-mini-login"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-body"> <button aria-hidden="true" data-dismiss="modal" class="close hidden-xs login-button-close" type="button">×</button> <form method="post"> <div class="form-group"> <input name="user-email" class="form-control" type="text" placeholder="Email"> </div> <div class="form-group"> <input name="user-password" class="form-control" type="password" placeholder="Password"> <input name="user-fp" class="fp" value="d8c893da5db7501669ffeeac46273ba6" type="hidden"> </div> <div class="form-group"> <a href="../../../reset-password.html" class="reset-password" style="font-weight: normal;"> Forgot your password? </a> </div> <div class="form-group"> <button type="submit" name="submit-login" value="1" class="btn btn-primary btn-block text-center"> <i class="fa fa-user fa-fw"></i> Log in </button> </div> </form> </div> </div> </div></div></div><div class="modal fade" id="show-topics" tabindex="-1" role="dialog"><div id="dialog-topics" class="modal-dialog modal-dialog-topics"><div class="modal-content"><div class="modal-body modal-body-topics"><div id="new-frontpage-toplevels-topics" class="frontpage-box"><div class="row"><div class="col-md-4 col-sm-4"><div class="item"><a href="../3016/free-access-weekend.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Free Access Weekend</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/free.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2696/start-here.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Start Here</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/start-here.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2902/examination-questions.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Examination Questions</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/exam.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2563/question-bank.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Question Bank</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/questionbank.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../537/algebra.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Algebra</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/algebra.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../539/functions.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Functions</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/function_s.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../540/geometry-trigonometry.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Geometry & Trigonometry</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/photo-by-a-href=httpsunsplash.com@pawel_czerwinskiutm_source=unsplash-utm_medium=referral-utm_content=creditcopytextpawel-czerwinskia-on-a-href=httpsunsplash.comutm_source=unsplash-utm_medium=referral-utm_co.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../549/stats-probability.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Stats & Probability</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/probability.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../550/calculus.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Calculus</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/calculus1.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../857/exam-tips.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Exam Tips</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/files/exam-tips/exam-tips.jpg'); height: 10vw;"></div></div></a></div></div></div></div></div></div></div></div><div class="container-fluid shadow-ith"><div class="row frontpage"> <div class="col-md-3" id="left-column"> <p class="visible-xs-block"></p><div class="dropdown user-menu hidden-xs" style="margin-bottom: 20px;"> <button type="button" class="btn btn-default dropdown-toggle" data-toggle="dropdown" style="border-radius: 0; padding: 6px 18px;"> <i class="fa fa-fw fa-user-circle"></i> <small><i class="fa fa-caret-down"></i></small> </button> <ul class="dropdown-menu"> <li class="dropdown-submenu"><a href="#"><i class="fa fa-fw fa-globe"></i> Subjects</a><ul id="user-subjects-menu" class="dropdown-menu"><li></li><li class=""><a href="../../../biology.html">DP Biology</a></li><li class=""><a href="../../../chemistry.html">DP Chemistry</a></li><li class=""><a href="../../../englishalanglit.html">DP English A: Language & Literature</a></li><li class="active"><a href="../../../mathsanalysis.html"><i class="fa fa-caret-right"></i> DP Maths: Analysis & Approaches</a></li><li class=""><a href="../../../mathsapplications.html">DP Maths: Applications & Interpretations SL</a></li><li class=""><a href="../../../physics.html">DP Physics</a></li><li class=""><a href="../../../spanishb.html">DP Spanish B</a></li></ul></li><li class="divider"></li><li><a href="../../../user.html"><i class="fa fa-fw fa-dashboard"></i> Dashboard</a></li><li><a href="../../../user/profile.html"><i class="fa fa-fw fa-cog"></i> My profile</a></li><li><a href="../../../user/messages.html"><i class="fa fa-fw fa-envelope"></i> Messages</a></li> <li class="divider"></li> <li> <a href="../../../index.htm?logout=1"> <i class="fa fa-fw fa-power-off"></i> Log out </a> </li> </ul></div> <div id="side-nav"><h4 class="side-nav-title dropdown"><a href="../550/calculus.html" class="dropdown-toggle toplevel-dropdown" data-toggle="dropdown"><i class="fa fa-ellipsis-v"></i></a> <a href="../550/calculus.html">Calculus</a><ul class="dropdown-menu"><li><a href="../../../mathsanalysis.html" style="padding-left: 5px; border-bottom: solid 1px #ddd"><i class="fa fa-fw fa-home"></i> Home</a></li><li><a href="../3016/free-access-weekend.html"><i class="fa fa-fw fa-caret-right"></i> Free Access Weekend</a></li><li><a href="../2696/start-here.html"><i class="fa fa-fw fa-caret-right"></i> Start Here</a></li><li><a href="../2902/examination-questions.html"><i class="fa fa-fw fa-caret-right"></i> Examination Questions</a></li><li><a href="../2563/question-bank.html"><i class="fa fa-fw fa-caret-right"></i> Question Bank</a></li><li><a href="../537/algebra.html"><i class="fa fa-fw fa-caret-right"></i> Algebra</a></li><li><a href="../539/functions.html"><i class="fa fa-fw fa-caret-right"></i> Functions</a></li><li><a href="../540/geometry-trigonometry.html"><i class="fa fa-fw fa-caret-right"></i> Geometry & Trigonometry</a></li><li><a href="../549/stats-probability.html"><i class="fa fa-fw fa-caret-right"></i> Stats & Probability</a></li><li><a href="../857/exam-tips.html"><i class="fa fa-fw fa-caret-right"></i> Exam Tips</a></li></ul><div class="pull-right"><a class="sidenav-expand" title="Expand all" href="#"><i class="fa fa-plus-circle"></i></a> <a class="sidenav-compress" title="Compress all" href="#"><i class="fa fa-minus-circle"></i></a></div></h4><ul class="side-nav level-0"><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2405/hl-mixed-differentiation.html">HL Mixed Differentiation</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../1165/mixed-integration.html">Mixed Integration</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../657/introducing-derivatives.html">Introducing Derivatives</a></label></li><li class="expanded parent selected"><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="graphs-and-derivatives.html">Graphs and Derivatives</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../742/chain-rule.html">Chain Rule</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../743/product-and-quotient-rule.html">Product and Quotient Rule</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../744/equation-of-tangent-and-normal.html">Equation of Tangent and Normal</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../695/optimisation.html">Optimisation</a></label></li><li class=""><label style="padding-left: 0px"><a class="expander" href="#" style="font-size: .9em"><i class="fa fa-fw fa-caret-right"></i></a><a href="../741/implicit-differentiation.html">Implicit Differentiation</a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../565/graphs-of-implicit-equations.html">Graphs of Implicit Equations</a></label></li></ul></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2872/related-rates-of-change.html">Related Rates of Change</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2829/lhopitals-rule.html">L'Hôpital's Rule</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../586/definite-integration.html">Definite Integration</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../730/area-between-graphs-hl.html">Area between Graphs HL</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../644/volume-of-revolution.html">Volume of Revolution</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../634/integration-by-substitution-hl.html">Integration by Substitution HL</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../635/integration-by-parts.html">Integration by Parts</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2107/differential-equations-separable-variables.html">Differential Equations - Separable Variables</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2109/differential-equations-integrating-factor.html">Differential Equations - Integrating Factor</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2108/differential-equations-homogeneous.html">Differential Equations - Homogeneous</a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> Graphs and Derivatives <a href="#" class="mark-page-favorite pull-right" data-pid="658" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../550/calculus.html">Calculus</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Graphs and Derivatives</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/differentiation/graphs/main-1.jpg" style="float: left; width: 100px; height: 100px;"> <img class="sibico" src="../../../img/sibico/hl-blue.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL"> <img class="sibico" src="../../../img/sibico/sl-blue.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL"> On the following page, we will look at graphs and derivatives. We will get some practice in sketching gradient functions and we will carefully consider stationary points (maximum, minimum and points of inflexion) as well as non-stationary points of inflexion.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p>On this page, you should learn about</p> <ul> <li>graphs of <span class="math-tex">\(f,f'\ \mathrm{and} \ f''\)</span></li> <li>local maximum and minimum points</li> <li>stationary points of inflexion</li> <li>non-stationary points of inflexion</li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <p>The following videos will help you understand all the concepts from this page</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Sketching the Gradient Function</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="384"> <p>In the following video we shall see how to sketch graphs of a gradient function</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/274353698"></iframe></div> <p>You can get some practice of sketching gradient functions by trying the activity below. This works best if you have a device with a tactile screen so that you can sketch the graphs on screen. If this is not the case, you can print a copy <a href="../../files/differentiation/graphs/sketching-derivatives.pdf" target="_blank">from here</a> and draw with a pencil.</p> <ul> <li>Choose the pen tool and sketch the gradient function</li> <li>Check your answer by clicking in the check box</li> <li>Turn off the answer and delete your sketch before moving on to the next question</li> </ul> <p style="text-align: center"><iframe height="550px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/UFUtuNSB/width/645/height/550/border/888888/smb/false/stb/true/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false" style="border:0px;" title="Sketching Derivatives" width="645px"></iframe></p> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Stationary Points</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="385"> <p>In the following section we consider stationary points, that is, points on a graph when the gradient = 0.</p> <p>There are three different types of stationary points</p> <ol> <li>Local Maxima</li> <li>Local Minima</li> <li>Point of Inflexion</li> </ol> <h4>Local Maxima and Local Minima</h4> <p>Drag the point and explore the gradient of this function</p> <p style="text-align: center"><iframe height="590px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/EJQfDsXx/width/618/height/590/border/888888/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="Local Maxima and Minima" width="618px"></iframe></p> <h4 style="text-align: center;"> </h4> <h4>Stationary Point of Inflexion</h4> <p>Drag the point and explore the gradient of this function</p> <p style="text-align: center"><iframe height="590px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/SSjkFNvF/width/618/height/590/border/888888/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="Point of Inflexion" width="618px"></iframe></p> <p>In the following video, we shall summarize the main features of graphs with stationary points and how we use the gradient function (and the gradient of the gradient function) to determine the nature of the stationary points.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/274339294"></iframe></div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Stationary Points Example</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="390"> <p>In the following video, we will look at an example in which we are required to find the coordinates of stationary points</p> <p><em>Find the co-ordinates of the stationary points on the curve <span class="math-tex">\(y = x^4 – 4x^3\)</span> and determine their nature. Sketch the curve.</em></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/274350858"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/graphs/example-finding_stationary_points.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/graphs/example-finding_stationary_points.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-expandable panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Non-stationary Points of Inflexion</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="391"> <p>Non-stationary points of inflexion exist where <span class="math-tex">\(\frac{d^2y}{dx^2}=0\)</span> and <span class="math-tex">\(\frac { dy }{ dx } \neq 0\)</span></p> <p>In the following video, we see why this is the case and we look at the following example in which we are required to find the coordinates of a point of inflexion.</p> <hr> <p><em>Find the coordinates of the point of inflexion on the curve <span class="math-tex">\(y=x^3-6x^2+13x-9\)</span></em></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/274892522"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/graphs/non-stationary-points-of-inflexion.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/graphs/non-stationary-points-of-inflexion.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/differentiation/graphs/revision-notes_graphs_and_derivatives.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/differentiation/graphs/revision-notes_graphs_and_derivatives.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#7ae54213"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="7ae54213"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Graphs and Derivatives <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-191-658" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The following diagram shows the graph fo the function <em><strong>f</strong></em></p><p><img alt="" src="../../files/differentiation/graphs/q1.jpg" style="width: 300px; height: 187px;"></p></div><div class="q-answer"><p>How many stationary points does <strong><em>f</em></strong> have? <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>Stationary points when gradient of the function = 0</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following diagram show a graph of <strong><em>f' </em></strong>, the derivative of <strong><em>f</em></strong></p><p><strong><em><img alt="" src="../../files/differentiation/graphs/q2.jpg" style="width: 300px; height: 236px;"></em></strong></p></div><div class="q-answer"><p>How many stationary points does <strong><em>f </em></strong> have? <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>Stationary points when <span class="math-tex">\(f' (x) = 0\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following diagram show a graph of <strong><em>f' </em></strong>, the derivative of <strong><em>f</em></strong></p><p><img alt="" src="../../files/differentiation/graphs/q3a.jpg" style="width: 300px; height: 219px;"></p></div><div class="q-answer"><p>How many stationary points does <strong><em>f </em></strong> have? <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p>There is no value of x for which <span class="math-tex">\(f' (x) = 0\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below is a graph of <strong><em>f</em></strong></p><p><strong><em><img alt="" src="../../files/differentiation/graphs/q3.jpg" style="width: 300px; height: 202px;"></em></strong></p><p>Fill in the missing parts in the following sentences.</p></div><div class="q-answer"><table border="1" cellpadding="0" cellspacing="0" style="width: 100%;"><tbody><tr><td>a. At the point A, the function is <input type="text" style="height: auto;" data-c="equal to zero"> <span class="review"></span></td><td><span class="q-text-draggable draggable" draggable="true">a point of inflexion</span></td></tr><tr><td>b. At the point B, there is <input type="text" style="height: auto;" data-c="a local maximum"> <span class="review"></span></td><td><span class="q-text-draggable draggable" draggable="true">even</span> <span class="q-text-draggable draggable" draggable="true">increasing</span></td></tr><tr><td>c. At point C, there is <input type="text" style="height: auto;" data-c="a point of inflexion"> <span class="review"></span></td><td><span class="q-text-draggable draggable" draggable="true">a local minimum</span> <span class="q-text-draggable draggable" draggable="true">equal to zero</span></td></tr><tr><td>d. Between A and B the function is <input type="text" style="height: auto;" data-c="increasing"> <span class="review"></span></td><td><span class="q-text-draggable draggable" draggable="true">negative</span> <span class="q-text-draggable draggable" draggable="true">concave up</span></td></tr><tr><td>e. Between C and D the function is <input type="text" style="height: auto;" data-c="concave up"> <span class="review"></span></td><td><span class="q-text-draggable draggable" draggable="true">a local maximum</span></td></tr></tbody></table></div><div class="q-explanation"><p> </p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following diagram shows the graph of the function <strong><em>f</em></strong></p><p><img alt="" src="../../files/differentiation/graphs/q5a.jpg" style="width: 300px; height: 238px;"></p><p>Fill in the blanks below.</p><ul><li>If more than one answer is required put your answers in alphabetical order</li><li>If an interval is required, put the largest possible interval</li></ul></div><div class="q-answer"><p>a. The gradient is zero at <input type="text" style="height: auto;" data-c="C"> <span class="review"></span> and <input type="text" style="height: auto;" data-c="E"> <span class="review"></span></p><p>b. There is a point of inflexion at <input type="text" style="height: auto;" data-c="D"> <span class="review"></span></p><p>c. The function is negative between <input type="text" style="height: auto;" data-c="F "> <span class="review"></span>and <input type="text" style="height: auto;" data-c="G"> <span class="review"></span></p><p>d. The function is increasing between <input type="text" style="height: auto;" data-c="C"> <span class="review"></span> and <input type="text" style="height: auto;" data-c="E"> <span class="review"></span></p><p>e. The function is concave up (positive concavity) between <input type="text" style="height: auto;" data-c="A"> <span class="review"></span> and <input type="text" style="height: auto;" data-c="D"> <span class="review"></span></p><p>f. The function is concave down (negative concavity) between <input type="text" style="height: auto;" data-c="D"> <span class="review"></span> and <input type="text" style="height: auto;" data-c="G"> <span class="review"></span></p><p>g. The solution to f(x) = 0 is at <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>h. The solution to f(0) is at <input type="text" style="height: auto;" data-c="B"> <span class="review"></span></p><p>i. The solution to f'(x) = 0 is at <input type="text" style="height: auto;" data-c="C"> <span class="review"></span> and <input type="text" style="height: auto;" data-c="E"> <span class="review"></span></p><p>j. f'(x) > 0 between <input type="text" style="height: auto;" data-c="C"> <span class="review"></span> and <input type="text" style="height: auto;" data-c="E"> <span class="review"></span></p></div><div class="q-explanation"><p>a. Gradient = 0 when graph of f is horizontal</p><p>b. D is a non-stationary point of inflexion</p><p>c. Graph is below the x axis</p><p>d. Gradient of f is positive</p><p>e. Gradient of f is increasing. Think about the graph being a bowl shape</p><p>f. Gradient of f is decreasing. Think about the graph being an upside bowl shape</p><p>g. Where graph crosses the x axis</p><p>h. Where the graph crosses the y axis</p><p>i. Where the gradient of f is zero - the graph is horizontal.</p><p>j. Where the gradient of f is positive - the graph is sloping up</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following diagram shows the graph of <strong><em>f'</em></strong>, the derivative of <strong><em>f</em></strong></p><p><strong><em><img alt="" src="../../files/differentiation/graphs/q6a.jpg" style="width: 300px; height: 336px;"></em></strong></p><p>Fill in the blanks below.</p><ul><li>If more than one answer is required put your answers in alphabetical order</li><li>If an interval is required, put the largest possible interval</li></ul></div><div class="q-answer"><p>a. There are stationary points at <input type="text" style="height: auto;" data-c="C"> <span class="review"></span> and <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>b. There is a stationary point of inflexion at <input type="text" style="height: auto;" data-c="C"> <span class="review"></span></p><p>c. There is a local minimum at <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>d. The function is concave down (negative concavity) between <input type="text" style="height: auto;" data-c="C"> <span class="review"></span> and <input type="text" style="height: auto;" data-c="E"> <span class="review"></span></p><p>e. f''(x) = 0 at <input type="text" style="height: auto;" data-c="C"> <span class="review"></span> and <input type="text" style="height: auto;" data-c="E"> <span class="review"></span></p><p>f. There is a non-stationary point of inflexion at <input type="text" style="height: auto;" data-c="E"> <span class="review"></span></p><p>g. The function is increasing in between <input type="text" style="height: auto;" data-c="F"> <span class="review"></span> and <input type="text" style="height: auto;" data-c="G"> <span class="review"></span></p></div><div class="q-explanation"><p>a. stationary points are when f'(x) = 0</p><p>b. C is a stationary point of inflexion since at C f'(x)=0. Before and after C the f'(x) < 0.</p><p>c. At F, f'(x)=0. Before F, f'(x) < 0 and after F, f'(x) > 0</p><p>d. For concave down f'(x) is decreasing</p><p>e. This is where the gradient of f'(x) = 0</p><p>f. gradient of f'(x) = 0. Before and after this point f'(x) < 0</p><p>g. The function is increasing where f'(x) > 0</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Let <span class="math-tex">\(f(x)=x^3-6x^2\)</span></p><p>The following diagram shows the graph of <strong><em>f</em></strong></p><p>There are x intercepts at x = 0 and x = c. There is a local minimum at B where x = b and a point of inflexion at A where x = a</p><p><img alt="" src="../../files/differentiation/graphs/q6.png" style="width: 400px; height: 234px;"></p></div><div class="q-answer"><p>a. Find the value of C <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p><p>b. Find the value of b <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p>c. Find the value of a <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>d. What is the rate of change of <strong><em>f </em></strong>at x = b <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p>a)</p><p><span class="math-tex">\(f(x)=x^3-6x^2\\ f(x) =x^2(x-6) \)</span></p><p>To find x interceps solve f(x) = 0</p><p>x²(x-6)=0</p><p>x=0 , x = 6</p><p>Hence C = 6</p><p>b)</p><p><span class="math-tex">\(f(x)=x^3-6x^2\\ f'(x) =3x^2-12x \)</span></p><p>To find stationary points solve <em>f'(x)=0</em></p><p><em>3x²-12x = 3x(x-4)=0</em></p><p><em>x=0 , x=4</em></p><p><em>Hence b = 4</em></p><p>c)</p><p><span class="math-tex">\(f'(x) =3x^2-12x\\ f''(x) = 6x - 12\)</span></p><p>To find point of inflexion folve f''(x)=0</p><p>6x - 12 = 0</p><p>x = 2</p><p>Since <span class="math-tex">\(f'(2)\neq 0\)</span> there is a non-stationary point of inflexion at x = 2</p><p>Hence a = 2</p><p>d) The gradient at x = 4 is zero</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="387"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The graph of y = f(x) is shown below, where B is a local maximum and C is a local minimum</p> <p><img alt="" src="../../files/differentiation/graphs/q1-question.jpg" style="width: 500px; height: 299px;"></p> <p>Sketch a graph of y = f'(x), clearly showing the images of the points B and C labellling them B' and C' respectively</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>Consider the gradient of each of the points.</p> <p>B and C are a stationary points - gradient = 0</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> <img alt="" src="../../files/differentiation/graphs/q1-answer.jpg" style="width: 500px; height: 294px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="386"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A function is given by <span class="math-tex">\(f(x)=-x^3+6x^2+4\)</span></p> <p>a) Find the coordinates of any stationary points and describe their nature</p> <p>b) Determine the values of x such that <em>f(x) </em>is a increasing function</p> <p>c) Find the coordinates of the point of inflexion</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) Solve f'(x)=0</p> <p><content>b) Draw a sketch of the graph</content></p> <p>c) point of inflexion is non-stationary solve f''(x)=0</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/graphs/esq_differentiation_graphs2a.pdf" target="_blank">here</a></p> <p><iframe frameborder="0" height="480" scrolling="no" src="../../files/differentiation/graphs/esq_differentiation_graphs2a.pdf" width="100%"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="388"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following diagram shows the graph of <span class="math-tex">\(f'\)</span>, the derivative of <em>f</em></p> <p><img alt="" src="../../files/differentiation/graphs/esq3a.png" style="width: 450px; height: 341px;"></p> <p>On the graph below, sketch the graph of y = f(x) given that f(0) = 0. Mark the images of A , B and C labelling them A' , B' and C'.</p> <p><img alt="" src="../../files/differentiation/graphs/esq3b.png" style="width: 450px; height: 341px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Consider where the gradient function is zero -> stationary points.</p> <p><content>Then consider the gradient before and after the stationary points.</content></p> <p>The gradient of f' is zero, what does this mean?</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> <img alt="" src="../../files/differentiation/graphs/esq3a_ans.png" style="width: 450px; height: 343px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="389"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Consider the function <span class="math-tex">\(f(x)=-x^3-3x^2+9x\)</span></p> <p>a) Find the coordinates of any stationary points and determine their nature</p> <p>b) Find the equation of the straight line that passes through both the local maximum and the local minimum points.</p> <p>c) Show that the point of inflexion lies on this line.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) Stationary points occur when f'(x)=0</p> <p><content>b) You can find the equation ( <span class="math-tex">\(y=mx+c\)</span> ) between two points by finding the gradient and using one of the points</content></p> <p><img alt="" src="../../files/differentiation/graphs/esq4.png" style="width: 300px; height: 232px;"></p> <p>c) Point of inflexion must be non-stationary. Solve <span class="math-tex">\(f''(x)=0\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/graphs/esq_differentiation_graphs4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/graphs/esq_differentiation_graphs4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="658"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Graphs and Derivatives</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer text-xs-center"> <span id="last-edited" class="mb-xs-3"> </span> <div class="actions mt-xs-3"> <button id="save-my-progress" type="button" class="btn btn-sm btn-primary text-center btn-xs-block"> <i class="fa fa-fw fa-floppy-o"></i> Save </button> </div> </div></div> <div id="modal-feedback" class="modal fade" tabindex="-1" role="dialog"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <h4 class="modal-title">Feedback</h4> <button type="button" class="close hidden-xs hidden-sm" data-dismiss="modal" aria-label="Close"> <span aria-hidden="true">×</span> </button> </div> <div class="modal-body"> <div class="errors"></div> <p><strong>Which of the following best describes your feedback?</strong></p> <form method="post" style="overflow: hidden"> <div class="form-group"> <div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Recommendation"> Recommend</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Problem"> Report a problem</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Improvement"> Suggest an improvement</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Other"> Other</label></div> </div> <hr> <div class="row"> <div class="col-md-6"> <div class="form-group"> <label for="feedback-name">Name</label> <input type="text" class="form-control" name="feedback-name" placeholder="Name" value=" "> </div> </div> <div class="col-md-6"> <div class="form-group"> <label for="feedback-email">Email address</label> <input type="email" class="form-control" name="feedback-email" placeholder="Email" value="@airmail.cc"> </div> </div> </div> <div class="form-group"> <label for="feedback-comments">Comments</label> <textarea class="form-control" name="feedback-comments" style="resize: vertical;"></textarea> </div> <input type="hidden" name="feedback-ticket" value="082b9c9c4ae3624d"> <input type="hidden" name="feedback-url" value="https://studyib.net/mathsanalysis/page/658/graphs-and-derivatives"> <input type="hidden" name="feedback-subject" value="11"> <input type="hidden" name="feedback-subject-name" value="Maths: Analysis & Approaches"> <div class="pull-left"> </div> </form> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary btn-xs-block feedback-submit mb-xs-3 pull-right"> <i class="fa fa-send"></i> Send </button> <button type="button" class="btn btn-default btn-xs-block m-xs-0 pull-left" data-dismiss="modal"> Close </button> </div> </div> </div></div> </article> <hr class="hidden-md hidden-lg"> <div class="hidden-md hidden-lg mt-xs-3"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <input type="hidden" id="user-id" value="38342"></div><input id="ticket" type="hidden" value="082b9c9c4ae3624d"><input id="tzoffset" type="hidden" value="new"><input id="fp" class="fp" type="hidden" value=""></div><div id="std-footer"> <div class="wmap"> <div class="layout-wrapper"> <p> <a href="https://www.inthinking.net"> © <span id="footer-year"></span> <em>InThinking</em> <script>document.getElementById("footer-year").innerHTML = new Date().getFullYear();</script> </a> | <a target="_self" href="../../../about.html"> About us </a> | <a target="_self" href="../../../terms-and-conditions.html"> Legal </a> | <a target="_self" href="../../../contact.html"> Contact </a> </p> <p> <a class="social" target="_blank" href="https://twitter.com/#!/inthinker"> <img src="../../../img/social/twitter-square.svg"> Twitter </a> <a class="social" target="_blank" href="https://www.facebook.com/inthinking.net"> <img src="../../../img/social/facebook-square.svg"> Facebook </a> <a class="social" target="_blank" href="https://www.linkedin.com/profile/view?id=139741989"> <img src="../../../img/social/linkedin-square.svg"> LinkedIn </a> </p> </div> </div></div><script src="../../../js/jquery-1.11.3.min.js"></script><script src="../../../js/bootstrap.min.js"></script><script src="../../../js/jquery-fancybox/jquery.fancybox.min.js"></script><script>var pageId = 658;</script><script type="text/javascript" src="../../../js/jqueryui/jquery-ui-custom.min.js"></script><script type="text/javascript" src="../../../js/jqueryui/jquery.ui.touch-punch.js"></script><script type="text/javascript" src="../../../js/std-quizzes-helpers.min.js?v=20220406"></script><script src="../../../ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script><script type="text/javascript" src="../../../js/slick-carousel/slick.min.js"></script><script type="text/javascript" src="../../../js/std-slide-quizzes.min.js?v=20220406"></script><script type="text/javascript" src="../../../js/jqueryui/jquery-ui.min.js"></script><script type="text/javascript" src="../../../js/jqueryui/jquery.ui.touch-punch.min.js"></script><script src="../../../js/user/page-my-progress.min.js?v=202211221000"></script><script>var sAJAX='/pages/subjects/activity/user-stats-page.php?t=082b9c9c4ae3624d&p=658&s=11&x=7374';var sData='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';var loopSecs = 30;var minSecsLog = 60;var lsKey = '11-83608-658';</script><script src="../../../js/subjects/activity/user-stats-page.min.js?v=202205311700"></script><script src="../../../js/subjects/activity/my-favorites.min.js?v=20220610"></script><script src="../../../js/subjects/feedback.min.js?v=202211221000"></script><script>var sessionUpdateSecs = 300;</script><script type="text/javascript" src="../../../js/session-updater.js"></script><script>(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)})(window,document,'script','https://www.google-analytics.com/analytics.js','ga');ga('create', 'UA-98480796-4', 'auto');ga('send', 'pageview');</script><script src="../../../js/devicefp/devicefp.min.js"></script><script src="../../../js/jq-quickfit/jq-quickfit.min.js?v=20220201"></script><script src="../../../js/frontpage-subjects.min.js?v=202211161300"></script><script type="text/javascript" src="../../../js/studyib-jq.min.js?v=202211241300"></script><script type="text/javascript">$(document).ready(function(){ $('#trigger-modal-video-overview').click(function() { if( $('#modal-video-overview .modal-body iframe').attr('src').length == 0 ) { $('#modal-video-overview .modal-body iframe').attr('src', 'https://www.youtube.com/embed/hxxtdiLdTFk?rel=0'); } }); $('#modal-video-overview').on('hidden.bs.modal', function() { $('#modal-video-overview .modal-body iframe').attr('src', ''); }); });</script></body></html>