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href="../../../mathsanalysis.html"><i class="fa fa-home"></i> Home</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Start Here</span></li> </ol> <article id="main-article"> <p><img alt="" src="../../images/start-here.jpg" style="float: left; width: 150px; height: 100px;">Welcome to the InThinking Mathematics Analysis and Approaches Revision Site! This site is designed to be your one-stop-virtual teacher for any help that you might need throughout the IB course. Whether you are trying to secure that all important grade 4, or wanting to change your 6s into 7s, this site has all you need:</p> <hr class="hidden-separator"> <h4>Site Overview</h4> <p>See a quick overview of the site and some suggestions about how best to use it from here</p> <hr class="hidden-separator"> <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/673885351"></iframe></div> <hr class="hidden-separator"> <div class="purple"> <ul> <li><strong>Video tutorials</strong> explaining <strong>key concepts</strong>.</li> <li>Concise<strong> revision notes</strong> providing you with reminders of all the<strong> key content </strong>you need</li> <li><strong>Onscreen quizzes </strong>giving you<strong> instant feedback </strong></li> <li><strong>Exam style questions </strong>with hints, detailed solutions and explanation</li> <li>A huge <strong><em><a href="../../test-your-knowledge.html">questionbank </a></em></strong>designed to test your knowledge</li> </ul> </div> <hr class="hidden-separator"> <h4>Track your Progress</h4> <p>You can track your own progress on the site</p> <p><img alt="" src="../../images/progress.png" style="width: 200px; height: 209px;"> <img alt="" src="../../images/progress-report.png" style="width: 479px; height: 209px;"></p> <hr class="hidden-separator"> <h4>Navigating the pages</h4> <p>Each of the topic pages is designed in the same way:</p> <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>Find out here exactly what you can expect to learn about on the page</p> </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>Had an explantion in class that you didn't quite understand? The short, high quality videos in this section will help you understand all the key concepts from each topic. <span id="docs-internal-guid-6e715e81-7fff-5538-f779-c9de05931317">Some are designed to explain concepts from the beginning and others are focussed on very specific skills and typical exam questions. All videos come with a set of notes that you can keep.</span></p> <p>Here is an example video about understanding logarithms</p> <div class="panel panel-orange 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>Example - Understanding Logarithms</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="472"> <p>One of the important things about using logarithms is to be able to see them as indices, but just written in another way <span></span><span class="math-tex">\(a^x=b \quad\Leftrightarrow \quad x=\log _{ a }{ b } \)</span></p> <p><span>Once you understand the conversion, then these questions become more about understanding indices.</span></p> <p>In the following video, we will look at these 5 examples</p> <p style="margin-left: 40px;"><span class="math-tex">\(1. \quad \log _{ 2 }{ 32 }=x \\ 2. \quad \log _{ 10}{ x }= 3\\ 3. \quad\ln{ 1 }=x \\ 4. \quad\log _{ 5 }{ \frac{1}{125} }=x \\ 5. \quad\log _{ 8}{ \frac{1}{4} } =x\\\)</span></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/281652939"></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/algebra/logarithms/converting-logs-to-indices.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/logarithms/converting-logs-to-indices.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>This section is like your own set of revision notes. Why not print them out so that you can add your own notes to them, then keep them together to make a complete revision guide for the whole course!</p> <p>Here are an example set of notes on the topic Introducing Derivatives</p> <p style="text-align: center;"><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/differentiation/introduction/revision-notes_the-derivative.pdf" width="640"></iframe></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>This section is all about getting practice. Before you try any exam-style questions, you will need a quick check to see if you have understood all the concepts from the page. These onscreen quizzes allow you to get just that. You can get instant feedback on your answers including advice about how to tackle the question and worked solutions.</p> <p>Here is an example quiz on the topic of Arrangement in Counting Principles</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#ad9782f7"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="ad9782f7"> <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%;"> Arrangements <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="11-434-2696" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/4-bowls-small.jpg" style="width: 400px; height: 300px;"></p><p>4 different coloured bowls are arranged in a line. How many <strong>different</strong> arrangements are there?</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="24"> <span class="review"></span></p></div><div class="q-explanation"><p>The objects are different, so there are 4! different arrangements</p><p>4! = 4 x 3 x 2 x 1 = 24</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><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/cards-1.jpg" style="width: 300px; height: 400px;"></p><p>5 cards are arranged in a line. How many <strong>different </strong>possible arrangements are there?</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="120"> <span class="review"></span></p></div><div class="q-explanation"><p>The objects are different, so there are 5! different arrangements</p><p>5! = 5 x 4 x 3 x 2 x 1 = 120</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><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/pirate-cards.jpg" style="width: 400px; height: 300px;"></p><p>The 6 cards are placed in a line. How many <strong>different </strong>arrangements are there?</p><p style="text-align: right;">*Note that the cards with a coin are identical</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="120"> <span class="review"></span></p></div><div class="q-explanation"><p>There are 6 objects and 3 of them are repeated.</p><p>There are <span class="math-tex">\(\frac{6!}{3!}\)</span>arrangements</p><p><span class="math-tex">\(\frac{6!}{3!}=\frac{6\cdot5\cdot4\cdot3\cdot 2\cdot1}{3\cdot2\cdot 1}=6\cdot5\cdot4=120\)</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><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/coloured-tiles.jpg" style="width: 400px; height: 300px;"></p><p>The coloured blocks are arranged in a line.</p><p>How many <strong>different </strong>arrangements are there?</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="180"> <span class="review"></span></p></div><div class="q-explanation"><p>There are 6 objects. The trapezia are repeated 2 times. The rhombus ar repeated 2 times.</p><p>There are <span class="math-tex">\(\frac{6!}{2!\cdot2!}=180\)</span> arrangements</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 letters from the word<strong> MATHEMATICS</strong> are placed in a line.</p><p>How many <strong>different</strong> arrangements are there?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{11!}{2!\cdot2!\cdot2!}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2!\cdot2!\cdot2!\cdot5!\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(11!\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{11!}{6!}\)</span></label></p></div><div class="q-explanation"><p>The are 11 objects. The <strong>M </strong>, <strong>A</strong> ,<strong>T</strong> are repeated 2 times each.</p><p>There are <span class="math-tex">\(\frac{11!}{2!\cdot2!\cdot2!}\)</span> arrangements</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><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/rods.jpg" style="width: 400px; height: 300px;"></p><p>The coloured rods are placed in a line. How many <strong>different</strong> arrangements are there?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\({3!\cdot2!\cdot2!}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(7!\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac{7!}{3!\cdot2!}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{7!}{5!}\)</span></span></label> </p></div><div class="q-explanation"><p>There are 7 objects.The purple rods are repeated 3 times. The green rods are repeated 2 times.</p><p>There are <span class="math-tex">\(\frac{7!}{3!\cdot2!}\)</span> arrangements</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><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/more-rods.jpg" style="width: 400px; height: 300px;"></p><p>8 different coloured rods are placed in a line.</p><p>How many <strong>different </strong>arrangements are there so that the blue and the green rods are <strong>together</strong>?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(2\cdot7!\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{7!}{2!}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{8!}{2!}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(6!\)</span></span></label> </p></div><div class="q-explanation"><p>Think about the blue and green rods as one object.</p><p>This means that there now are 7 objects.</p><p>These 7 objects can be arranged in 7! ways.</p><p>For each of these arrangements, it is possible to reverse the order of the blue and green rods</p><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/rods2-ans.png" style="width: 400px; height: 107px;"></p><p>There are 2x7! different arrangements</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><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/10-cards.jpg" style="width: 300px; height: 400px;"></p><p>10 cards are placed in a line. How many <strong>different </strong>ways can they be arranged so that the cards 1, 2 & 3 are <strong>together</strong>?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(3\cdot8!\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{8!}{3!}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(7!\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(3!\cdot8!\)</span></label></p></div><div class="q-explanation"><p>Think of the cards 1, 2 & 3 as one object.</p><p>This makes 8 objects with 8! arrangements.</p><p>There are 3! ways of arranging the numbers 1, 2 & 3</p><p>For each of these arrangements, it is possible to rearrange the items 1 , 2 & 3</p><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/10-cards-ans.png" style="width: 400px; height: 184px;"></p><p>There are 3!x8!</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><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/6-bowls.jpg" style="width: 400px; height: 300px;"></p><p>6 different coloured bowls are placed in a line. How many arrangements are there in which the pink and red bowls are <strong>separated</strong>?</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="480"> <span class="review"></span></p></div><div class="q-explanation"><p>The pink and red bowls are either <strong>separated </strong>or <strong>together</strong>.</p><p>Find the number of arrangements in total = 6!</p><p>Find the number of arrangements in which the pink and red bowls are <strong>together</strong> = 2x5!</p><p>The number of arrangements in which the pink and red bowls are <strong>separated</strong> = 6 - 2x5! = 480</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>Three men and two women stand together in a line.</p><p>In how many arrangements will all the men stand next to each and all the women stand next to each other?</p></div><div class="q-answer"><p>Arrangements = <input type="text" style="height: auto;" data-c="24"> <span class="review"></span></p></div><div class="q-explanation"><p>Think of the men as 1 object. Think of the women as 1 object.</p><p>There are 2 ways of arranging these 2 objects</p><p>There are 3! arrangements of men. There are 2! arrangements of women.</p><p><img alt="" src="../../files/algebra/counting-principles/arrangements-quiz/men-women.png" style="width: 250px; height: 165px;"></p><p>There are 3! x 2! x 2 = 24 arrangements</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"> <p>This is the section where you can practise questions just like you the ones that you will get in the exam. If you need some help getting started, there are hints. Each question has a full solution with explanation</p> <p>Each of the questions is marked by difficulty level</p> <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-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> Easy</p> <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-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> Medium</p> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> Hard</p> <p>Here is an example question from the topic of Optimisation</p> <div> <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>Example - Optimisation</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="888"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The diagram below shows the graph of the functions f(x) = sinx and g(x) = 2sinx</p> <p><img alt="" src="../../files/differentiation/optimisation/esq3_1.jpg" style="width: 500px; height: 376px;"></p> <p>A rectangle ABCD is placed in between the two functions as shown so that B and C lie on <strong><em>g</em></strong> , BC is parallel to the x axis and the local minima of the function <em>f</em> lies on AD.</p> <p>Let NA = <strong><em>x</em></strong></p> <p>a) Find an expression for the height of the rectangle AB</p> <p>b) Show that the area of the rectangle, <strong><em>A</em></strong> can be given by A = 4xcosx - 2x</p> <p>c) Find <span class="math-tex">\(\frac{dA}{dx}\)</span></p> <p>d) Find the maximum value of the area of the rectangle.</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) The y coordinate of B is <span class="math-tex">\(g(\frac{\pi}{2}+x)=2sin(\frac{\pi}{2}+x)\)</span></p> <p>b) Note that <span class="math-tex">\(sin(x+\frac{\pi}{2})=cosx\)</span><content></content></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>a) The y cordinate of A is 1 (the maximum value of the function f(x) = sinx)</p> <p>The y coordinate of B is <span class="math-tex">\(g(\frac{\pi}{2}+x)=2sin(\frac{\pi}{2}+x)\)</span></p> <p>The height of the rectangle, AB = <span class="math-tex">\(2sin(\frac{\pi}{2}+x)-1\)</span></p> <p>Note that <span class="math-tex">\(sin(x+\frac{\pi}{2})=cosx\)</span></p> <p>Therfore, AB = 2cosx - 1</p> <p><img alt="" src="../../files/differentiation/optimisation/esq3a.jpg" style="width: 400px; height: 248px;"></p> <p>b) The width of the rectangle = 2x</p> <p>Hence, the area of the rectangle, A = 2x(2cosx - 1) = 4xcosx - 2x</p> <p>c) To find <span class="math-tex">\(\frac{dA}{dx}\)</span>, we need to use the <a href="../743/product-and-quotient-rule.html">Product Rule</a></p> <p><span class="math-tex">\(\frac{dA}{dx}=4cosx - 4xsinx - 2\)</span></p> <p>d) The maximum value of the area of the rectangle occurs where <span class="math-tex">\(\frac{dA}{dx}=0\)</span></p> <p>We can use our graphical calculator to solve this. The value occurs when x <span class="math-tex">\(\approx\)</span> 0.592</p> <p>A<sub>max</sub> <span class="math-tex">\(\approx\)</span> 0.781</p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> </div> <hr class="thin"> <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 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