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id="main-article"> <p><img alt="" src="../../files/differentiation/tangents-and-normals/main2-1.jpg" style="float: left; width: 100px; height: 100px;">On this page we look at how to find the equation of a tangent and also the normal to a curve. A tangent is a straight line that touches a curve at one point and has the same gradient as the curve at that point. A normal is straight line that is perpendicular to the tangent to the curve. We use the gradient function and need to know a point on the line.</p> <hr class="hidden-separator"> <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>On this page, you should learn about</p> <ul> <li>equations of a tangent to a curve at a given point</li> <li>equations of a normal to a curve at a given point</li> </ul> </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>The following videos will help you understand all the concepts from this page</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>Equation of a Tangent</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="382"> <p>A tangent is a straight line that touches a curve at one and only one point.</p> <p style="text-align: center"><iframe height="450px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/EVbNzBjQ/width/500/height/450/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="What is a tangent" width="500px"></iframe></p> <p>In the video below we are going to look at tangents to curves and how we can use differentiation to find the gradient to a tangent and therefore the equation of a tangent. We will look at the example</p> <p><em>Find the equation of the tangent to the curve y = x² at the point P(3, 9)</em></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/273065133"></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/differentiation/tangents-and-normals/equation-of-a-tangent.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/tangents-and-normals/equation-of-a-tangent.pdf" width="640"></iframe></p> </section> </div> </div> </div> <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>Equation of a Normal</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="383"> <p>A normal line to a curve is the line that is <strong>perpendicular</strong> to the tangent to a curve at a particular point.</p> <p style="text-align: center;"><img alt="" class="gifffer" data-gifffer="/media/mathsanalysis/files/differentiation/tangents-and-normals/what-is-a-normal.gif" style="width: 400px; height: 403px;"></p> <p>In the following video we are going to look at normals to curves, how we can use differentiation to find the gradient of a normal and therefore how to find the equation of a normal. In the example, we will also look at how to find the intersection of a normal with the curve</p> <ol style="list-style-type:lower-alpha;"> <li><em>Find the equation of the normal to the curve <span class="math-tex">\(f(x)= \frac{1}{x}\)</span>at the point P<span class="math-tex">\((-2,-\frac{1}{2})\)</span></em></li> <li><em>Find the co-ordinates of the point where this normal meets the curve again.</em></li> </ol> <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/273071592"></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/differentiation/tangents-and-normals/equation-of-a-normal.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/tangents-and-normals/equation-of-a-normal.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><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/differentiation/tangents-and-normals/revision-notes_tangents_and_normals.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/differentiation/tangents-and-normals/revision-notes_tangents_and_normals.pdf" target="_blank">here</a></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>Before you attempt to answer questions about equations of tangents and normals, you should be confident with finding the equation of a straight line and perpendicular lines. Here is a quiz that practices that knowledge</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#134e748e"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="134e748e"> <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%;"> Equation of a Straight Line <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="11-189-744" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>What is the gradient and y intercept of the following line?</p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q1.jpg" style="width: 300px; height: 441px;"></p></div><div class="q-answer"><p>gradient = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>y intercept = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>gradient = <span class="math-tex">\(\frac{rise}{run}\)</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>The equation of the following line can be written in the form <span class="math-tex">\(y=ax+b\)</span></p><p>Find <em><strong>a</strong></em> and <strong><em>b</em></strong></p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q2a.jpg" style="width: 400px; height: 467px;"></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>a = gradient</p><p>b = y intercept</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>Which of the following is the correct equation of the following straight line</p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q3.jpg" style="width: 244px; height: 450px;"></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>y = 3x - 1</span></label> </p><p><label class="radio"> <input type="radio"> <span>y = -3x - 1</span></label> </p><p><label class="radio"> <input type="radio"> <span>y = -x +3</span></label> </p><p><label class="radio"> <input type="radio"> <span>y = 2x + 1</span></label> </p></div><div class="q-explanation"><p>gradient = 3</p><p>y intercept = -1</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>Which of the following lines passes through the point (2 , 1)</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> y = 2x - 1</label></p><p><label class="radio"><input type="radio"> y = 2x + 1</label></p><p><label class="radio"><input class="c" type="radio"> y = -x + 3</label></p><p><label class="radio"><input type="radio"> y = x - 2</label></p></div><div class="q-explanation"><p>y = -x + 3</p><p>1 = -2 + 3</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>Which of the following lines is parallel to <span class="math-tex">\(y = \frac{1}{2}x+2\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>x + 2y + 3 = 0</span></label> </p><p><label class="radio"> <input type="radio"> <span>2x - y + 3 = 0</span></label> </p><p><label class="radio"> <input type="radio"> <span>2x + y + 3 = 0</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>x - 2y + 3 = 0</span></label> </p></div><div class="q-explanation"><p>x - 2y + 3 = 0 can be rearranged into the form y = mx + c</p><p>x + 3 = 2y</p><p><span class="math-tex">\(y = \frac{1}{2}x+\frac{3}{2}\)</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>Which of he following gradients could represent the gradients of 2 perpendicular lines</p><p>A) 2</p><p>B) <span class="math-tex">\(\frac{2}{5}\)</span></p><p>C) 5</p><p>D) <span class="math-tex">\(-\frac{5}{2}\)</span></p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B and C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A and D</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A and B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> B and D</label></p></div><div class="q-explanation"><p>If a line has gradient <strong><em>m </em></strong>then a perpendicular line has gradient <span class="math-tex">\(\frac{-1}{m}\)</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>What is the gradient of the line that is <strong>perpendicular</strong> to the line below</p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q1.jpg" style="width: 300px; height: 441px;"></p></div><div class="q-answer"><p>gradient = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></div><div class="q-explanation"></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 gradient a line <strong>perpendicular </strong>to <span class="math-tex">\(y = -\frac{2}{3}x-2\)</span> can be written in the form <span class="math-tex">\(\frac{a}{b}\)</span></p><p>Find <em><strong>a</strong></em> and <strong><em>b</em></strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>If a line has gradient <strong><em>m </em></strong>then a perpendicular line has gradient <span class="math-tex">\(\frac{-1}{m}\)</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>The equation line below can be written in the form <strong><em><span class="math-tex">\(ax+by+c=0\)</span></em></strong></p><p>Find <strong><em>a </em></strong>,<strong><em> b</em></strong> and <strong><em>c</em></strong></p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q7.jpg" style="width: 600px; height: 449px;"></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p></div><div class="q-explanation"><p>gradient = <span class="math-tex">\(-\frac{2}{5}\)</span></p><p>y intercept = -2</p><p><span class="math-tex">\(y = -\frac{2}{5}x-2\\ 5y=-2x-10\\ 5y+2x+10=0\\ 2x+5y+10=0\)</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>The equation of the line that is <strong>perpendicular</strong> to the line below and that passes through the point (1,1) can be written in the form <strong><em><span class="math-tex">\(ax+2y+b=0\)</span></em></strong></p><p>Find <strong><em>a </em></strong>and<strong><em> b</em></strong></p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q7.jpg" style="width: 600px; height: 449px;"></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>gradient of line in graph = <span class="math-tex">\(-\frac{2}{5}\)</span></p><p>gradient of perpendicular line = <span class="math-tex">\(\frac{5}{2}\)</span></p><p><span class="math-tex">\(y = \frac{5}{2}x+c \)</span></p><p>passes through (1,1)</p><p><span class="math-tex">\(1 = \frac{5}{2}\cdot1+c\\ -\frac{3}{2} = c \\ y = \frac{5}{2}x-\frac{3}{2} \\ 2y=5x-3\\ -5x+2y+3=0 \)</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> </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> <hr class="hidden-separator"> <p>Here is a quiz about finding the equation of a tangent to a curve and the equation of a normal to a curve</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#20d23b1b"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="20d23b1b"> <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%;"> Equation of Tangent and Normal <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="11-190-744" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>f(x) = x<sup>3</sup>+2x</p><p>Find the gradient of the curve y = f(x) at the point (-1,-3)</p></div><div class="q-answer"><p>gradient = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>f'(x)=3x²+2</p><p>f '(-1) = 3(-1)²+2 = 5</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 gradient of the curve <span class="math-tex">\(y = \frac{1}{x^2}\)</span> at point A is <span class="math-tex">\(\frac{1}{4}\)</span>. Find the x coordinate at A</p></div><div class="q-answer"><p>x = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p>y = x<sup>-2</sup></p><p><span class="math-tex">\(\frac{dy}{dx}\)</span>=-2x<sup>-3</sup></p><p><span class="math-tex">\(\frac{-2}{x^3}=\frac{1}{4}\)</span></p><p>-8 = x<sup>3</sup></p><p>x = -2</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 gradient of the tangent below is <span class="math-tex">\(\frac {a}{4}\)</span> . Find <em><strong>a</strong></em></p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz2/q3.jpg" style="width: 300px; height: 176px;"></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=x+\sqrt x\)</span></p><p><span class="math-tex">\(f(x)=x+{ x }^{ \frac { 1 }{ 2 } }\)</span></p><p><span class="math-tex">\(f'(x)=1+{ \frac { 1 }{ 2 } x }^{ -\frac { 1 }{ 2 } }\)</span></p><p><span class="math-tex">\(f'(x)=1+{ \frac { 1 }{ 2 \sqrt x }} \)</span></p><p><span class="math-tex">\(f'(4)=1+{ \frac { 1 }{ 2 \sqrt 4 }} \)</span></p><p><span class="math-tex">\(f'(4)=1+{ \frac { 1 }{ 4 }} \)</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>The equation of the tangent to the curve at a point P is given by <span class="math-tex">\(y=-\frac { 1 }{ 2 } x+5\)</span></p><p>What is the gradient to the normal at the point P</p></div><div class="q-answer"><p>Gradient =<span new="" roman="" style="font-size:12.0pt;line-height:107%;
font-family:" times=""> </span> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>The normal is perpendicular to the tangent.</p><p><span class="math-tex">\(gradient\ of \ normal\ =\frac { -1 }{ gradient\ of\ tangent } \)</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>The gradient of the normal to the curve y = 2x(x - 1) at the point (2,4) is</p></div><div class="q-answer"><p>gradient = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>y = 2x²-2x</p><p><span class="math-tex">\(\frac { dy }{ dx } =4{ x }-2\)</span></p><p>when x = 2 , <span class="math-tex">\(\frac { dy }{ dx }= 4 \cdot2-2=6\)</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>The equation of the tangent to the curve at A is given by y = <strong><em>m</em></strong> x + <strong><em>c</em></strong></p><p>Find<strong><em> m </em></strong>and<strong><em> c</em></strong></p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz2/q6.jpg" style="width: 300px; height: 260px;"></p></div><div class="q-answer"><p>m = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="-8"> <span class="review"></span></p></div><div class="q-explanation"><p>The gradient of the tangent = 6</p><p>y = 6x + c</p><p>Tangent passes through (1,-2)</p><p>-2=6(1)+c</p><p>c=-8</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 gradient of the tangent to the curve at (1,-3) is -2</p><p>The equation of the normal to the curve at A is given by <strong>x + <em>a</em>y +<em> b</em> = 0</strong></p><p>Find<strong><em> a </em></strong>and<strong><em> b</em></strong></p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz2/q7a.jpg" style="width: 300px; height: 248px;"></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p>gradient of tangent = -2</p><p>gradient of normal = <span class="math-tex">\(\frac{1}{2}\)</span></p><p>y = <span class="math-tex">\(\frac{1}{2}\)</span>x + c</p><p>normal passes through the point (1,-3)</p><p>-3 = <span class="math-tex">\(\frac{1}{2}\)</span>+c</p><p>c = <span class="math-tex">\(3\frac{1}{2}=\frac{7}{2}\)</span></p><p>y = <span class="math-tex">\(\frac{1}{2}\)</span>x + <span class="math-tex">\(\frac{7}{2}\)</span></p><p>2y = x + 7</p><p>0 = x - 2y + 7</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 equation of the tangent to the curve f(x)=x<sup>3 </sup>at the point (1,1) is given by y = <strong><em>m</em></strong>x -2</p><p>Find <strong><em>m</em></strong></p></div><div class="q-answer"><p>m = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>f(x) = x<sup>3 </sup></p><p>f '(x) = 3x²</p><p>f '(1) = 3</p><p>Gradient of the tangent = 3</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 equation of the normal to the curve f(x) = <span class="math-tex">\(\sqrt x\)</span> at the point (9,3) is given by y = <strong><em>m</em></strong>x + 57</p><p>Find <strong><em>m</em></strong></p></div><div class="q-answer"><p>m = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span></p></div><div class="q-explanation"><p>f(x) = <span class="math-tex">\(\sqrt x\)</span> =<span class="math-tex">\(x^{\frac{1}{2}}\)</span></p><p>f '(x) = <span class="math-tex">\(\frac{1}{2}x^{-\frac{1}{2}}\)</span> = <span class="math-tex">\(\frac{1}{2 \sqrt{x}}\)</span></p><p>f '(9) =<span class="math-tex">\(\frac{1}{2 \sqrt{9}}=\frac{1}{6}\)</span></p><p>gradient of tangent = <span class="math-tex">\(\frac{1}{6}\)</span></p><p>gradient of normal = -6</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 equation of the normal to the curve y = <span class="math-tex">\(\frac {2}{x}\)</span> at (2,1) is y = 2x - 3</p><p>Find the coordinates (a,b) of the point where the normal crosses the curve again.</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac {2}{x}=2x-3\)</span></p><p>2 = 2x² - 3x</p><p>2x² - 3x - 2 = 0</p><p>(2x + 1)(x - 2) = 0</p><p>x = <span class="math-tex">\(-\frac {1}{2}\)</span> , x = 2</p><p>when x = <span class="math-tex">\(-\frac {1}{2}\)</span>, y = <span class="math-tex">\(\frac {2}{-\frac{1}{2}}=-4\)</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> </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"> <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>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="379"> <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-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let f(x) = (x - 1)(x - 4)(x + 2). The diagram below shows the graph of <strong><em>f</em></strong> and the point P where the graph crosses the x axis.</p> <p><img alt="" src="../../files/differentiation/tangents-and-normals/esq1a.jpg" style="width: 300px; height: 332px;"></p> <p>The line L is the tangent to the graph of <strong><em>f</em></strong> at the point P.</p> <p>The line L intersects the graph of <strong><em>f</em></strong> at another point Q, as shown below</p> <p><img alt="" src="../../files/differentiation/tangents-and-normals/esq1b.jpg" style="width: 300px; height: 349px;"></p> <p>a) Find the coordinates of P</p> <p>b) Show that <span class="math-tex">\(f(x)=x^3-3x^2-6x+8\)</span></p> <p>c) Find the equation of L in the form y = ax + b</p> <p>d) Find the x coordinate of Q.</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>b) Expand the brackets carefully. For example, you could expand (x - 4)(x + 2) first. Once done multiple (x - 1) by your result.</p> <p><content>c) The gradient of the tangent = f '(0)</content></p> <p>d) Solve <span class="math-tex">\(−6x+8=x^3−3x^2−6x+8\)</span></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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/tangents-and-normals/esq_equation_of_tangent1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/tangents-and-normals/esq_equation_of_tangent1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border 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> <div class="smart-object center" data-id="380"> <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-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let <span class="math-tex">\(f(x)=\frac{x^4-4x^2}{4}\)</span> .</p> <p>C(2 , 0) lies on the graph of y = f(x)</p> <p>a) The tangent to the graph of y = f(x) at C cuts the y axis at A. Find the coordinates of A.</p> <p>b) The normal to the graph of y = f(x) at C cuts the y axis at B. Find the area of the triangle ABC.</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>It helps to visualise the problem</p> <p><content><img alt="" src="../../files/differentiation/tangents-and-normals/esq2.jpg" style="width: 500px; height: 690px;"></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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/tangents-and-normals/esq_differentiation_tangents2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/tangents-and-normals/esq_differentiation_tangents2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border 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> <div class="smart-object center" data-id="381"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The function <span class="math-tex">\(f(x)=x^3-x^2-9x+9\)</span> intersects the x axis at A, B and C.</p> <p>The x coordinate of the point D is the mean of the x coordinates of B and C.</p> <p><img alt="" src="../../files/differentiation/tangents-and-normals/esq3.jpg" style="width: 400px; height: 369px;"></p> <p>a) Find the coordinates of A, B and C.</p> <p>b) Find the equation of the tangent to the curve at D.</p> <p>c) Find the point of the intersection of the tangent with the curve. Interpret your result.</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) Use the Factor Theorem to find the coordinates of A, B and C</p> <p><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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/tangents-and-normals/esq_differentiation_tangents3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/tangents-and-normals/esq_differentiation_tangents3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div 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