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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"> 5.1 & 5.3 Introduction to Calculus <a href="#" class="mark-page-favorite pull-right" data-pid="2400" 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">5.1 & 5.3 Introduction to Calculus</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 40 minutes"><i class="fa fa-clock-o"></i> 40'</span> </ol> <article id="main-article"> <h2><img alt="" src="../../files/CLS/CC/CC-small-1.jpg" style="width: 100px; height: 100px; float: left;">Differentiating</h2> <p>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 whole principle is based around examining 'Rates of change'.</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"> <p><meta charset="utf-8"></p> <p dir="ltr">In this unit you should learn to</p> <ul dir="ltr"> <li>Understand the basic principles of differential calculus as rate of change</li> <li>Differentiate polynomial functions and use this work out gradients</li> <li>Differentiate functions involving negative indices</li> <li>Use basic differentiation to solve problems about gradients</li> </ul> </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-260" class="dynamic-gallery carousel slide" data-id="260"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-260/cc-nologos001.jpeg" data-fancybox="gallery-260" title="" data-caption=""><img alt="" src="../../../std-galleries/15-260/cc-nologos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-260/cc-nologos002.jpeg" data-fancybox="gallery-260" title="" data-caption=""><img alt="" src="../../../std-galleries/15-260/cc-nologos002.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-260/cc-nologos003.jpeg" data-fancybox="gallery-260" title="" data-caption=""><img alt="" 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title="Click to view" src="../../../std-galleries/15-260/cc-nologos009-thumb128.jpg"><li><li data-index="9"><img title="Click to view" src="../../../std-galleries/15-260/cc-nologos010-thumb128.jpg"><li><li data-index="10"><img title="Click to view" src="../../../std-galleries/15-260/cc-nologos011-thumb128.jpg"><li><li data-index="11"><img title="Click to view" src="../../../std-galleries/15-260/cc-nologos012-thumb128.jpg"><li><li data-index="12"><img title="Click to view" src="../../../std-galleries/15-260/cc-nologos013-thumb128.jpg"><li><li data-index="13"><img title="Click to view" src="../../../std-galleries/15-260/cc-nologos014-thumb128.jpg"><li><li data-index="14"><img title="Click to view" src="../../../std-galleries/15-260/cc-nologos015-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. Introduction - what is calculus about</p> </div> </div> <div class="panel-body"> <div> <p>This video puts the whole idea in to some context and shows how calculus builds carefully on work already done.</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/594576839"></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 Gradient function</p> </div> </div> <div class="panel-body"> <div> <p>Here we introduce the idea of deriving a function which tells about the gradient of the function we started with.</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/594577463"></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. Differentiating polynomials</p> </div> </div> <div class="panel-body"> <div> <p>At this point we move on to the general pattern for deriving the gradient function.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/594617804"></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><span style="font-size: 16.8px;"><b>4. Negative indices and differentiating</b></span></p> </div> </div> <div class="panel-body"> <div> <p>This is a focus on dealing with negative indices. They are a potential pitfall, but if you are careful there should be no problem.</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/594618522"></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. Finding gradients</p> </div> </div> <div class="panel-body"> <div> <p>Here we apply the techniques to find the gradient of a given function at any 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/594620108"></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>Review these condensed 'key point' flashcards to help you check and keep ideas fresh in your mind.</p> <div id="carousel-261" class="dynamic-gallery carousel slide" data-id="261"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-261/cc-nologos001.jpeg" data-fancybox="gallery-261" title="" data-caption=""><img alt="" src="../../../std-galleries/15-261/cc-nologos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-261/cc-nologos002.jpeg" data-fancybox="gallery-261" title="" data-caption=""><img alt="" src="../../../std-galleries/15-261/cc-nologos002.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-261" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-261" 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/15-261/cc-nologos001-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-261/cc-nologos002-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="15-319-2400"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Identify the gradient/rate of change of the following functions</p></div><div class="q-answer"><p>y = 3x +2 <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p><p>y= -5x <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span> </p><p>y = 7 <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> </p><p>y = 0.5x - 4 <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span> </p></div><div class="q-explanation"><p>These are all straight line (linear) functions in the form y = mx + c, where m is the gradient. In the case of y = 7, we can think of this as y = 0x + 7 and so the gradient is 0. Graphs like this are horizontal lines so it is logical to conclude that they have gradient = 0.</p></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">2</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 x to the power of 5" 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»2«/mn»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-9.php?formula=cfe5d309eb8164946c463240c3dc5b82.png"> for which <img align="middle" alt="f apostrophe left parenthesis x right parenthesis equals a x to the power of b" 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»«mi»a«/mi»«msup»«mi»x«/mi»«mi»b«/mi»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-10.php?formula=e671c34160ee0d23060d2d296a1eabf1.png">. What are the values of a and b?</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="i f space f left parenthesis x right parenthesis space equals space a x to the power of n comma space f apostrophe left parenthesis x right parenthesis space equals space a n x to the power of n minus 1 end exponent" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«mi»f«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»a«/mi»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«mo»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»a«/mi»«mi»n«/mi»«msup»«mi»x«/mi»«mrow»«mi»n«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-11.php?formula=50dfafd9e6d11660b4838c85ab299f98.png"></p></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 negative 3.5 x to the power of 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»«mo»-«/mo»«mn»3«/mn»«mo».«/mo»«mn»5«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-12.php?formula=bd4ea5fedb2ce850aa66fc29de0be219.png"> for which <img align="middle" alt="f apostrophe left parenthesis x right parenthesis equals a x to the power of b" 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»«mi»a«/mi»«msup»«mi»x«/mi»«mi»b«/mi»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-10.php?formula=e671c34160ee0d23060d2d296a1eabf1.png">. What are the values of a and b?</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-14"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="i f space f left parenthesis x right parenthesis space equals space a x to the power of n comma space f apostrophe left parenthesis x right parenthesis space equals space a n x to the power of n minus 1 end exponent" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«mi»f«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»a«/mi»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«mo»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»a«/mi»«mi»n«/mi»«msup»«mi»x«/mi»«mrow»«mi»n«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-11.php?formula=50dfafd9e6d11660b4838c85ab299f98.png"></p></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 x cubed plus 5 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»5«/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-12.php?formula=1621e2a59b1481e17270f4d4cd7d6687.png">, for which <img align="middle" alt="f apostrophe left parenthesis x right parenthesis equals a x squared plus 10 x to the power of b plus c" 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»«mi»a«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»10«/mn»«msup»«mi»x«/mi»«mi»b«/mi»«/msup»«mo»+«/mo»«mi»c«/mi»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-13.php?formula=e03bbc90d09a4c96c3ac4bb786d88039.png">. What are the values of a, b and c</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> , b= <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> , c = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="i f space f left parenthesis x right parenthesis space equals space a x to the power of n comma space f apostrophe left parenthesis x right parenthesis space equals space a n x to the power of n minus 1 end exponent" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«mi»f«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»a«/mi»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«mo»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»a«/mi»«mi»n«/mi»«msup»«mi»x«/mi»«mrow»«mi»n«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-11.php?formula=50dfafd9e6d11660b4838c85ab299f98.png"></p></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 2 over 3 x to the power of 4 minus 3 x squared minus 7 x" 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»3«/mn»«/mfrac»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»7«/mn»«mi»x«/mi»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-13.php?formula=6fcbea3a0fb500d228b9c6989a6f1393.png">, for which <img align="middle" alt="f apostrophe left parenthesis x right parenthesis equals 8 over a x cubed minus b x to the power of c plus d" 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»«mfrac»«mn»8«/mn»«mi»a«/mi»«/mfrac»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mi»b«/mi»«msup»«mi»x«/mi»«mi»c«/mi»«/msup»«mo»+«/mo»«mi»d«/mi»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-14.php?formula=af37c90ac17fec4c1b3a0b142b2cf178.png">. What are the values of a, b, c and d</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> , b= <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> , c = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> , d = <input type="text" style="height: auto;" data-c="-7"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="i f space f left parenthesis x right parenthesis space equals space a x to the power of n comma space f apostrophe left parenthesis x right parenthesis space equals space a n x to the power of n minus 1 end exponent" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«mi»f«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»a«/mi»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«mo»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»a«/mi»«mi»n«/mi»«msup»«mi»x«/mi»«mrow»«mi»n«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-11.php?formula=50dfafd9e6d11660b4838c85ab299f98.png">, watch out for signs.</p></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">6</div><div class="exercise shadow-bottom"><div class="q-question"><p> <img align="middle" alt="fraction numerator 3 over denominator 4 x squared end fraction" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-26.php?formula=3f5876199d89e8062c6b977d4c78bf12.png"> can also be written as <img align="middle" alt="a over b x to the power of c" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»a«/mi»«mi»b«/mi»«/mfrac»«msup»«mi»x«/mi»«mi»c«/mi»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-27.php?formula=2c0f920604e9f773be17569b0db301f0.png">, what are the values of a, b and c?</p></div><div class="q-answer"><p><label class="checkbox">a = </label> <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> , c = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="1 over x to the power of n equals x to the power of negative n end exponent space a n d space s o space fraction numerator a over denominator b x to the power of n end fraction equals a over b x to the power of negative n end exponent" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«/mfrac»«mo»=«/mo»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mi»n«/mi»«/mrow»«/msup»«mo»§#160;«/mo»«mi»a«/mi»«mi»n«/mi»«mi»d«/mi»«mo»§#160;«/mo»«mi»s«/mi»«mi»o«/mi»«mo»§#160;«/mo»«mfrac»«mi»a«/mi»«mrow»«mi»b«/mi»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mi»a«/mi»«mi»b«/mi»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mi»n«/mi»«/mrow»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-28.php?formula=3ab8d0caf1618c915d46088bfd3ab98a.png"></p></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 <img align="middle" alt="f left parenthesis x right parenthesis equals 1 over x plus fraction numerator 3 over denominator 4 x squared end fraction" 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»1«/mn»«mi»x«/mi»«/mfrac»«mo»+«/mo»«mfrac»«mn»3«/mn»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-15.php?formula=3bdce127c358f891c8528925318b0adb.png">, for which <img align="middle" alt="f apostrophe left parenthesis x right parenthesis equals a x to the power of b minus c over 2 x to the power of negative d end exponent" 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»«mi»a«/mi»«msup»«mi»x«/mi»«mi»b«/mi»«/msup»«mo»-«/mo»«mfrac»«mi»c«/mi»«mn»2«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mi»d«/mi»«/mrow»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-16.php?formula=a16ac5da8502af7cedadcfee3d0f4c07.png">. What are the values of a, b, c and d? (do it on paper and be careful with the signs)</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-1"> <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="3"> <span class="review"></span> , d = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="i f space f left parenthesis x right parenthesis space equals space x to the power of negative 1 end exponent plus 3 over 4 x to the power of negative 2 end exponent space s o space f apostrophe left parenthesis x right parenthesis space equals space minus x to the power of negative 2 end exponent minus 6 over 4 x to the power of negative 3 end exponent space space space space a n d space 6 over 4 equals 3 over 2" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«mi»f«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«mo»+«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»2«/mn»«/mrow»«/msup»«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»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»2«/mn»«/mrow»«/msup»«mo»-«/mo»«mfrac»«mn»6«/mn»«mn»4«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»3«/mn»«/mrow»«/msup»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»a«/mi»«mi»n«/mi»«mi»d«/mi»«mo»§#160;«/mo»«mfrac»«mn»6«/mn»«mn»4«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-17.php?formula=c2448d84731771fe1458cfb514785f49.png"></p></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>What is the gradient of the function <img align="middle" alt="f left parenthesis x right parenthesis space equals space 4 x squared plus 3 x" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-15.php?formula=1740d4882b338149f0242a7d8ae4127d.png"> when x = 2?</p></div><div class="q-answer"><p>Gradient = <input type="text" style="height: auto;" data-c="19"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="f apostrophe left parenthesis x right parenthesis space equals space 8 x plus 3 comma space s u b s t i t u t e space x space equals space 2f apostrophe left parenthesis 2 right parenthesis space equals space 19" 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»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»8«/mn»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»s«/mi»«mi»u«/mi»«mi»b«/mi»«mi»s«/mi»«mi»t«/mi»«mi»i«/mi»«mi»t«/mi»«mi»u«/mi»«mi»t«/mi»«mi»e«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»2«/mn»«mspace linebreak=¨newline¨/»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mn»2«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»19«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-16.php?formula=849a1c49afd0a00461f15b26b389254b.png"></p></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>What is the gradient of the function <img align="middle" alt="f left parenthesis x right parenthesis space equals space minus 2 x cubed minus 4 x squared plus 5 x minus 3" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mi»x«/mi»«mo»-«/mo»«mn»3«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-17.php?formula=a24857962e9bb2ca0fe4a34fee38233c.png"> when x = -3?</p></div><div class="q-answer"><p>Gradient = <input type="text" style="height: auto;" data-c="-25"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="f apostrophe left parenthesis x right parenthesis space equals space minus 6 x squared minus 8 x plus 5 comma space w h e n space x space equals space minus 3f apostrophe left parenthesis negative 3 right parenthesis space equals space minus 54 space minus space minus space 24 space plus space 5 space equals space minus 25" 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»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»8«/mn»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»w«/mi»«mi»h«/mi»«mi»e«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»3«/mn»«mspace linebreak=¨newline¨/»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mo»-«/mo»«mn»3«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»54«/mn»«mo»§#160;«/mo»«mo»-«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mo»§#160;«/mo»«mn»24«/mn»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»25«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-18.php?formula=d0f74f080faa5734a3256fb187fac8c6.png"></p></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>Given <img align="middle" alt="f left parenthesis x right parenthesis space equals space 2 x to the power of 5 plus 3 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»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»+«/mo»«mfrac»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-19.php?formula=a7e56e1ebeac4896ffa0e851f19f4dd3.png"> find <img align="middle" alt="f apostrophe left parenthesis 2 right parenthesis" 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»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-20.php?formula=d8bf974c05ac3955cc6541e3a7676fc6.png"> to 3sf</p></div><div class="q-answer"><p><img align="middle" alt="f apostrophe left parenthesis 2 right parenthesis" 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»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-20.php?formula=d8bf974c05ac3955cc6541e3a7676fc6.png"> = <input type="text" style="height: auto;" data-c="159"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="f left parenthesis x right parenthesis space equals space 2 x to the power of 5 plus 3 x to the power of negative 2 end exponentf apostrophe left parenthesis x right parenthesis space equals space 10 x to the power of 4 minus 6 x to the power of negative 3 end exponent comma space s u b s t i t u t e space x space equals space 2f apostrophe left parenthesis 2 right parenthesis space equals space 159.25" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»2«/mn»«/mrow»«/msup»«mspace linebreak=¨newline¨/»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»10«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»3«/mn»«/mrow»«/msup»«mo»,«/mo»«mo»§#160;«/mo»«mi»s«/mi»«mi»u«/mi»«mi»b«/mi»«mi»s«/mi»«mi»t«/mi»«mi»i«/mi»«mi»t«/mi»«mi»u«/mi»«mi»t«/mi»«mi»e«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»2«/mn»«mspace linebreak=¨newline¨/»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mn»2«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»159«/mn»«mo».«/mo»«mn»25«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-21.php?formula=570083abf861d14bdfb4347b6739a976.png"> 159 to 3sf (use the table function on your GDC</p></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"> <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 style="text-align: center;"><img alt="" src="../../files/CLS/CC/CCQ1.jpg" style="width: 700px; height: 280px;"></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/595181002"></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 style="text-align: center;"><img alt="" src="../../files/CLS/CC/CCQ2.jpg" style="width: 700px; height: 276px;"></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/595182911"></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 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