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class="fa fa-fw"></i><a href="../2127/23-24-key-features-of-functions-and-graphing-.html">2.3 & 2.4 Key Features of Functions And Graphing </a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2207/25-linear-models.html">2.5 Linear models</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2256/25-quadratic-models.html">2.5 Quadratic Models</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2257/25-exponential-models.html">2.5 Exponential Models</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2186/25-trigonometric-models-.html">2.5 Trigonometric Models </a></label></li></ul></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="../1070/geometry-and-trigonometry.html">Geometry and Trigonometry</a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2398/31-volume-and-surface-area.html">3.1 Volume and surface area</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2260/32-right-angled-trigonometry.html">3.2 Right angled Trigonometry</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2261/32-non-right-angled-trigonometry.html">3.2 Non-right angled trigonometry</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2196/33-applications-of-trigonometry.html">3.3 Applications of trigonometry</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../1862/34-circles.html">3.4 Circles</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2120/35-36-perpendicular-bisectors-voronoi-diagrams-.html"> 3.5 & 3.6 Perpendicular Bisectors & Voronoi Diagrams </a></label></li></ul></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="../1065/statistics-and-probability.html">Statistics and Probability</a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2188/41-statistical-concepts.html">4.1 Statistical concepts</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2099/42-43-cumulative-frequency-box-plots-.html">4.2 & 4.3 Cumulative Frequency & Box Plots </a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2189/43-central-tendency-and-dispersion.html">4.3 Central tendency and Dispersion</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2266/44-bivariate-data-linear-correlation.html">4.4 Bivariate data & Linear Correlation</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2258/45-46-probability.html">4.5 & 4.6 Probability</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2190/47-discrete-random-variables.html">4.7 Discrete random variables</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2403/48-binomial-distribution.html">4.8 Binomial Distribution</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2191/49-normal-distribution.html">4.9 Normal Distribution</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2407/410-spearmans-rank-correlation-coefficient.html">4.10 Spearman's Rank Correlation Coefficient</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2259/411-chi-squared-independence-tests.html">4.11 Chi Squared Independence tests</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2413/411-chi-squared-goodness-of-fit-and-the-t-test.html">4.11 Chi Squared Goodness of Fit and the T-Test</a></label></li></ul></li><li class="expanded parent selected"><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="calculus.html">Calculus</a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><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=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2401/52-54-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: 14px"><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: 14px"><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></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="../2800/pre-ib-preparation-.html">Pre-IB Preparation </a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2812/pre-ib-weekly-quiz-1.html">Pre-IB Weekly Quiz 1</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2814/pre-ib-weekly-quiz-2.html">Pre IB Weekly Quiz 2</a></label></li><li class=""><label 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0px"><i class="fa fa-fw"></i><a href="../2928/for-teachers.html">For Teachers</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"> Calculus <a href="#" class="mark-page-favorite pull-right" data-pid="1078" 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> Home</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Calculus</span></li> </ol> <article id="main-article"> <div class="greyBg"> <h2><img alt="" src="../../images/dans-images/calc.png" style="width: 150px; height: 120px; float: left;">Rates of change & Infinite Sums </h2> <p>This section is your window on to the hugely signifiant branch of mathematics called Calculus. Here the focus is very much on looking at the gradients (rates of change) of functions and how they are changing. By understanding what is happening with gradients we can understand how things are changing at instantaneous moments and works towards 'optimisation' where we look to isolate the maxima and minima of functions. Imagine this is the context of a profit function with the aim if creating the circumstances where we want to maximise the profit. Working backwards is referred to as 'anti differentiation' or 'integration' and this allows for further analysis of the area under graphs through infinite summation. For example, the gradient of a distance time graph is the speed and the area under a speed time graph is the distance travelled. It is a sophisticated development of your mathematical development and we have you covered here.</p> </div> <h2>What is in this section?</h2> <div class="index-list"> <div class="item"> <div class="bg-cover cover-96" onclick="window.location='/mathsapplications/page/2400/51-53-introduction-to-calculus'" style="background-image: url('../../files/CLS/CC/CC-small.jpg')" title="5.1 & 5.3 Introduction to Calculus"> </div> <div class="content"> <h4 class="head"><a href="../2400/51-53-introduction-to-calculus.html">5.1 & 5.3 Introduction to Calculus</a></h4> <p class="body">This is the first of three sections on differential calculus. This is significant branch of mathematics with lots of applications. It builds very nicely on other concepts in the course and those you are likely to have covered previously. The...</p> </div> </div> <div class="item"> <div class="bg-cover cover-96" onclick="window.location='/mathsapplications/page/2401/52-54-tangents-normals-and-the-second-derivative'" style="background-image: url('../../files/CLS/GTN/GTN-small.jpg')" title="5.2 & 5.4 Tangents, normals and the second derivative"> </div> <div class="content"> <h4 class="head"><a href="../2401/52-54-tangents-normals-and-the-second-derivative.html">5.2 & 5.4 Tangents, normals and the second derivative</a></h4> <p class="body">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> </div> </div> <div class="item"> <div class="bg-cover cover-96" onclick="window.location='/mathsapplications/page/2402/56-57-stationary-points-and-optimisation'" style="background-image: url('../../files/CLS/spo/spo.png')" title="5.6 & 5.7 Stationary Points and Optimisation"> </div> <div class="content"> <h4 class="head"><a href="../2402/56-57-stationary-points-and-optimisation.html">5.6 & 5.7 Stationary Points and Optimisation</a></h4> <p class="body">This is the part where we learn about how we use calculus to solve different kinds of problems. So far we have spent a lot of time learning about how calculus helps us to work out the gradient of given function for any given point. In this...</p> </div> </div> <div class="item"> <div class="bg-cover cover-96" onclick="window.location='/mathsapplications/page/2197/55-58-integration-and-trapezoidal-rule'" style="background-image: url('../../images/page-pic.jpg')" title="5.5 & 5.8 Integration and Trapezoidal rule"> </div> <div class="content"> <h4 class="head"><a href="../2197/55-58-integration-and-trapezoidal-rule.html">5.5 & 5.8 Integration and Trapezoidal rule</a></h4> <p class="body">In this section of the course you will learn about an area of calculus called integration. This an be considered in multiple ways, both as the inverse of differentiation (anti-differentiation) and in finding areas under curves. We will also...</p> </div> </div> </div> <h2>Key Questions</h2> <p>The following are list of key questions that you might be asking about these topics with links to the places where you can get the answers!</p> <h2>What is calculus for?</h2> <p>Calculus is used in a huge number of situations, to find rates of change, areas, and optimise solutions. <a href="calculus.html" title="Calculus">Calculus</a> <a href="../2400/51-53-calculus-concept.html" title="5.1 & 5.3 Calculus Concept">5.1 & 5.3 Calculus Concept</a></p> <h2>How do you find tangents and normals</h2> <p>To find the tangents and normals to a curve, you must first find the gradient at that point. <a href="../2401/52-54-calculus-part-2.html" title="5.2 & 5.4 Calculus part 2">5.2 & 5.4 Calculus part 2</a></p> <h2>How do you find stationary points</h2> <p>Stationary points are where the gradient of a function is zero. This is a really useful property, that helps us to solve all sorts of problems. <a href="../2402/56-57-calculus-part-3.html" title="5.6 & 5.7 Calculus Part 3">5.6 & 5.7 Calculus Part 3</a></p> <h2>What is the opposite of differentiation</h2> <p>The opposite of differentiation is integrating, and it is used all the time! <a href="../2197/55-58-integration-and-trapezoidal-rule.html" title="5.5 & 5.8 Integration and Trapezoidal rule">5.5 & 5.8 Integration and Trapezoidal rule</a></p> <hr class="thin"><div class="row"><div class="col-md-12"><br><ul class="resources"><li class="resource-item"><div class="img-container"><img class="thumb thumb-cover" src="../../files/CLS/CC/CC-small.jpg"></div><div class="media-body"><h4 style="font-size: 19px;"><a class="text-normal" href="../2400/51-53-introduction-to-calculus.html">5.1 & 5.3 Introduction to Calculus</a></h4><p class="body">This is the first of three sections on differential calculus. This is significant branch of mathematics with lots of applications. It builds very nicely on other concepts in the course and those you are likely to have covered previously. The...</p></div></li><li class="resource-item"><div class="img-container"><img class="thumb thumb-cover" src="../../files/CLS/GTN/GTN-small.jpg"></div><div class="media-body"><h4 style="font-size: 19px;"><a class="text-normal" href="../2401/52-54-tangents-normals-and-the-second-derivative.html">5.2 & 5.4 Tangents, normals and the second derivative</a></h4><p class="body">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></div></li><li class="resource-item"><div class="img-container"><img class="thumb thumb-cover" src="../../files/CLS/spo/spo.png"></div><div class="media-body"><h4 style="font-size: 19px;"><a class="text-normal" href="../2402/56-57-stationary-points-and-optimisation.html">5.6 & 5.7 Stationary Points and Optimisation</a></h4><p class="body">This is the part where we learn about how we use calculus to solve different kinds of problems. So far we have spent a lot of time learning about how claculus helps us to work out the gradient of given function for any given point. In this...</p></div></li><li class="resource-item"><div class="img-container"><img class="thumb thumb-cover" src="../../images/page-pic.jpg"></div><div class="media-body"><h4 style="font-size: 19px;"><a class="text-normal" href="../2197/55-58-integration-and-trapezoidal-rule.html">5.5 & 5.8 Integration and Trapezoidal rule</a></h4><p class="body">In this section of the course you will learn about an area of calculus called integration. This an be considered in multiple ways, both as the inverse of differentiation (anti-differentiation) and in finding areas under curves. We will also...</p></div></li></ul></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 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