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class="col-md-9" id="main-column"> <h1 class="page_title"> Graphs of Implicit Equations <a href="#" class="mark-page-favorite pull-right" data-pid="565" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../550/calculus.html">Calculus</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../741/implicit-differentiation.html">Implicit Differentiation</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Graphs of Implicit Equations</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/differentiation/implicit-differentiation/main2.png" style="float: left; width: 100px; height: 100px;">Implicit Equations are often used to represent relations that cannot be expressed as functions (one to many relations or many to many relations). These often have really interesting graphs. This page will allow to explore some of these graphs and consider why we need to use implicit diffrentiation. If you are looking for help with the techniques of implicit differentiation, examples and exam-style questions then you should visit <a href="../741/implicit-differentiation.html" title="Implicit Differentiation">Implicit Differentiation</a> .</p> <hr class="hidden-separator"> <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"> <div> <p>Here are the graphs of some implicit relations</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Circle</p> </div> </div> <div class="panel-body"> <div> <p>A circle is the locus of a point that is a fixed distance from a point</p> <p>The equation can be written<strong> implicity</strong> like this <span class="math-tex">\(x^2+y^2=a^2\)</span> where <em><strong>a</strong></em> is the radius of the circle. Drag the point around the circle to consider the gradient of this relation.</p> <p style="text-align: center"><iframe height="553px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/USNNrta2/width/767/height/553/border/888888/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="Circle" width="767px"></iframe></p> <p>We can write the eqution of this circle <strong>explicity </strong><span class="math-tex">\(y=\pm\sqrt{a^{2}-x^{2}}\)</span></p> <p>This creates 2 derivatives: one for when y>0 and one for when y<0.</p> <p>Let’s take the case when the radius if the circle is equal to 1. The graph of <span class="math-tex">\(y=\pm\sqrt{1^{2}-x^{2}} \)</span>for y>0 is plotted below.</p> <p>Sketch the graph of the gradient function. You can check your answer by clicking in the box.</p> <p style="text-align: center"><iframe height="555px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/u2dW9Bf8/width/769/height/555/border/888888/smb/false/stb/true/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false" style="border:0px;" title="Sketching gradient function of a circle" width="769px"></iframe></p> <p>You can find the gradient function of this explicit function using the chain rule. Try it for yourself!</p> <p>Answer</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="math-tex">\(y=\sqrt{1-x^{2}}\)</span></p> <p><span class="math-tex">\(y=(1-x^{2})^{\frac{1}{2}}\)</span></p> <p><span class="math-tex">\(y=u^{\frac{1}{2}}\qquad\qquad u=1-x^{2}\)</span></p> <p><span class="math-tex">\(\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}\qquad\frac{du}{dx}=-2x\)</span></p> <p><span class="math-tex">\(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}\)</span></p> <p><span class="math-tex">\(\frac{dy}{dx}=\frac{1}{2}\frac{1}{\sqrt{1-x^{2}}}(-2x)\)</span></p> <p><span class="math-tex">\(\frac{dy}{dx}=\frac{-x}{\sqrt{1-x^{2}}}\)</span></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Ellipse</p> </div> </div> <div class="panel-body"> <div> <p>An ellipse is the locus of a point in which the <strong>sum</strong> of the distances to two fixed points is constant.</p> <p>The equation can be written<strong> implicity</strong> like this <span class="math-tex">\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \)</span></p> <p>We can write it <strong>explicitly</strong> like this <span class="math-tex">\(y=\pm b\sqrt{1-\frac{x^{2}}{a^{2}}}\)</span></p> <p>A graph of an ellipse can be seen below. You can drag the blue point to check the property of the locus.</p> <p>It is possible to find the gradient! Consider what the gradient function might look like.</p> <p style="text-align: center"><iframe height="555px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/kGwucnx9/width/769/height/555/border/888888/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false" style="border:0px;" title="An Ellipse" width="769px"></iframe></p> <h4> </h4> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Casini Oval</p> </div> </div> <div class="panel-body"> <div> <p>A Cassini Oval has a beautiful graph. It is defined as the locus of a point in which the <strong>product</strong> of the distances to two fixed points is constant.</p> <p>The equation can be written<strong> implicity</strong> like this <span class="math-tex">\(((x-a)^2+y^2)((x+a)^2+y^2)=b^4\)</span></p> <p>Here is the graph. Adjust the parameters <strong><em>a </em></strong>and <strong><em>b</em></strong> using the sliders and explore the gradient of the graph.</p> <p>It is rather difficult to write the function of this graph explicitly, but we can still describe the gradient!</p> <p style="text-align: center"><iframe height="555px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/wvmdADtf/width/769/height/555/border/888888/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false" style="border:0px;" title="Cassini Oval" width="769px"></iframe></p> <h4> </h4> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Another Beautiful Implicit Relation </p> </div> </div> <div class="panel-body"> <div> <h4> </h4> <p><span class="math-tex">\(sin(x+y) – cos(xy) – 1 = 0\)</span> is another implicit relation with a beautiful graph. It is impossible to write explicitly, but we can still find its gradient.</p> <p>Try plotting this implicit function by typing the equation in the input bar below</p> <p style="text-align: center"><iframe frameborder="0" height="433px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/uVNt8ztt/width/769/height/555/border/888888/smb/false/stb/false/stbh/false/ai/true/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false" style="border:0px;" title="Blank axes" width="600px"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <h4> </h4> <div class="page-container panel-self-assessment" data-id="565"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Graphs of Implicit Equations</strong> have you understood?</p><div class="slider-container 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