File "52-54-tangents-normals-and-the-second-derivative.html"
Path: /StudyIB/mathsapplications/page/2401/52-54-tangents-normals-and-the-second-derivativehtml
File size: 72.83 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>5.2 & 5.4 Tangents, normals and the second derivative</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: Applications & Interpretations SL: 5.2 & 5.4 Tangents, normals and the second derivative"> <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="In this unit we look at find the equations of tangents and normals to curves, increasing and decreasing functions and the second derivative. All this builds on the fundamental notion that calculus tells us about the rate of of change of a function."> <meta prefix="og: http://ogp.me/ns#" property="og:url" name="og:url" content="https://studyib.net/mathsapplications/page/2401/52-54-tangents-normals-and-the-second-derivative"> <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="In this unit we look at find the equations of tangents and normals to curves, increasing and decreasing functions and the second derivative. All this builds on the fundamental notion that calculus tells us about the rate of of change of a function."> <meta name="image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="keywords" content="Tangents, Normals, Gradient, Rate of change, Second derivative, Increasing, Decreasing, IB, IBDP, InThinking, International Baccalaureate, Revision, Revision websites, Student Sites, Students, Learning, Guidance, Exam, Questions, Practice problems"> <meta itemprop="name" content="StudyIB Maths: Applications & Interpretations SL: 5.2 & 5.4 Tangents, normals and the second derivative"> <meta itemprop="description" content="In this unit we look at find the equations of tangents and normals to curves, increasing and decreasing functions and the second derivative. All this builds on the fundamental notion that calculus tells us about the rate of of change of a function."> <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: Applications & Interpretations SL: 5.2 & 5.4 Tangents, normals and the second derivative"> <meta name="twitter:description" content="In this unit we look at find the equations of tangents and normals to curves, increasing and decreasing functions and the second derivative. All this builds on the fundamental notion that calculus tells us about the rate of of change of a function."> <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"><style type="text/css">.filter-hl-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 mathsapplications"> <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="../../../mathsapplications.html"> <span style="font-family: 'Helvetica Narrow','Arial Narrow',Tahoma,Arial,Helvetica,sans-serif;font-stretch: condensed;letter-spacing: -1px">IBDP Maths: Applications & Interpretations SL</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>Jim Noble, Rebecca Baker & Dan Pearcy</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="../../../mathsapplications.html"> <strong class="title" style="white-space: nowrap;font-size: 18px;">Maths: A&I (SL)</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=""><a href="../../../mathsanalysis.html">DP Maths: Analysis & Approaches</a></li><li class="active"><a href="../../../mathsapplications.html"><i class="fa fa-caret-right"></i> 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="../../../mathsapplications.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="../../../mathsapplications.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="" href="../../blogs.html"><i class="fa fa-fw fa-paperclip"></i> Blog</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="../../../mathsapplications.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="mathsapplications/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="mathsapplications/search"> <input class="form-control nav-search" name="glob" type="search" placeholder="Search Maths: A&I (SL)..."> <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="../3010/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.jpeg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../1053/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/2022-01-09_13-28-49.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../3011/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.jpeg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../1094/practice-questions.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Practice Questions</h3></div><div class="body"><div class="cropper" style="background-image: url('../../../img/gen-pics/mathsapplications/08.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2105/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('/media/mathsapplications/multiple-choice.png'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../1054/your-graphical-display-calculator.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Your Graphical Display Calculator</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/gdc-4.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../1057/number-and-algebra.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Number & Algebra</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsapplications/images/number-and-algebra-pic.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../1061/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('../../images/dans-images/functions.png'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../1070/geometry-and-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/dans-images/trig.png'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../1065/statistics-and-probability.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Statistics & Probability</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsapplications/images/prob.png'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../1078/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('../../images/dans-images/calc.png'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2800/pre-ib-preparation-.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Pre-IB Preparation</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/dans-images/pexels-keira-burton-6146978.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../1082/exams-assessment.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Exams & Assessment</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsapplications/images/exams2.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2928/for-teachers.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">For Teachers</h3></div><div class="body"><div class="cropper" style="background-image: url('../../georgia.png'); 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=""><a href="../../../mathsanalysis.html">DP Maths: Analysis & Approaches</a></li><li class="active"><a href="../../../mathsapplications.html"><i class="fa fa-caret-right"></i> 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="../1078/calculus.html" class="dropdown-toggle toplevel-dropdown" data-toggle="dropdown"><i class="fa fa-ellipsis-v"></i></a> <a href="../1078/calculus.html">Calculus</a><ul class="dropdown-menu"><li><a href="../../../mathsapplications.html" style="padding-left: 5px; border-bottom: solid 1px #ddd"><i class="fa fa-fw fa-home"></i> Home</a></li><li><a href="../3010/free-access-weekend.html"><i class="fa fa-fw fa-caret-right"></i> Free Access Weekend!</a></li><li><a href="../1053/start-here.html"><i class="fa fa-fw fa-caret-right"></i> Start here</a></li><li><a href="../3011/examination-questions.html"><i class="fa fa-fw fa-caret-right"></i> Examination Questions</a></li><li><a href="../1094/practice-questions.html"><i class="fa fa-fw fa-caret-right"></i> Practice Questions</a></li><li><a href="../2105/question-bank.html"><i class="fa fa-fw fa-caret-right"></i> Question Bank</a></li><li><a href="../1054/your-graphical-display-calculator.html"><i class="fa fa-fw fa-caret-right"></i> Your Graphical Display Calculator</a></li><li><a href="../1057/number-and-algebra.html"><i class="fa fa-fw fa-caret-right"></i> Number and Algebra</a></li><li><a href="../1061/functions.html"><i class="fa fa-fw fa-caret-right"></i> Functions</a></li><li><a href="../1070/geometry-and-trigonometry.html"><i class="fa fa-fw fa-caret-right"></i> Geometry and Trigonometry</a></li><li><a href="../1065/statistics-and-probability.html"><i class="fa fa-fw fa-caret-right"></i> Statistics and Probability</a></li><li><a href="../2800/pre-ib-preparation-.html"><i class="fa fa-fw fa-caret-right"></i> Pre-IB Preparation </a></li><li><a href="../1082/exams-assessment.html"><i class="fa fa-fw fa-caret-right"></i> Exams & Assessment</a></li><li><a href="../2928/for-teachers.html"><i class="fa fa-fw fa-caret-right"></i> For Teachers</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="../2400/51-53-introduction-to-calculus.html">5.1 & 5.3 Introduction to Calculus</a></label></li><li class="expanded parent selected"><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="52-54-tangents-normals-and-the-second-derivative.html">5.2 & 5.4 Tangents, normals and the second derivative</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2402/56-57-stationary-points-and-optimisation.html">5.6 & 5.7 Stationary Points and Optimisation</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2197/55-58-integration-and-trapezoidal-rule.html">5.5 & 5.8 Integration and Trapezoidal rule</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"> 5.2 & 5.4 Tangents, normals and the second derivative <a href="#" class="mark-page-favorite pull-right" data-pid="2401" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1078/calculus.html">Calculus</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">5.2 & 5.4 Tangents, normals and the second derivative</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"> <h2><img alt="" src="../../files/CLS/GTN/GTN-small-1.jpg" style="width: 100px; height: 100px; float: left;">Tangents, Normals, and the next level</h2> <p>In this unit we look at find the equations of tangents and normals to curves, increasing and decreasing functions and the second derivative. All this builds on the fundamental notion that calculus tells us about the rate of of change of a function.</p> <hr> <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"> <div> <p><span id="docs-internal-guid-9246535a-7fff-3745-7604-c51a984a7508"></span></p> <p dir="ltr"><span id="docs-internal-guid-9246535a-7fff-3745-7604-c51a984a7508">In this unit you should learn to…</span></p> <ul style="list-style-type:disc;"> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">find the equations of straight lines</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">find the equations of tangents and normals</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">recognise when functions are increasing and decreasing</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">find the rate of change of the rate of change.</p> </li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <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"> <div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Slides Gallery</p> </div> </div> <div class="panel-body"> <div> <p>Use these slides to review the material and key points covered in the videos.</p> <div id="carousel-262" class="dynamic-gallery carousel slide" data-id="262"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-262/gtn-nologos001.jpeg" data-fancybox="gallery-262" title="" data-caption=""><img alt="" src="../../../std-galleries/15-262/gtn-nologos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-262/gtn-nologos002.jpeg" data-fancybox="gallery-262" title="" data-caption=""><img alt="" src="../../../std-galleries/15-262/gtn-nologos002.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-262/gtn-nologos003.jpeg" data-fancybox="gallery-262" title="" data-caption=""><img alt="" src="../../../std-galleries/15-262/gtn-nologos003.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-262/gtn-nologos004.jpeg" data-fancybox="gallery-262" title="" data-caption=""><img alt="" src="../../../std-galleries/15-262/gtn-nologos004.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-262/gtn-nologos005.jpeg" data-fancybox="gallery-262" title="" data-caption=""><img alt="" src="../../../std-galleries/15-262/gtn-nologos005.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-262/gtn-nologos006.jpeg" data-fancybox="gallery-262" title="" data-caption=""><img alt="" src="../../../std-galleries/15-262/gtn-nologos006.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-262/gtn-nologos007.jpeg" data-fancybox="gallery-262" title="" data-caption=""><img alt="" src="../../../std-galleries/15-262/gtn-nologos007.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-262/gtn-nologos008.jpeg" data-fancybox="gallery-262" title="" data-caption=""><img alt="" src="../../../std-galleries/15-262/gtn-nologos008.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-262/gtn-nologos009.jpeg" data-fancybox="gallery-262" title="" data-caption=""><img alt="" src="../../../std-galleries/15-262/gtn-nologos009.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-262/gtn-nologos010.jpeg" data-fancybox="gallery-262" title="" data-caption=""><img alt="" src="../../../std-galleries/15-262/gtn-nologos010.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-262/gtn-nologos011.jpeg" data-fancybox="gallery-262" title="" data-caption=""><img alt="" src="../../../std-galleries/15-262/gtn-nologos011.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-262" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-262" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-262/gtn-nologos001-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-262/gtn-nologos002-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-262/gtn-nologos003-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/15-262/gtn-nologos004-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/15-262/gtn-nologos005-thumb128.jpg"><li><li data-index="5"><img title="Click to view" src="../../../std-galleries/15-262/gtn-nologos006-thumb128.jpg"><li><li data-index="6"><img title="Click to view" src="../../../std-galleries/15-262/gtn-nologos007-thumb128.jpg"><li><li data-index="7"><img title="Click to view" src="../../../std-galleries/15-262/gtn-nologos008-thumb128.jpg"><li><li data-index="8"><img title="Click to view" src="../../../std-galleries/15-262/gtn-nologos009-thumb128.jpg"><li><li data-index="9"><img title="Click to view" src="../../../std-galleries/15-262/gtn-nologos010-thumb128.jpg"><li><li data-index="10"><img title="Click to view" src="../../../std-galleries/15-262/gtn-nologos011-thumb128.jpg"><li></li></ol> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Finding equations of straight lines</p> </div> </div> <div class="panel-body"> <div> <p>This video covers the technique required to find the equation of a straight line given its gradient and a point on the line. This is key for finding the equations of tangents and normals.</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/595205244"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. The equation of the tangent</p> </div> </div> <div class="panel-body"> <div> <p>Here we look at how you can use differentiation to find the gradient of a curve at a given point and then work out the equation of the tangent at that point.</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/595210568"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. The equation of the normal</p> </div> </div> <div class="panel-body"> <div> <p>This is the same technique as we used for the tangent. They key here is the relationship between gradients of perpendicular lines.</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/595212496"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>4. Increasing and decreasing functions</p> </div> </div> <div class="panel-body"> <div> <p>Definition and understanding of these key terms. </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/595213387"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>5. The second derivative</p> </div> </div> <div class="panel-body"> <div> <p>Here we look at the 'rate of change' of the 'rate of change'. A key idea that helps us identify local maximums and minimums.</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/595214225"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p>This section of the page can be used for quick review. The flashcards help you go over key points and the quiz lets you practice answering questions on this subtopic.</p> <h4>Revision Cards</h4> <p>Review these condensed 'key point' flashcards to help you check and keep ideas fresh in your mind.</p> <div id="carousel-263" class="dynamic-gallery carousel slide" data-id="263"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-263/gtn-nologos001.jpeg" data-fancybox="gallery-263" title="" data-caption=""><img alt="" src="../../../std-galleries/15-263/gtn-nologos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-263/gtn-nologos002.jpeg" data-fancybox="gallery-263" title="" data-caption=""><img alt="" src="../../../std-galleries/15-263/gtn-nologos002.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-263/gtn-nologos003.jpeg" data-fancybox="gallery-263" title="" data-caption=""><img alt="" src="../../../std-galleries/15-263/gtn-nologos003.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-263" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-263" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-263/gtn-nologos001-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-263/gtn-nologos002-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-263/gtn-nologos003-thumb128.jpg"><li></li></ol> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <h4>Self Checking Quiz</h4> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above.</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-320-2401"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>The equation of the straight line that goes through the point (3, 4) with a gradient of 2 is given as y = ax + b. Find the values of a and b</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> </p></div><div class="q-explanation"><p>You are given the gradient. So you know y = 2x + b, now substitute x = 4 and y = 4 (from the given point) to get the value of b.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><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><p>......</p><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given the function <img align="middle" alt="f left parenthesis x right parenthesis equals x cubed plus 3 x squared minus 2 x plus 4" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-35.php?formula=ebfad0ff17c376f68c81c689be800f8b.png">, what is the gradient when x = -2</p></div><div class="q-answer"><p>Gradient = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="f apostrophe left parenthesis x right parenthesis equals 3 x squared plus 6 x minus 2 comma space f apostrophe left parenthesis negative 2 right parenthesis space equals space minus 2" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mo»-«/mo»«mn»2«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»2«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-36.php?formula=561f972ba7d352e8300be15465d30d28.png"><img align="middle" alt="f apostrophe left parenthesis x right parenthesis equals 3 x squared plus 6 x minus 2 comma space f apostrophe left parenthesis negative 2 right parenthesis space equals space minus 2" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mo»-«/mo»«mn»2«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»2«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-36.php?formula=561f972ba7d352e8300be15465d30d28.png"></p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <img align="middle" alt="f left parenthesis x right parenthesis equals x cubed plus 3 x squared minus 2 x plus 4" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-35.php?formula=ebfad0ff17c376f68c81c689be800f8b.png">, The tangent to this curve at x = -2 is given by y = ax +b. Find the values of a and b</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="f apostrophe left parenthesis x right parenthesis equals 3 x squared plus 6 x minus 2 comma space s o space f apostrophe left parenthesis negative 2 right parenthesis space equals space minus 2" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»s«/mi»«mi»o«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mo»-«/mo»«mn»2«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»2«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-37.php?formula=a80318687df6f0d60c0508b89f5e68fe.png"> Therefore the gradient, a = -2. So you know y = -2x + b, <img align="middle" alt="f left parenthesis negative 2 right parenthesis space equals space 12" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mo»-«/mo»«mn»2«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»12«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-38.php?formula=6063e7d60711827998ab397761e82ff3.png"> which means the line goes through the point (-2, 12). N<span style="background-color: initial;">ow substitute x = -2 and y = 12 (from the gievn point) to get the value of b.</span></p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <img align="middle" alt="f left parenthesis x right parenthesis equals 2 over 5 x squared plus 1 over x squared" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mfrac»«mn»2«/mn»«mn»5«/mn»«/mfrac»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mfrac»«mn»1«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-39.php?formula=4721396f32322ab1099e37b463346213.png">, The tangent to this curve at x = 2 is given by y = ax +b. Find the values of a and b.</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="1.35"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="-0.85"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="f left parenthesis x right parenthesis equals 0.4 x squared plus x to the power of negative 2 end exponent comma space s o space f apostrophe left parenthesis x right parenthesis equals 0.8 x squared minus 2 x to the power of negative 3 end exponent space a n d space f apostrophe left parenthesis 2 right parenthesis space equals space 1.35" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mn»0«/mn»«mo».«/mo»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»2«/mn»«/mrow»«/msup»«mo»,«/mo»«mo»§#160;«/mo»«mi»s«/mi»«mi»o«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mn»0«/mn»«mo».«/mo»«mn»8«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»3«/mn»«/mrow»«/msup»«mo»§#160;«/mo»«mi»a«/mi»«mi»n«/mi»«mi»d«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mn»2«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»1«/mn»«mo».«/mo»«mn»35«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-40.php?formula=334bc7ac031982a59584d38e2c30c479.png"> You can read this from the table function on your GDC. <img align="middle" alt="f left parenthesis 2 right parenthesis equals 1.85" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mn»2«/mn»«mo»)«/mo»«mo»=«/mo»«mn»1«/mn»«mo».«/mo»«mn»85«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-29.php?formula=2aff32144589066b9558ff792a609dbc.png">, so the line goes through the point (2, 1.85). Use this to work out the value of c when y = 1.35x + c (since you know that the gradient is 1.35)</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <img align="middle" alt="f left parenthesis x right parenthesis equals x cubed plus 3 x squared minus 2 x plus 4" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-35.php?formula=ebfad0ff17c376f68c81c689be800f8b.png">, The normal to this curve at x = -2 is given by y = ax +b. Find the values of a and b</p></div><div class="q-answer"><p> a = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="13"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)={ \quad x }^{ 3 }\quad +\quad 3{ x }^{ 2 }\quad -2x\quad +\quad 4\\ f(-2)=\quad -8\quad +\quad 12\quad +\quad 4\quad +\quad 4\quad =\quad 12\\ f'(x)={ \quad 3x }^{ 2 }\quad +\quad { 6x }\quad -2\\ f'(-2)\quad =\quad 12\quad -\quad 12\quad -\quad 2\quad =\quad -\quad 2\\ So\quad at\quad x\quad =\quad -\quad 2\quad the\quad gradient\quad of\quad the\quad function\quad is\quad -\quad 2\\ The\quad gradient\quad of\quad the\quad normal,\quad { m }_{ n }\times -2\quad =\quad -1\\ so\quad { m }_{ n }\quad =\quad 0.5\\ The\quad equation\quad of\quad the\quad normal\quad will\quad there\quad for\quad be\\ y\quad =\quad 0.5x\quad +\quad c\quad and\quad goes\quad through\quad the\quad point\quad (-2,\quad 12)\\ so,\quad 12\quad =\quad 0.5\times -2\quad +\quad c\\ therefore,\quad c\quad =\quad 13\\ Equation\quad of\quad Normal\quad is\quad y\quad =\quad 0.5x\quad +\quad 13\)</span></p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><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><p>......</p><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <span class="math-tex">\(f(x)\quad =\quad 3{ x }^{ 2 }+\frac { 5 }{ { x }^{ 2 } } \)</span> at the point x = 2.</p><p>a) The function can be re written in this form <span class="math-tex">\(f(x)\quad =\quad 3{ x }^{ 2 }+\quad p{ x }^{ q }\)</span></p><p>What are the values of p and q?</p><p>b) The differential is given as <span class="math-tex">\(f'(x)\quad =\quad r{ x }+s{ x }^{ t }\)</span> </p><p>What are the values of r, s and t?</p><p>c) What is the gradient of the function at x = 2? (give the exact answer)</p><p>The equation of the normal is given by y = ax + b.</p><p>d) Use your GDC or otherwise ind the values of a and b (correct to 3sf)</p></div><div class="q-answer"><p>a) p = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span> , q = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> </p><p>b) r = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> , s = <input type="text" style="height: auto;" data-c="-10"> <span class="review"></span> , t = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span> </p><p>c) Gradient at x = 2 is <input type="text" style="height: auto;" data-c="10.75"> <span class="review"></span> </p><p>d) a = <input type="text" style="height: auto;" data-c="-0.0930"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="13.4"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)\quad =\quad 3{ x }^{ 2 }+\frac { 5 }{ { x }^{ 2 } } \\ f(x)\quad =\quad 3{ x }^{ 2 }+\quad 5{ x }^{ -2 }\\ f(2)\quad =\quad 12\quad +\quad 1.25\quad =\quad 13.25\\ f'(x)\quad =\quad 6{ x }-10{ x }^{ -3 }\\ f'(2)\quad =\quad 12\quad -\quad 1.25\quad =\quad 10.75\\ So\quad at\quad x\quad =\quad 2\quad the\quad gradient\quad of\quad the\quad function\quad is\quad 10.75\\ The\quad gradient\quad of\quad the\quad normal,\quad { m }_{ n }\times 10.75\quad =\quad -1\\ so\quad { m }_{ n }\quad =\quad -\frac { 1 }{ 10.75 } \\ The\quad equation\quad of\quad the\quad normal\quad will\quad there\quad for\quad be\\ y\quad =\quad -\frac { 1 }{ 10.75 } x\quad +\quad c\quad and\quad goes\quad through\quad the\quad point\quad (2,\quad 13.25)\\ so,\quad 13.25\quad =\quad -\frac { 1 }{ 10.75 } \times 2\quad +\quad c\\ therefore,\quad c\quad =\quad 13.44\\ Equation\quad of\quad Normal\quad is\quad y\quad =\quad -\frac { 1 }{ 10.75 } x\quad +\quad 13.44\)</span></p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function, <span class="math-tex">\(f(x)\quad =\quad { x }^{ 3 }+\frac { 3 }{ 2 } { x }^{ 2 }-18x\)</span>shown in the diagram below</p><p style="text-align: center;"><img alt="" src="../../files/CLS/GTN/q7.jpg" style="width: 600px; height: 452px;"></p><p>The derivative of the function is given as <span class="math-tex">\(f'(x)\quad =\quad a{ x }^{ 2 }+{ bx }+c\)</span></p><p>a) What are the values of a, b and c?</p><p>b) What is the gradient of the function at x = -3</p><p>c) What is the gradient of the function at x = 2</p><p>At any given value of x on the function we can say it is either A - increasing or B - decreasing.</p><p>d) For the intervals and x values below enter either A or B as appropriate</p></div><div class="q-answer"><p>a) a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> , c = <input type="text" style="height: auto;" data-c="-18"> <span class="review"></span> </p><p>b) Gradient at x = -3 is <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> </p><p>c) Gradient at x = -3 is <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> </p><p>d) For the intervals and x values below enter either A (increasing) or B (decreasing) as appropriate</p><p><span class="math-tex">\(x=-2\)</span> <input type="text" style="height: auto;" data-c="B"> <span class="review"></span> </p><p><span class="math-tex">\(x=4\)</span> <input type="text" style="height: auto;" data-c="A"> <span class="review"></span> </p><p><span class="math-tex">\(x=-4\)</span> <input type="text" style="height: auto;" data-c="A"> <span class="review"></span> </p><p><span class="math-tex">\(x<-3\)</span> <input type="text" style="height: auto;" data-c="A"> <span class="review"></span> </p><p><span class="math-tex">\(-3<x<2\)< span=""> <input type="text" style="height: auto;" data-c="B"> <span class="review"></span> </x<2\)<></span></p><p><span class="math-tex">\(x>2\)</span> <input type="text" style="height: auto;" data-c="A"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) <span class="math-tex">\(f(x)\quad =\quad { x }^{ 3 }+\frac { 3 }{ 2 } { x }^{ 2 }-18x\\ f'(x)\quad =\quad 3{ x }^{ 2 }+3{ x }-18\)</span></p><p>b) <span class="math-tex">\(f'(x)\quad =\quad 3{ x }^{ 2 }+3{ x }-18\\ f'(-3)\quad =\quad 27\quad -9\quad -18\quad =\quad 0\)</span></p><p>c) <span class="math-tex">\(f'(x)\quad =\quad 3{ x }^{ 2 }+3{ x }-18\\ f'(2)\quad =\quad 12\quad +6\quad -18\quad =\quad 0\)</span></p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below shows the function <span class="math-tex">\(f(x)\quad =\quad { x }^{ 4 }+8{ x }^{ 3 }+6{ x }^{ 2 }-40x\)</span></p><p style="text-align: center;"><img alt="" src="../../files/CLS/GTN/q8.jpg" style="width: 700px; height: 466px;"></p></div><div class="q-answer"><p>The derivative of the function is given as <span class="math-tex">\(f'(x)\quad =\quad { ax }^{ 3 }+b{ x }^{ 2 }+c{ x }-d\)</span></p><p>a) What are the values of a, b, c and d?</p><p>a = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="24"> <span class="review"></span> , c = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span> , d = <input type="text" style="height: auto;" data-c="40"> <span class="review"></span> </p><p>b) What is the gradient where x = -5, x = -2 and x = 1</p><p>Gradient at x = -5 is <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> </p><p>Gradient at x = -2 is <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> </p><p>Gradient at x = 1 is <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> </p><p>c) What is the equation of the tangent at x = -2? y = <input type="text" style="height: auto;" data-c="56"> <span class="review"></span> </p><p>d) What is the equation of the normal at x = 1, x = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> </p><p>The function is said to be decreasing for the following intervals reading from left to right...</p><p><span class="math-tex">\(x<a,\quad b=""> what are the values of a, b and c?</a,\quad></span></p><p>a = <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> , c = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) <span class="math-tex">\(f(x)\quad =\quad { x }^{ 4 }+8{ x }^{ 3 }+6{ x }^{ 2 }-40x\\ f'(x)\quad =\quad { 4x }^{ 3 }+24{ x }^{ 2 }+12{ x }-40\)</span></p><p>b) <span class="math-tex">\(f'(x)\quad =\quad { 4x }^{ 3 }+24{ x }^{ 2 }+12{ x }-40\\ f'(-5)\quad =\quad -500\quad +\quad 600\quad -60\quad -40\quad =\quad 0\\ f'(-2)\quad =\quad -32\quad +\quad 96\quad -24\quad -40\quad =\quad 0\\ f'(1)\quad =\quad 4+24+12-40\quad =\quad 0\)</span> or using the table function on the GDC</p><p>c) at x = -2, tangent has gradient 0 and as such the equation y = 56</p><p>d) Since the gradient at x = 1 is 0, the normal is a vertical line with equation x = 1</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <span class="math-tex">\(f(x)\quad =\quad { x }^{ 3 }+\frac { 3 }{ 2 } { x }^{ 2 }-18x\)</span> that has two points at which the gradient is zero. (-3, 40.5) and ( 2, 22).</p><p>The second derivative is given as <span class="math-tex">\(f''(x)\quad =\quad ax\quad +\quad b\)</span></p><p>a) What are the values of a and b?</p><p>b) Now consider the value of the second derivative at x = -3 and use this to tell you if this point is a local maximum (max) or a local minimum (min)</p><p>c) Now consider the value of the second derivative at x = -2 and use this to tell you if this point is a local maximum (max) or a local minimum (min)</p></div><div class="q-answer"><p>a) a = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p><p>b) at x = -3, f''(x) = <input type="text" style="height: auto;" data-c="-15"> <span class="review"></span> so point is <input type="text" style="height: auto;" data-c="max"> <span class="review"></span> (enter max/min)</p><p>c) at x = 2, f''(x) = <input type="text" style="height: auto;" data-c="15"> <span class="review"></span> so point is <input type="text" style="height: auto;" data-c="min"> <span class="review"></span> (enter max/min)</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(f(x)\quad =\quad { x }^{ 3 }+\frac { 3 }{ 2 } { x }^{ 2 }-18x\\ f'(x)\quad =\quad 3{ x }^{ 2 }+3{ x }-18\\ f''(x)\quad =\quad 6x\quad +\quad 3\)</span></p><p>b) <span class="math-tex">\(f'(x)\quad =\quad 3{ x }^{ 2 }+3{ x }-18\\ f''(x)\quad =\quad 6x\quad +\quad 3\\ f''(-3)\quad =\quad -15\)</span> </p><p>Point is a maximum since the second derivative is negative, we unerstand that the gradient is changing from positive to negative as occurs at local maximum points.</p><p>c) <span class="math-tex">\(f''(x)\quad =\quad 6x\quad +\quad 3\\ f''(2)\quad =\quad 15\)</span></p><p>Point is a minimum since the second derivative is positive, we unerstand that the gradient is changing from negative to positive as occurs at local minimum points.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <span class="math-tex">\(f(x)\quad =\quad \frac { { x }^{ 4 } }{ 4 } -\frac { 4{ x }^{ 3 } }{ 3 } +\frac { { x }^{ 2 } }{ 2 } +6x\)</span> which has gradient = 0 at x = -1, 2 and 3. By finding the second derivative determine the nature of each of those points.</p></div><div class="q-answer"><p>f''(-1) = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span>so point is <input type="text" style="height: auto;" data-c="min"> <span class="review"></span> (enter max/min)</p><p>f''(2) = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span> so point is <input type="text" style="height: auto;" data-c="max"> <span class="review"></span> (enter max/min)</p><p>f''(3) = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> so point is <input type="text" style="height: auto;" data-c="min"> <span class="review"></span> (enter max/min)</p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)\quad =\quad \frac { { x }^{ 4 } }{ 4 } -\frac { 4{ x }^{ 3 } }{ 3 } +\frac { { x }^{ 2 } }{ 2 } +6x\\ f'(x)\quad =\quad { x }^{ 3 }-{ 4{ x }^{ 2 } }+x+6\\ f''(x)\quad =\quad 3{ x }^{ 2 }-8x+1\\ f''(-1)\quad =\quad 3+8+1=12\quad (positive\quad so\quad min)\\ f''(2)\quad =\quad 12-16+1=-3\quad (negative\quad so\quad max)\\ f''(3)\quad =\quad 27-24+1=4\quad (positive\quad so\quad min)\)</span></p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><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="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> </div> </div> <div class="panel-footer"> <div> </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> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.</p> <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>Question 1</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../files/CLS/GTN/CP2Q1.jpg" style="width: 700px; height: 286px;"></p> <p>Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <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/595214730"></iframe></div> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default 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> <p><img alt="" src="../../files/CLS/GTN/CP2Q2.jpg" style="width: 700px; height: 215px;"></p> <p>Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <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/595215672"></iframe></div> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <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> <p><img alt="" src="../../files/CLS/GTN/CP2Q3.jpg" style="width: 700px; height: 557px;"></p> <p>Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <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/595216158"></iframe></div> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default 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> <p><img alt="" src="../../files/CLS/GTN/CP2Q4.jpg" style="width: 700px; height: 328px;"></p> <p>Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <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/595217388"></iframe></div> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <p><span id="cke_bm_805E" style="display: none;"> </span></p> <div class="page-container panel-self-assessment" data-id="2401"> <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>5.2 & 5.4 Tangents, normals and the second derivative</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/mathsapplications/page/2401/52-54-tangents-normals-and-the-second-derivative"> <input type="hidden" name="feedback-subject" value="15"> <input type="hidden" name="feedback-subject-name" value="Maths: Applications & Interpretations SL"> <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 = 2401;</script><script type="text/javascript" src="../../../js/std-galleries.min.js"></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 type="text/javascript" src="../../../js/std-quizzes.min.js?v=2022041300"></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=2401&s=15&x=22434';var sData='k4iA6GSB03BgPm14adQyIxiA6GSB0bfq7Y9FwFiA6GSBPFAF7YcHEKvrZ0fREzP5Pm92OFQnPd92axp96GqHEGqr74AlWnv4WGcR74psWnClSYsBPVaB63v2ExpgE0ijSncrZ0EBPFAF7YcHEKvr6GSDPmjn6GyDEzAFWGcrE0ij6GyDPmjn6GyDEzAF7bKFEYv86GB2Pm1FPFAF7YcrZxP5PBAhWGcrZVas7VpRZGasSYBhWHa7IbpsE3K7IDPrOdc7ID94IU9rI0fsWnSBWHfDIGqh7nLsWVO86GqgI0f1EzLDEGahWnC8EYK4Z0EsSYB3Eziv';var loopSecs = 30;var minSecsLog = 60;var lsKey = '15-83609-2401';</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-98480796-1', '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>