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class="gray">Calculus Examination Questions HL</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>On this page you can find examination questions from the topic of calculus</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>Introducing Derivatives</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="377"> <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"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find the value(s) of x for which the graph <span class="math-tex">\(y=x^3-8x+2\)</span> has gradient 4</p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Find where <span class="math-tex">\(\frac{dy}{dx}=4\)</span><content></content></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/introduction/esq_differentiation_power_rule1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/introduction/esq_differentiation_power_rule1.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="376"> <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>Find <span class="math-tex">\(f'(4)\)</span> for the function <span class="math-tex">\(f(x)=2x+\frac { 8 }{ \sqrt { x } } +\frac { 32 }{ x } \)</span></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"><span class="math-tex">\(\frac { 1 }{ \sqrt { x } } ={ x }^{ -\frac { 1 }{ 2 } }\)</span><content> </content></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/introduction/esq_differentiation_power_rule2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/introduction/esq_differentiation_power_rule2.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="378"> <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>The gradient of <em><span class="math-tex">\(y=x^2+ax+b\)</span></em> at the point (1,-3) is -4.</p> <p>Find <em><strong>a</strong></em> and <em><strong>b</strong></em></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><content> </content>We know 2 facts about this graph:</p> <p>x=1 when y = -3</p> <p><span class="math-tex">\(\frac{dy}{dx}=-4\)</span> when x = 1</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/introduction/esq_differentiation_power_rule3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/introduction/esq_differentiation_power_rule3.pdf" width="640"></iframe></p> </section> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Graphs and Derivatives</p> </div> </div> <div class="panel-body"> <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="387"> <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>The graph of y = f(x) is shown below, where B is a local maximum and C is a local minimum</p> <p><img alt="" src="../../files/differentiation/graphs/q1-question.jpg" style="width: 500px; height: 299px;"></p> <p>Sketch a graph of y = f'(x), clearly showing the images of the points B and C labellling them B' and C' respectively</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><content> </content>Consider the gradient of each of the points.</p> <p>B and C are a stationary points - gradient = 0</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> <img alt="" src="../../files/differentiation/graphs/q1-answer.jpg" style="width: 500px; height: 294px;"></p> </section> <h4> </h4> </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="386"> <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>A function is given by <span class="math-tex">\(f(x)=-x^3+6x^2+4\)</span></p> <p>a) Find the coordinates of any stationary points and describe their nature</p> <p>b) Determine the values of x such that <em>f(x) </em>is a increasing function</p> <p>c) Find the coordinates of the point of inflexion</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) Solve f'(x)=0</p> <p><content>b) Draw a sketch of the graph</content></p> <p>c) point of inflexion is non-stationary solve f''(x)=0</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/graphs/esq_differentiation_graphs2a.pdf" target="_blank">here</a></p> <p><iframe frameborder="0" height="480" scrolling="no" src="../../files/differentiation/graphs/esq_differentiation_graphs2a.pdf" width="100%"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="388"> <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>The following diagram shows the graph of <span class="math-tex">\(f'\)</span>, the derivative of <em>f</em></p> <p><img alt="" src="../../files/differentiation/graphs/esq3a.png" style="width: 450px; height: 341px;"></p> <p>On the graph below, sketch the graph of y = f(x) given that f(0) = 0. Mark the images of A , B and C labelling them A' , B' and C'.</p> <p><img alt="" src="../../files/differentiation/graphs/esq3b.png" style="width: 450px; height: 341px;"></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>Consider where the gradient function is zero -> stationary points.</p> <p><content>Then consider the gradient before and after the stationary points.</content></p> <p>The gradient of f' is zero, what does this mean?</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> <img alt="" src="../../files/differentiation/graphs/esq3a_ans.png" style="width: 450px; height: 343px;"></p> </section> <h4> </h4> </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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="389"> <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>Consider the function <span class="math-tex">\(f(x)=-x^3-3x^2+9x\)</span></p> <p>a) Find the coordinates of any stationary points and determine their nature</p> <p>b) Find the equation of the straight line that passes through both the local maximum and the local minimum points.</p> <p>c) Show that the point of inflexion lies on this line.</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) Stationary points occur when f'(x)=0</p> <p><content>b) You can find the equation ( <span class="math-tex">\(y=mx+c\)</span> ) between two points by finding the gradient and using one of the points</content></p> <p><img alt="" src="../../files/differentiation/graphs/esq4.png" style="width: 300px; height: 232px;"></p> <p>c) Point of inflexion must be non-stationary. Solve <span class="math-tex">\(f''(x)=0\)</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/graphs/esq_differentiation_graphs4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/graphs/esq_differentiation_graphs4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </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>Product and Quotient Rule</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="395"> <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 <span class="math-tex">\(y=xe^x\)</span></p> <p>a) Find <span class="math-tex">\(\frac{dy}{dx}\)</span></p> <p>b) Show that <span class="math-tex">\(\frac{d^2y}{dx^2}=e^x(2+x)\)</span></p> <p>c) Find the coordinates of the stationary point and show that it is a local minimum.</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 Product Rule to find <span class="math-tex">\(\frac{dy}{dx}\)</span><content></content></p> <p>c) Solve <span class="math-tex">\(\frac{dy}{dx}=0\)</span> to find position of stationary point. Show that <span class="math-tex">\(\frac{d^2y}{dx^2}>0\)</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/product-quotient/esq_differentiation_product-quotient_rule1a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/esq_differentiation_product-quotient_rule1a.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="392"> <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 <span class="math-tex">\(f(x)={ x }^{ 2 }{ (2x-3) }^{ 3 }\)</span></p> <p>a) Find <span class="math-tex">\(f'(x)\)</span></p> <p>b) The graph of y = f(x) has stationary points at x = 0, x= <span class="math-tex">\(\frac{3}{2}\)</span> and x = <strong>a</strong>. Find the value of <strong>a</strong></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) You need to use the chain rule to differentiate <span class="math-tex">\({ (2x-3) }^{ 3 }\)</span></p> <p><content>b) Factorise your answer to part a)</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/product-quotient/esq_differentiation_product-quotient_rule1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/esq_differentiation_product-quotient_rule1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-has-colored-body panel-expandable panel-default"> <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="394"> <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 <span class="math-tex">\(f(x)=\frac{lnx}{x},x>0\)</span></p> <p>a) Show that <span class="math-tex">\(f'(x)=\frac{1-lnx}{x^2}\)</span></p> <p>b) Find <span class="math-tex">\(f''(x)\)</span></p> <p>c) The graph of <em>f</em> has a point of inflexion at A. Find the x-coordinate of A.</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 Quotient Rule to find <span class="math-tex">\(f'(x)\)</span><content></content></p> <p>c) Solve <span class="math-tex">\(f''(x)=0\)</span> to find point of inflexion</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/product-quotient/esq_differentiation_product-quotient_rule3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/esq_differentiation_product-quotient_rule3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-has-colored-body panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="393"> <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">\(f(x)=tanx\)</span>. A<span class="math-tex">\((\frac{\pi}{3},\sqrt{3})\)</span> is a point that lies on the graph of <span class="math-tex">\(y=f(x)\)</span></p> <p>a) Given that <span class="math-tex">\(tanx=\frac{sinx}{cosx}\)</span> find <span class="math-tex">\(f'(x)\)</span></p> <p>b) Show that <span class="math-tex">\(f'(\frac{\pi}{3})=4\)</span></p> <p>c) Find the equation of the normal to the curve y = f(x) at the point A</p> <p>d) Show that the normal crosses the y axis at <span class="math-tex">\(\sqrt{3}+\frac{\pi}{12}\)</span></p> <h4>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 quotient rule to differentiate tanx</p> <p><content>c) <span class="math-tex">\(gradient \ of \ normal =-\frac{1}{gradient \ of \ tangent}\)</span></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/product-quotient/esq_differentiation_product-quotient_rule2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/esq_differentiation_product-quotient_rule2.pdf" width="640"></iframe></p> </section> <h4> </h4> </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 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="396"> <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>Let <span class="math-tex">\(f(x)=e^{2x}cosx\)</span></p> <p>a) Find <span class="math-tex">\(f'(x)\)</span></p> <p>b) Show that <span class="math-tex">\(f''(x)=4f'(x)-5f(x)\)</span></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 Product Rule to find <span class="math-tex">\(f'(x)\)</span></p> <p>b) Use Product Rule again to find <span class="math-tex">\(f''(x)\)</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/product-quotient/esq_differentiation_product-quotient_rulehl5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/esq_differentiation_product-quotient_rulehl5.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </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>Equation of Tangent and Normal</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 class="panel-footer"> <div> </div> </div> </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>Optimisation</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="495"> <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><img alt="" src="../../files/differentiation/optimisation/esq1.jpg" style="float: left; width: 640px; height: 254px;"></p> <hr class="hidden-separator"> <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">Area of a cylinder , <span class="math-tex">\(A=2\pi r^2+2\pi rh\)</span><content></content></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/optimisation/esq_optimisation1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/optimisation/esq_optimisation1.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="889"> <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>A container is made from a cylinder and a hemisphere. The radius of the cylinder is <strong><em>r</em></strong> m and the height is <strong><em>h</em></strong>. The volume of the container is <span class="math-tex">\(45\pi\)</span></p> <p><img alt="" src="../../files/differentiation/optimisation/esq2.jpg" style="width: 261px; height: 329px;"></p> <p>a) Find an expression for the height of the cylinder in terms of <strong><em>r</em></strong></p> <p>b) Show that the surface area of the container, <span class="math-tex">\(A=\frac{5 \pi r^2}{3}+\frac{90 \pi}{r}\)</span></p> <p>c) Hence, find the values of <strong><em>r</em></strong> and <strong><em>h</em></strong> that give the minimum surface area of the container</p> <p>* Volume of a sphere = <span class="math-tex">\(\frac{4}{3}\pi r^3\)</span></p> <p>** Surface area of a sphere = <span class="math-tex">\(4\pi r^2\)</span></p> <hr class="hidden-separator"> <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>The volume of a hemisphere = <span class="math-tex">\(\frac{2}{3}\pi r^3\)</span><content></content></p> <p>The curved surface area of a hemisphere = <span class="math-tex">\(2\pi r^2\)</span></p> <p>The surface area of the container = <span class="math-tex">\(2\pi r^2+\pi r^2+2\pi rh\)</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>a) The volume of a hemisphere = <span class="math-tex">\(\frac{2}{3}\pi r^3\)</span></p> <p>The volume of the cylinder = <span class="math-tex">\(\pi r^2h\)</span></p> <p>Hence, the volume of the container = <span class="math-tex">\(\frac{2}{3}\pi r^3+\pi r^2h\)</span></p> <p><span class="math-tex">\(\frac{2}{3}\pi r^3+\pi r^2h=45\pi\)</span></p> <p>We can divide through by <span class="math-tex">\(\pi\)</span> and rearrange to make <strong><em>h</em></strong> the subject of the formula</p> <p><span class="math-tex">\(\frac{2}{3} r^3+ r^2h=45\)</span></p> <p><span class="math-tex">\(r^2h=45-\frac{2}{3} r^3\)</span></p> <p><span class="math-tex">\(h=\frac{45}{r^2}-\frac{2}{3} r\)</span></p> <p>b) The surface area of the container = <span class="math-tex">\(2\pi r^2+\pi r^2+2\pi rh\)</span></p> <p><span class="math-tex">\(A=3\pi r^2+2\pi rh\)</span></p> <p><span class="math-tex">\(A=3\pi r^2+2\pi r(\frac{45}{r^2}-\frac{2}{3} r)\)</span></p> <p><span class="math-tex">\(A=\frac{9\pi r^2}{3}-\frac{4 \pi r^2}{3}+\frac{90 \pi}{r}\)</span></p> <p><span class="math-tex">\(A=\frac{5\pi r^2}{3}+\frac{90 \pi}{r}\)</span></p> <p>c) <span class="math-tex">\(\frac{dA}{dr}=\frac{10\pi r}{3}-\frac{90 \pi}{r^2}\)</span></p> <p>Minimimum occurs when <span class="math-tex">\(\frac{dA}{dr}=0\)</span></p> <p><span class="math-tex">\(\frac{10\pi r}{3}-\frac{90 \pi}{r^2}=0\)</span></p> <p><span class="math-tex">\(\frac{10\pi r}{3}=\frac{90 \pi}{r^2}\)</span></p> <p><span class="math-tex">\(r^3=27\)</span></p> <p>r = 3</p> <p><span class="math-tex">\(h=\frac{45}{3^2}-\frac{2}{3} \times 3\)</span></p> <p>h = 5 - 2</p> <p>h = 3</p> <p>We can verify that this gives the minimum value by finding the 2nd derivative</p> <p><span class="math-tex">\(\frac{dA}{dr}=\frac{10\pi r}{3}-\frac{90 \pi}{r^2}\)</span></p> <p><span class="math-tex">\(\frac{d^2A}{dr^2}=\frac{180 \pi}{r^3}\)</span></p> <p>When r = 3, <span class="math-tex">\(\frac{d^2A}{dr^2}>0\)</span> , hence <em><strong>A</strong></em> is a minimum</p> </section> <h4> </h4> </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="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 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>Implicit Differentiation</p> </div> </div> <div class="panel-body"> <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="511"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find the equation of the tangent to the curve <span class="math-tex">\(x^3 + y^3 -6xy = 0\)</span> at the point (3,3)</p> <p><img alt="" src="../../files/differentiation/implicit-differentiation/qu1.png" style="width: 300px; height: 284px;"></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>This is a question about implicit differentiation.</p> <p>Differentiate <span class="math-tex">\(x^3 + y^3 -6xy = 0\)</span> with respect to x and find the gradient function</p> <content> </content></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></span><span class="math-tex">\(3x^2 +3y^2 \frac{dy}{dx} – (6y + 6x \frac{dy}{dx}) = 0\)</span><span></span></p> <p>Substitute x=3 and y=3 to find the gradient of the tangent</p> <p><span class="math-tex">\(3\times3^2 +3\times3^2 \frac{dy}{dx} – (6 \times3+ 6\times3 \frac{dy}{dx} ) = 0 \)</span></p> <p><span class="math-tex">\(27 +27 \frac{dy}{dx} – (18+ 18 \frac{dy}{dx}) = 0\)</span></p> <p><span class="math-tex">\(9 \frac{dy}{dx} +9 = 0\)</span></p> <p>Our question becomes: find the equation of the tangent with gradient -1 which passes through the point (3,3)</p> <p><span class="math-tex">\(-1=\frac{y-3}{x-3}\)</span></p> <p><span class="math-tex">\(-x+3=y-3\)</span></p> <p><span class="math-tex">\(y=-x+6\)</span></p> </section> <h4> </h4> </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="512"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find the gradient of the curve <span class="math-tex">\(x^2+2e^{(x+2y)}=3\)</span> at the point when x=-1</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>This is a question about implicit differentiation.</p> <p><span calibri="" style="font-size:11.0pt;line-height:107%; font-family:">You will need to find the y coordinate when x=-1 and then find </span><span class="math-tex">\(\frac{dy}{dx} \)</span> at this point.</p> <p>To find <span class="math-tex">\(\frac{dy}{dx} \)</span> <span calibri="" style="font-size:11.0pt;line-height:107%; font-family:">, differentiate <span class="math-tex">\(x^2+2e^{(x+2y)}=3\)</span> with respect to x.</span></p> <p style="margin:0cm;margin-bottom:.0001pt">From the Chain Rule, we know that <span class="math-tex">\(\frac{d}{dx}(e^{f(x)})=f'(x)e^{f(x)}\)</span></p> <content> </content></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>Substitute x=-1 into equation</p> <span class="math-tex">\((-1)^2+2e^{(-1+2y)}=3\\ 1+2e^{(-1+2y)}=3\\ 2e^{(-1+2y)}=2\\ e^{(-1+2y)}=1\\ -1+2y=0\\ y=0.5\\ \)</span> <p>Find <span class="math-tex">\(\frac{dy}{dx}\)</span> by differentiating <span class="math-tex">\( x^2+2e^{(x+2y)}=3\)</span> with respect to x</p> <p>From the Chain Rule, we know that <span class="math-tex">\(\frac{d}{dx}(e^{f(x)})=f'(x)e^{f(x)}\)</span></p> <p><span class="math-tex">\(2x+2(1+2\frac{dy}{dx})e^{(x+2y)}=0\)</span></p> <p>Substitute x=-1, y=0.5</p> <p><span class="math-tex">\(2(-1)+2(1+2\frac{dy}{dx})e^{(-1+2(0.5)}=0\\ -2+2(1+2\frac{dy}{dx})e^{(0)}=0\\ 2(1+2\frac{dy}{dx})=2\\ 1+2\frac{dy}{dx}=1\\ 2\frac{dy}{dx}=0\\ \frac{dy}{dx}=0 \)</span></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="513"> <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>A curve is defined by <span class="math-tex">\(x\arcsin y=e^{2y}\)</span></p> <p>Show that <span class="math-tex">\(\frac{dy}{dx}=\frac{\sqrt{1-y^{2}}\arcsin y}{2e^{2y}\sqrt{1-y^{2}}-x}\)</span></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"><content> </content> <p>This is a question about implicit differentiation.</p> <p>Differentiate <span class="math-tex">\(x\arcsin y=e^{2y}\)</span> with respect to x and find the gradient function.</p> <p>You might need to look up <span class="math-tex">\(\frac{d}{dx}( arcsin x)\)</span> in the information booklet.</p> <p>You will need to use the Product Rule, since <span class="math-tex">\(x\arcsin y\)</span> is the product of two functions.</p> <p>Careful rearrangement will be required!</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>Note that <span class="math-tex">\(\frac{d}{dx}(\arcsin x)=\frac{1}{\sqrt{1-x^{2}}}\)</span></p> <p>Therefore, <span class="math-tex">\(\frac{d}{dx}(\arcsin y)=\frac{1}{\sqrt{1-y^{2}}}\frac{dy}{dx}\)</span></p> <hr class="hidden-separator"> <p>Differentiate <span class="math-tex">\(x\arcsin y=e^{2y}\)</span> with respect to x</p> <p><span class="math-tex">\(1\arcsin y+x\frac{1}{\sqrt{1-y^{2}}}\frac{dy}{dx}=2e^{2y}\frac{dy}{dx}\)</span></p> <p>Rearrange the equation to collect terms with <span class="math-tex">\(\frac{dy}{dx}\)</span> on one side</p> <p><span class="math-tex">\(x\frac{1}{\sqrt{1-y^{2}}}\frac{dy}{dx}-2e^{2y}\frac{dy}{dx}=-\arcsin y\)</span></p> <p>Multiply both sides by -1</p> <p><span class="math-tex">\(2e^{2y}\frac{dy}{dx}-x\frac{1}{\sqrt{1-y^{2}}}\frac{dy}{dx}=\arcsin y\)</span></p> <p>Factorise</p> <p><span class="math-tex">\((2e^{2y}-\frac{x}{\sqrt{1-y^{2}}})\frac{dy}{dx}=\arcsin y\)</span></p> <p>Divide both sides through by <span class="math-tex">\(2e^{2y}-\frac{x}{\sqrt{1-y^{2}}}\)</span></p> <p><span class="math-tex">\(\frac{dy}{dx}=\frac{\arcsin y}{2e^{2y}-\frac{x}{\sqrt{1-y^{2}}}}\)</span></p> <p>Multiply numerator and denominator by <span class="math-tex">\({\sqrt{1-y^{2}}}\)</span></p> <p><span class="math-tex">\(\frac{dy}{dx}=\frac{\sqrt{1-y^{2}}\arcsin y}{2e^{2y}\sqrt{1-y^{2}}-x}\)</span></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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="514"> <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> <ol style="list-style-type:lower-alpha;"> <li>Find the coordinates of the stationary point on the graph <span class="math-tex">\(x^2y+2x=1\)</span></li> <li value="2">Show that <span class="math-tex">\(2y + 4x\frac{dy}{dx}+ x^2\frac{d^2y}{dx^2}=0\)</span> and hence justify that the stationary point is a local minima.</li> </ol> <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>This is a question about implicit differentiation. If you need a reminder about stationary points visit <em>stationary points page</em></p> <p>a) Differentiate <span class="math-tex">\(x^2y+2x=1\)</span> with respect to x. The stationary point exists where <span class="math-tex">\( \frac{dy}{dx}=0\)</span>. You can find an equation for y in terms of x by substituting this into your equation. Can you use this and the initial equation to find the coordinates?</p> <p>b) Differentiate the equation for a second time. Remember that local minima occurs when <span class="math-tex">\(\frac{d^2y}{dx^2}>0 \)</span></p> <content> </content></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 style="margin-left:18.0pt;">a)</p> <p style="margin-left:18.0pt;">Differentiate <span class="math-tex">\(x^2y+2x=1\)</span> with respect to x.</p> <p style="margin-left:18.0pt;"><span class="math-tex">\(2xy + x^2\frac{dy}{dx}+2=0\)</span></p> <p style="margin-left:18.0pt;">Substitute <span class="math-tex">\(\frac{dy}{dx}=0 \)</span></p> <p style="margin-left:18.0pt"><span class="math-tex">\(2xy + 0 + 2 = 0\)</span></p> <p style="margin-left:18.0pt"><span class="math-tex">\(xy = -1\)</span></p> <p style="margin-left:18.0pt"><span class="math-tex">\(y = \frac{-1}{x}\)</span></p> <p style="margin-left:18.0pt">Substitute <span class="math-tex">\(y = \frac{-1}{x}\)</span> into initial implicit equation.</p> <p style="margin-left:18.0pt"><span class="math-tex">\(x^2(\frac{-1}{x})+2x=1\\ -x + 2x = 1\\ x=1\\ y=-1 \)</span></p> <p style="margin-left:18.0pt">Hence, a stationary point exists at (1, -1)</p> <p style="margin-left:18.0pt">b)</p> <p><span class="math-tex">\(2xy + x^2\frac{dy}{dx}+2=0\)</span></p> <p>Differentiate <span class="math-tex">\(2xy + x^2\frac{dy}{dx}+2=0\)</span> with respect to x</p> <p><span class="math-tex">\(2y + 2x\frac{dy}{dx}+ 2x\frac{dy}{dx}+ x^2\frac{d^2y}{dx^2}=0\)</span></p> <p>Hence</p> <p><span class="math-tex">\(2y + 4x\frac{dy}{dx}+ x^2\frac{d^2y}{dx^2}=0\)</span></p> <p>Substitute x=1 , y= -1, <span class="math-tex">\(\frac{dy}{dx}=0\)</span></p> <p><span class="math-tex">\(-2 + 0 + \frac{d^2y}{dx^2}=0\\ \frac{d^2y}{dx^2}=2 \)</span></p> <p>Since <span class="math-tex">\(\frac{d^2y}{dx^2}>0\)</span> , stationary point at (1, -1) is a local minima</p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Related Rates of Change</p> </div> </div> <div class="panel-body"> <div> <div class="panel 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="951"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The radius of a sphere is increasing at 2.5 cms<sup>-1</sup></p> <p>Find the rate at which the volume of the sphere is increasing when the radius is 8 cm.</p> <p>Give your answer in terms of <span class="math-tex">\(\large \pi\)</span></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>The rate at which the volume of the sphere is increasing is <span class="math-tex">\(\large \frac{dV}{dt}\)</span><content> </content></p> <p>To find it, use the chain rule</p> <p style="margin-left: 80px;"><span class="math-tex">\(\large \frac{dV}{dt}=\)</span><span style="color:#B22222;"><span class="math-tex">\(\large \frac{dV}{dr}\)</span></span><span class="math-tex">\(\large \times\)</span><span style="color:#0000FF;"><span class="math-tex">\(\large \frac{dr}{dt}\)</span></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>The volume of a sphere, <span class="math-tex">\(\large V = \frac{4}{3}\pi r^3\)</span></p> <p>Differentiate with respect to <strong><em>r</em></strong> , <span class="math-tex">\(\large \frac{dV}{dr}=4\pi r^2\)</span></p> <p>When <strong><em>r</em></strong> = 8 cm , <span style="color:#B22222;"><span class="math-tex">\(\large \frac{dV}{dr}=4\pi \times 8^2=256 \pi\)</span></span></p> <hr class="hidden-separator"> <p>The radius of a sphere is increasing at 2.5 cms<sup>-1</sup>, <span style="color:#0000FF;"><span class="math-tex">\(\large \frac{dr}{dt}=2.5\)</span></span></p> <hr class="hidden-separator"> <p>Using the chain rule</p> <p style="margin-left: 80px;"><span class="math-tex">\(\large \frac{dV}{dt}=\)</span><span style="color:#B22222;"><span class="math-tex">\(\large \frac{dV}{dr}\)</span></span><span class="math-tex">\(\large \times\)</span><span style="color:#0000FF;"><span class="math-tex">\(\large \frac{dr}{dt}\)</span></span></p> <p style="margin-left: 80px;"><span class="math-tex">\(\large \frac{dV}{dt}=\)</span><span style="color:#B22222;"><span style="color:#B22222;"><span class="math-tex">\(\large 256 \pi\)</span></span></span><span class="math-tex">\(\large \times\)</span><span style="color:#0000FF;"><span style="color:#0000FF;"><span class="math-tex">\(\large 2.5\)</span></span></span></p> <p style="margin-left: 80px;"><span class="math-tex">\(\large \frac{dV}{dt}=640 \pi\)</span></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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="952"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><img alt="" src="../../files/differentiation/related-rates/esq2-cone.png" style="float: right; width: 250px; height: 248px;">The diagram shows a container in the form of a right circular cone. The height of the cone is equal to its diameter. Initially the cone is empty, then water is added at a rate of <span class="math-tex">\(\large 18\pi\)</span> cm<sup>3 </sup>per minute. the depth of water in the container at the time is given by <strong><em>h</em></strong> cm.</p> <p>a) Show that the volume, <strong><em>V</em></strong> cm<sup>3</sup> , of water in the container when the depth is <strong><em>x</em></strong> cm is given by</p> <p style="margin-left: 80px;"><span class="math-tex">\(\large V=\frac{1}{12} \pi x^3\)</span></p> <p>b) Find the rate at which the depth of the water is increasing at the instant when the depth is 12 cm</p> <hr class="hidden-separator"> <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) height = diameter</p> <p>Therefore, <span class="math-tex">\(\large r=\frac{h}{2}\)</span></p> <p>In this case, the height of water = <strong><em>x</em></strong></p> <p>b) The rate at which the depth of the water is increasing is <span class="math-tex">\(\large \frac{dx}{dt}\)</span><content> </content></p> <p>To find it, use the chain rule</p> <p style="margin-left: 80px;"><span style="color:#0000FF;"><span class="math-tex">\(\large \frac{dV}{dt}\)</span>=</span><span style="color:#B22222;"><span class="math-tex">\(\large \frac{dV}{dx}\)</span></span><span class="math-tex">\(\large \times\)</span><span class="math-tex">\(\large \frac{dx}{dt}\)</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>a) The volume of a cone, <span class="math-tex">\(\large V = \frac{1}{3}\pi r^2h\)</span></p> <p>In this case the height = diameter</p> <p>Therefore, <span class="math-tex">\(\large r=\frac{h}{2}\)</span></p> <p>The volume of a cone, <span class="math-tex">\(\large V = \frac{1}{3}\pi (\frac{h}{2})^2h\\ \large V = \frac{1}{3}\pi (\frac{h^2}{4})h\\ \large V = \pi \frac{h^3}{12}\)</span></p> <p>When the depth (height) of water= <strong><em>x</em></strong>, then</p> <p><span class="math-tex">\(\large V=\frac{1}{12} \pi x^3\)</span></p> <hr class="hidden-separator"> <p>b) <span class="math-tex">\(\large V=\frac{1}{12} \pi x^3\)</span></p> <p>Differentiate with respect to <strong><em>x</em></strong> , <span class="math-tex">\(\large \frac{dV}{dx}=\frac{1}{4}\pi r^2\)</span></p> <p>When <strong><em>x</em></strong> = 12 cm , <span style="color:#B22222;"><span class="math-tex">\(\large \frac{dV}{dx}=\frac{1}{4}\pi \times12^2=36\pi\)</span></span></p> <hr class="hidden-separator"> <p>Water is added at a rate of <span class="math-tex">\(\large 18\pi\)</span> cm<sup>3 </sup>per minute, <span style="color:#0000FF;"><span class="math-tex">\(\large \frac{dV}{dt}=18 \pi\)</span></span></p> <hr class="hidden-separator"> <p>Using the chain rule</p> <p style="margin-left: 80px;"><span style="color:#0000FF;"><span class="math-tex">\(\large \frac{dV}{dt}\)</span>=</span><span style="color:#B22222;"><span class="math-tex">\(\large \frac{dV}{dx}\)</span></span><span class="math-tex">\(\large \times\)</span><span class="math-tex">\(\large \frac{dx}{dt}\)</span></p> <p style="margin-left: 80px;"><span style="color:#0000FF;"><span style="color:#0000FF;"><span class="math-tex">\(\large 18 \pi\)</span> </span>=</span><span style="color:#B22222;"><span style="color:#B22222;"><span class="math-tex">\(\large36\pi\)</span></span></span><span class="math-tex">\(\large \times\)</span><span class="math-tex">\(\large \frac{dx}{dt}\)</span></p> <p style="margin-left: 80px;"><span style="color:#B22222;"></span><span style="color:#0000FF;"></span><span class="math-tex">\(\large \frac{dx}{dt}=\frac{18\pi}{36\pi}\)</span></p> <p style="margin-left: 80px;"><span style="color:#B22222;"><span style="color:#B22222;"></span></span><span style="color:#0000FF;"><span style="color:#0000FF;"></span></span><span class="math-tex">\(\large \frac{dx}{dt}=0.5\)</span> cm per minute</p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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="953"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>A searchlight rotates at 2 revolutions per minute. The beam hits a wall 30m away and produces a spot of light that moves horizontally along the wall. How fast is the spot moving along the wall when the angle, <span class="math-tex">\(\large \theta\)</span> between the beam and the line through the spotlight perpendicular to the wall is 45°?</p> <p style="text-align: center;"><img alt="" src="../../files/differentiation/related-rates/esq3_1-searchlight.png" style="width: 300px; height: 253px;"></p> <hr class="hidden-separator"> <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>If we let <strong><em>x </em></strong>be the distance along the wall, then the speed of the spotlight is<span class="math-tex">\(\large \frac{dx}{dt}\)</span><content> </content></p> <p>To find it, use the chain rule</p> <p style="margin-left: 80px;"><span style="color:#0000FF;"></span><span class="math-tex">\(\large \frac{dx}{dt}\)</span><span style="color:#0000FF;">=</span><span style="color:#B22222;"><span class="math-tex">\(\large \frac{dx}{d\theta}\)</span></span><span class="math-tex">\(\large \times\)</span><span style="color:#0000FF;"><span class="math-tex">\(\large \frac{d\theta}{dt}\)</span></span></p> <p>To find <span style="color:#B22222;"><span class="math-tex">\(\large \frac{dx}{d\theta}\)</span></span> use right-angled triangle trigonometry to find <strong><em>x</em></strong> in terms of <span class="math-tex">\(\large \theta\)</span></p> <p><img alt="" src="../../files/differentiation/related-rates/esq3_2-searchlight.png" style="width: 200px; height: 134px;"></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>The searchlight is rotating at 2 revolutions per minute</p> <p>1 revolution is <span class="math-tex">\(\large 2\pi \)</span> radians</p> <p>Therefore, <span style="color:#0000FF;"><span class="math-tex">\(\large \frac{d\theta}{dt}=4\pi\)</span></span> radians per minute</p> <hr class="hidden-separator"> <p>Let the distance fom from the perpendicular be <strong><em>x</em></strong></p> <p><img alt="" src="../../files/differentiation/related-rates/esq3_2-searchlight.png" style="width: 200px; height: 134px;"></p> <p><span class="math-tex">\(\large \sin \theta=\frac{x}{30}\)</span></p> <p><span class="math-tex">\(\large x=30 \sin\theta\)</span></p> <p>Differentiate with respect to <span class="math-tex">\(\large \theta\)</span></p> <p><span class="math-tex">\(\large \frac{dx}{d\theta}=30 \cos \theta\)</span></p> <p>When <span class="math-tex">\(\theta\)</span>=45°, <span style="color:#B22222;"><span class="math-tex">\(\large \frac{dx}{d\theta}=30 \cos45°=15\sqrt{2}\)</span></span></p> <hr class="hidden-separator"> <p>Use the chain rule</p> <p style="margin-left: 80px;"><span style="color:#0000FF;"></span><span class="math-tex">\(\large \frac{dx}{dt}\)</span><span style="color:#0000FF;">=</span><span style="color:#B22222;"><span class="math-tex">\(\large \frac{dx}{d\theta}\)</span></span><span class="math-tex">\(\large \times\)</span><span style="color:#0000FF;"><span class="math-tex">\(\large \frac{d\theta}{dt}\)</span></span></p> <p style="margin-left: 80px;"><span class="math-tex">\(\large \frac{dx}{dt}\)</span><span style="color:#0000FF;">=</span><span style="color:#B22222;"><span style="color:#B22222;"><span class="math-tex">\(\large 15\sqrt{2}\)</span></span></span><span class="math-tex">\(\large \times\)</span><span style="color:#0000FF;"><span style="color:#0000FF;"><span class="math-tex">\(\large 4\pi\)</span></span></span></p> <p style="margin-left: 80px;"><span class="math-tex">\(\large \frac{dx}{dt}=60\sqrt{2}\pi\approx267\)</span>m per minute</p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="954"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>A ladder AB of length 8m has one endA on horizontal ground and the other end B resting against a vertical wall.</p> <p>The end A slips away from the wall at a constant speed of 0.5 ms<sup>-1</sup> and the end B slips down the wall.</p> <p>Determine the speed of the end B is slipping down the wall when the top of the ladder is 5 m above the ground.</p> <p style="text-align: center;"><img alt="" src="../../files/differentiation/related-rates/esq4_1-ladder.png" style="width: 257px; height: 326px;"></p> <hr class="hidden-separator"> <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>If we let <strong><em>x </em></strong>be the distance along the ground, then the speed end A slips away from the wall is <span class="math-tex">\(\large \frac{dx}{dt}\)</span><content> </content></p> <p>If w let <strong><em>y</em></strong> be the distance up the wall, then the speed end B slips down the wall is <span class="math-tex">\(\large \frac{dy}{dt}\)</span></p> <p>Use Pythagoras' Theorem</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>The speed end A slips away from the wall is <span class="math-tex">\(\large \frac{dx}{dt}\)</span><content> </content></p> <p><span class="math-tex">\(\large \frac{dx}{dt}=0.5\)</span> ms<sup>-1</sup></p> <hr class="hidden-separator"> <p>The speed end B slips down the wall is <span class="math-tex">\(\large \frac{dy}{dt}\)</span></p> <hr class="hidden-separator"> <p>Using Pythagoras' Theorem</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large x^2+y^2=8^2\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large x^2+y^2=64\)</span></p> <p>Differentiate with respect to <strong><em>t </em></strong>(implicit differentiation)</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large 2x \frac{dx}{dt}+2y\frac{dy}{dt}=0\)</span></p> <hr class="hidden-separator"> <p>When <strong><em>y</em></strong> =5 ,</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large x^2+5^2=64\\ \large x^2=39\\ \large x=\sqrt {39}\)</span></p> <hr class="hidden-separator"> <p>Substitute these values in the differential equation</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large 2x \frac{dx}{dt}+2y\frac{dy}{dt}=0\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large 2\sqrt{39}\times 0.5+2\times 5\frac{dy}{dt}=0\\ \large 10\frac{dy}{dt}=-\sqrt{39}\\ \large \frac{dy}{dt}=-\frac{\sqrt{39}}{10}\\ \large \frac{dy}{dt}=-0.624\)</span></p> <p>Notice that the value is negative. This represents the <strong>velocity </strong>and shows that it is slipping <strong>down</strong><strong>.</strong></p> <p><strong>The speed is 0.624 ms<sup>-1</sup></strong></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="955"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p> </p> <p style="text-align: center;"><img alt="" src="../../files/differentiation/related-rates/esq5-solid.png" style="width: 350px; height: 394px;"></p> <p> </p> <p>A solid is made up of a cylinder and a hemisphere as shown in the diagram.</p> <p>a) Write down a formula for the volume of the solid.</p> <p>b) At the time when the radius is 6cm, the volume of the solid is <span class="math-tex">\(\large 684\pi\)</span> cm<sup>3</sup> , the radius is changing at a rate of 1.5 cm/minute and the volume is changing at a rate of <span class="math-tex">\(\large 1800\pi\)</span> cm<sup>3</sup> /min. Find the rate of change of the height at this time.</p> <hr class="hidden-separator"> <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>We can find the volume of the solid to be <span class="math-tex">\(\large V=\frac{2}{3}\pi r^3+\pi r^2h\)</span></p> <p>We know <span class="math-tex">\(\large \frac{dr}{dt}\)</span> and <span class="math-tex">\(\large \frac{dV}{dt}\)</span>. We are required to find <span class="math-tex">\(\large \frac{dh}{dt}\)</span>.</p> <p>Use implicit differentiation and differentiate the formula for <strong><em>V</em></strong> with respect to <strong><em>t</em></strong></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 volume of a hemisphere is <span class="math-tex">\(\large \frac{2}{3}\pi r^3\)</span></p> <p>The volume of a cylinder is <span class="math-tex">\(\large \pi r^2h\)</span></p> <p>The volume of the solid, <span class="math-tex">\(\large V=\frac{2}{3}\pi r^3+\pi r^2h\)</span></p> <hr class="hidden-separator"> <p>b) We can find the value of h when r = 6 and V = <span class="math-tex">\(\large 684\pi\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large 684\pi=\frac{2}{3}\pi \times 6^3+\pi \times 6^2\times h\\ \large 684\pi=144\pi+36\pi h\\ \large 540\pi=36\pi h\\ \large h = 15\)</span></p> <hr class="hidden-separator"> <p>The radius is changing at a rate of 1.5 cm/minute, <span class="math-tex">\(\large \frac{dr}{dt}=1.5\)</span></p> <p>The volume is changing at a rate of <span class="math-tex">\(\large 1800\pi\)</span> cm<sup>3</sup> /min, <span class="math-tex">\(\large \frac{dV}{dt}=1800\pi\)</span></p> <p>We are required to find <span class="math-tex">\(\large \frac{dh}{dt}\)</span>.</p> <p>Use implicit differentiation and differentiate the formula for <strong><em>V</em></strong> with respect to <strong><em>t</em></strong></p> <p><span class="math-tex">\(\large V=\frac{2}{3}\pi r^3+\pi r^2h\)</span></p> <p>Notice that the second part of the expression <img alt="" src="../../files/differentiation/related-rates/esq5-ans-solid.png" style="width: 190px; height: 50px;"> is a product (both <strong><em>r</em></strong> and <strong><em>h</em></strong> are variables). Therefore we need to use the Product Rule for differentiation</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{dV}{dt}=2\pi r^2\frac{dr}{dt}+\pi r^2\frac{dh}{dt}+2\pi r^2\frac{dr}{dt}\cdot h\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large 1800\pi=2\pi \times 6^2\times 1.5+\pi \times 6^2\frac{dh}{dt}+2\pi \times 6^2\times 1.5\times 15\\ \large 1800\pi=108\pi+36\pi \cdot \frac{dh}{dt}+1620\pi\\ \large 72\pi=36\pi \cdot \frac{dh}{dt}\\ \large \frac{dh}{dt}=2\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{dh}{dt}=2\)</span> cm/min</p> </section> </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-default panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>L'Hôpital's Rule</p> </div> </div> <div class="panel-body"> <div> <div class="panel 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="947"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Use l’Hôpital’s rule to determine the value of <span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}( x \ln x)\)</span></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"><content> </content>Write the limit in the correct form <span class="math-tex">\(\large \lim\limits_{x\rightarrow 0} (x \ln x)= \lim\limits_{x\rightarrow 0} \frac{\ln x}{\frac{1}{x}}\)</span></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="math-tex">\(\large \lim\limits_{x\rightarrow 0} (x \ln x)= \lim\limits_{x\rightarrow 0} \frac{\ln x}{\frac{1}{x}}=\frac{-\infty}{\infty}\)</span></p> <p>This is in the indeterminate form</p> <p>Let <span class="math-tex">\(\large f(x) = \ln x\)</span> and <span class="math-tex">\(\large g(x) = x^{-1}\)</span></p> <p>Then, <span class="math-tex">\(\large f'(x) = \frac{1}{x}\)</span> and <span class="math-tex">\(\large g'(x) = -x^{-2}=-\frac{1}{x^2}\)</span></p> <p>Therefore,</p> <p><span class="math-tex">\( \lim\limits_{x\rightarrow 0} \frac{\ln x}{\frac{1}{x}}= \lim\limits_{x\rightarrow 0} \frac{\frac{1}{x}}{-\frac{1}{x^2}}\)</span></p> <p>We can simplify this</p> <p><span class="math-tex">\(\lim\limits_{x\rightarrow 0} \frac{\frac{1}{x}}{-\frac{1}{x^2}}=\lim\limits_{x\rightarrow 0} \frac{-x^2}{x}=\lim\limits_{x\rightarrow 0}(-x)=0\)</span></p> <p>Therefore, <span class="math-tex">\(\large \lim\limits_{x\rightarrow 0} (x \ln x)= 0\)</span></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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="949"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Use l’Hôpital’s rule to determine the value of <span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{e^{x^2}-1}{\sin x^2}\)</span></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><content> </content>When you apply l'Hôpital's Rule, simplify your answer.</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="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{e^{x^2}-1}{\sin x^2}=\frac{1-1}{0}=\frac{0}{0}\)</span></p> <p>This is in the indeterminate form</p> <p>Let <span class="math-tex">\(\large f(x) = e^{x^2}-1\)</span> and <span class="math-tex">\(\large g(x) =\sin x^2\)</span></p> <p>Then, <span class="math-tex">\(\large f'(x) = 2xe^{ x^2}\)</span> and <span class="math-tex">\(\large g'(x) = 2x\cos x^2\)</span></p> <p>Therefore,</p> <p><span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{e^{x^2}-1}{\sin x^2}= \lim\limits_{x\rightarrow 0}\frac{2xe^{x^2}}{2x\cos x^2}\)</span></p> <p>We can simplify this</p> <p><span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{e^{x^2}}{\cos x^2}=\frac{1}{1}=1\)</span></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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="948"> <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>Use l’Hôpital’s rule to determine the value of <span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{1-\cos x^2}{x^4}\)</span></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><content> </content>You will need to apply l'Hôpital's rule twice.</p> <p>The second application will need simplifying</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="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{1-\cos x^2}{x^4}=\frac{1-1}{0}=\frac{0}{0}\)</span></p> <p>This is in the indeterminate form</p> <p>Let <span class="math-tex">\(\large f(x) = 1-\cos x^2\)</span> and <span class="math-tex">\(\large g(x) = x^4\)</span></p> <p>Then, <span class="math-tex">\(\large f'(x) = 2x\sin x^2\)</span> and <span class="math-tex">\(\large g'(x) = 4x^3\)</span></p> <p>Therefore,</p> <p><span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{1-\cos x^2}{x^4}= \lim\limits_{x\rightarrow 0}\frac{2x\sin x^2}{4x^3}\)</span></p> <p>We can simplify this</p> <p><span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{\sin x^2}{2x^2}=\frac{0}{0}\)</span></p> <p>We can apply l'Hôpital's rule again</p> <p>Let <span class="math-tex">\(\large f(x) = \sin x^2\)</span> and <span class="math-tex">\(\large g(x) = 2x^2\)</span></p> <p>Then, <span class="math-tex">\(\large f'(x) = 2x\cos x^2\)</span> and <span class="math-tex">\(\large g'(x) = 4x\)</span></p> <p>Therefore, <span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{\sin x^2}{2x^2}= \lim\limits_{x\rightarrow 0}\frac{2x\cos x^2}{4x}\)</span></p> <p>We can simplify this</p> <p><span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{\cos x^2}{2}=\frac{1}{2}\)</span></p> <p>Therefore, <span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{1-\cos x^2}{x^4}=\frac{1}{2}\)</span></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="950"> <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>Use l’Hôpital’s rule to determine the value of <span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{6\tan x-6x}{x^3}\)</span></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><content> </content>You will need to apply l'Hôpital's rule three times.</p> <p>Ensure that you simplify the limit if it is possible.</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="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{6\tan x-6x}{x^3}=\frac{0-0}{0}=\frac{0}{0}\)</span></p> <p>This is in the indeterminate form</p> <p>Let <span class="math-tex">\(\large f(x) = 6\tan x-6x\)</span> and <span class="math-tex">\(\large g(x) = x^3\)</span></p> <p>Then, <span class="math-tex">\(\large f'(x) = 6 \sec ^2x-6\)</span> and <span class="math-tex">\(\large g'(x) = 3x^2\)</span></p> <p>Therefore,</p> <p><span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{6\tan x-6x}{x^3}= \lim\limits_{x\rightarrow 0}\frac{\frac{6}{(\cos x)^2}-6}{3x^2}=\frac{0}{0}\)</span></p> <p>We can apply l'Hôpital's rule again</p> <p>Let</p> <p><span class="math-tex">\(\large f(x) = \frac{6}{(\cos x)^2}-6\\ \large f(x) =6(\cos x)^{-2}-6\)</span> and <span class="math-tex">\(\large g(x) = 3x^2\)</span></p> <p>Then,</p> <p><span class="math-tex">\(\large f'(x) =6(-2)(-\sin x)(\cos x)^{-3}\\ \large f'(x)=\frac{12\sin x}{(\cos x)^3}\)</span> and <span class="math-tex">\(\large g'(x) = 6x\)</span></p> <hr class="hidden-separator"> <p>Therefore, <span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{6\tan x-6x}{x^3}= \lim\limits_{x\rightarrow 0}\frac{\frac{12\sin x}{(\cos x)^3}}{6x}\)</span></p> <p>We can simplify this</p> <p><span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{2\sin x}{x(\cos x)^3}\)</span></p> <p>We can apply l'Hôpital's rule again</p> <p>Let</p> <p><span class="math-tex">\(\large f(x) = 2\sin x\)</span> and <span class="math-tex">\(\large g(x) = x(\cos x)^3\)</span></p> <p>Then,</p> <p><span class="math-tex">\(\large f'(x) =2\cos x\)</span> and <span class="math-tex">\(\large g'(x) = 1(\cos x)^3+x\cdot3(-\sin x)(\cos x)^2\\ \large g'(x) = (\cos x)^3-3x\sin x(\cos x)^2\)</span></p> <p>Therefore, <span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{6\tan x-6x}{x^3}=\lim\limits_{x\rightarrow 0}\frac{2\cos x}{ (\cos x)^3-3x\sin x(\cos x)^2}=\frac{2}{1-0}=2\)</span></p> <hr class="hidden-separator"> <p><span class="math-tex">\(\large \lim\limits_{x\rightarrow 0}\frac{6\tan x-6x}{x^3}=2\)</span></p> <hr class="hidden-separator"></section> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Definite Integration</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="538"> <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>Given that <span class="math-tex">\(\int _{ 4 }^{ 8 }{ \frac { 1 }{ 2x-4 } dx= } ln\sqrt { a } \)</span> , find the value of <strong><em>a</em></strong></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"><content> </content><span class="math-tex">\(\int { \frac { 1 }{ ax+b } dx=\frac { 1 }{ a } ln(ax+b)+C } \)</span></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/integration/definite-integral-and-area/esq_calculus_definiteintegral1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/definite-integral-and-area/esq_calculus_definiteintegral1.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="539"> <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>Consider a function f(x) such that <span class="math-tex">\(\int _{ 0 }^{ 4 }{ f(x)dx } \)</span> = 6</p> <p>Find</p> <p>a) <span class="math-tex">\(\int _{ 0 }^{ 4 }{3 f(x)dx } \)</span></p> <p>b) <span class="math-tex">\(\int _{ 0 }^{ 4 }{[ f(x)+3]dx } \)</span></p> <p>c) <span class="math-tex">\(\int _{ -3 }^{ 1 }{\frac{1}{3} f(x+3)dx } \)</span></p> <p>d) <span class="math-tex">\(\int _{ 0 }^{ 4 }{ [f(x)+x]dx } \)</span></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><content> </content>The following properties will be useful</p> <p><span class="math-tex">\(\int _{ a }^{ b }{k f(x)dx } =k\int _{ a }^{ b }{ f(x)dx } \)</span></p> <p><span class="math-tex">\(\int _{ a }^{ b }{ [f(x)+g(x)]dx } =\int _{ a }^{ b }{ f(x)dx } +\int _{ a }^{ b }{ g(x)dx } \)</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/integration/definite-integral-and-area/esq_calculus_definiteintegral2a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/definite-integral-and-area/esq_calculus_definiteintegral2a.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="537"> <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>Given that <span class="math-tex">\(\int _{ 2 }^{ 5 }{ ln(sinx)dx=A } \)</span></p> <p>show that <span class="math-tex">\(\int _{ 2 }^{ 5 }{ ln({ e }^{ x }sinx)dx=A } +\frac { 21 }{ 2 } \)</span></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"><content> </content><span class="math-tex">\(\int _{ 2 }^{ 5 }{ ln({ e }^{ x }sinx)dx= } \int _{ 2 }^{ 5 }{ ln({ e }^{ x })dx+\int _{ 2 }^{ 5 }{ ln(sinx)dx } } \)</span></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/integration/definite-integral-and-area/esq_calculus_definiteintegral3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/definite-integral-and-area/esq_calculus_definiteintegral3.pdf" width="640"></iframe></p> </section> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Area between Graphs</p> </div> </div> <div class="panel-body"> <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="280"> <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)=sinx, for <span class="math-tex">\(0\le x\le 2\pi \)</span></p> <p>The following diagram shows the graph of <strong><em>f</em></strong></p> <p><img alt="" src="../../files/integration/area-between/esq1.png" style="width: 498px; height: 190px;"></p> <p>The shaded region R is enclosed by the graph of <strong><em>f</em></strong>, the line x=a , where a<<span class="math-tex">\(\pi\)</span> and the x-axis.</p> <p>The area of R is <span class="math-tex">\(\left( 1-\frac { \sqrt { 3 } }{ 2 } \right) \)</span>. Find the value of <strong><em>b</em></strong>.</p> <p> </p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <p> </p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content>Area = <span class="math-tex">\(\int _{ a }^{ \pi }{ \sin { x } \ dx } \)</span></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/integration/area-between/esq-sol-areabetweengraphs1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/area-between/esq-sol-areabetweengraphs1.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="269"> <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><em>Show that the area bounded by the graphs of y = f(x) and y = g(x) in the interval <span class="math-tex">\(0\le x\le \pi \)</span> is given by 2 - <span class="math-tex">\(\frac { \pi }{ 2} \)</span></em></p> <p><em>f(x) = sinx </em></p> <p><em>g(x) = sin²x </em></p> <p><img alt="" src="../../files/integration/area-between/esqdiagram3.png" style="width: 530px; height: 355px;"></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"> <ol> <li>You need to find the area between the two graphs y = sinx and y=sin²x</li> <li>The tricky part is integrating sin²x. To do this you will need to use the trig identity for cos2x</li> </ol> </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/integration/area-between/esq_ans_areabetweengraphs2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/area-between/esq_ans_areabetweengraphs2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="282"> <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">\(f(x)=ln\left( \frac { x }{ x-1 } \right) \)</span> for x>1</p> <p>a) Find f ' (x)</p> <p>b) <strong>Hence</strong>, show that the area bounded by g(x) = <span class="math-tex">\(\frac { 1 }{ x(x-1) } \)</span> , the x axis, x = 2 and x = e is given by <span class="math-tex">\(ln\left( \frac { 2e-2 }{ e } \right) \)</span></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><content> </content>a) use log laws to make this function easier to differentiate (log<span class="math-tex">\(\frac{a}{b}\)</span>=loga - logb)</p> <p>b) <em><strong>'Hence'</strong></em> is important here. This suggests that the way to do part b) follows from what you have found in part a). You should notice that the function to integrate is the same as the answer to part a)</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/integration/area-between/esq-sol-areabetweengraphs3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/area-between/esq-sol-areabetweengraphs3.pdf" width="640"></iframe></p> </section> </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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="278"> <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 graph below shows the function <strong>f(x) = sinx </strong>where a < b <span class="math-tex">\(\le \frac { \pi }{ 2 } \)</span></p> <p><img alt="" src="../../files/integration/area-between/esq3.png" style="width: 400px; height: 233px;"></p> <p>a) Write an expression for the area A, bounded by the curve y = f(x), the x axis, x = a and x = b</p> <p>b) <strong>Hence</strong>, without using any further integration, show that the area B = <span class="math-tex">\(bsinb - asina - cosb + cosa\)</span></p> <p>c) Area B can <strong><u>also </u></strong>be found by finding the area bounded by the curve x = f(y), the y axis, y = f(a) and y = f(b). Find the area B using this method and <strong>show</strong> that your answer is the same as the one you found in part b)</p> <p> </p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <p> </p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>b) Area B can be found by considering the 2 rectangles. You can find the area of these two rectangles without doing any further integration. For example, the height of the larger rectangle is <em>sinb</em></p> <p>c) Find the area bounded by x = arcsiny, y= sina, y= sinb and the y axis.</p> <p>You can use integration by parts to find <span class="math-tex">\(\int { 1\bullet \arcsin { y}_\ dy } \)</span> . This method is similar to integrating <a href="../635/integration-by-parts.html#eg3" target="_blank">lnx</a></p> <p>You will then be required to find <span class="math-tex">\(\int { \frac { y }{ \sqrt { 1-{ y }^{ 2 } } } dy } \)</span> which you can do using the method of <a href="../635/index.htm" target="_blank">integration by substitution</a> or recognition.</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><img alt="" src="../../files/integration/area-between/esq3.png" style="width: 400px; height: 233px;"></p> <p>a) Area A</p> <p><span class="math-tex">\(= \int _{ a }^{ b }{ sinx } dx\\ ={ [-cosx] }_{ a }^{ b }\\ =-cosb\ +\ cosa\)</span></p> <p>b)</p> <p><img alt="" src="../../files/integration/area-between/esq3---partb.png" style="width: 400px; height: 233px;"></p> <p>Area B = area large rectangle - area small rectangle - area A</p> <p>= bsinB - a sinA - (-cosb + cosa)</p> <p>= bsinB - a sinA + cosb - cosa</p> <p>c) Graph is y = sinx which is equivalent to <strong>x = arcsiny</strong> for <span class="math-tex">\(0\le \ x\le \frac { \pi }{ 2} \ \)</span></p> <p>Area B <span class="math-tex">\(=\int _{ sina }^{ sinb }{ \arcsin { y } }\ dy\)</span></p> <p>Let's consider <span class="math-tex">\(\int { \arcsin { y } }\ dy\)</span></p> <p>We can use integration by parts</p> <p><span class="math-tex">\(\int {u\frac{{dv}}{{dx}}}\ dx = uv - \int {\frac{{du}}{{dx}}}\ vdx\)</span></p> <p>since our variable is y , this becomes</p> <p><span class="math-tex">\(\int {u\frac{{dv}}{{dy}}} \ dy = uv - \int {\frac{{du}}{{dy}}} v \ dy\)</span></p> <p>We make our integral a product of two parts (this is like <a href="../635/integration-by-parts.html#eg3" target="_blank">integrating <em>lnx</em>)</a></p> <p><span class="math-tex">\(\int 1\cdot { \arcsin { y } } \ dy\)</span></p> <p style="margin-left: 160px;"><span class="math-tex">\(u = arcsiny \quad \frac { dv }{ dy } = 1\\ \frac { du }{ dy } = \frac { 1 }{ \sqrt { 1-{ y }^{ 2 } } }\quad v = y\\\)</span></p> <p><span class="math-tex">\(=y \cdot arcsiny - \int \frac { y }{ \sqrt { 1-{ y }^{ 2 } } } \ dy\)</span></p> <p>To find <span class="math-tex">\(\int \frac { y }{ \sqrt { 1-{ y }^{ 2 } } } \ dy\)</span> , we need to use integration by substitution (or recognition)</p> <p style="margin-left: 200px;">Let w = 1 - y<sup>2</sup></p> <p style="margin-left: 200px;"><span class="math-tex">\( \frac { dw }{dy } = -2y\)</span></p> <p><span class="math-tex">\(=y \cdot arcsiny + \int \frac { 1 }{ 2 } { w }^{ -\frac { 1 }{ 2 } }\ dw\)</span></p> <p><span class="math-tex">\(=y \cdot arcsiny + { w }^{ \frac { 1 }{ 2 } } \ +c\)</span></p> <p><span class="math-tex">\(=y \cdot arcsiny + \sqrt {1-{ y }^2 } \ +c\)</span></p> <p>So, let's go back to the area of B</p> <p>Area B <span class="math-tex">\(=\int _{ sina }^{ sinb }{ \arcsin { y } }\ dy\)</span></p> <p><span class="math-tex">\(={ \left[ { \ y \cdot arcsiny + \sqrt {1-{ y }^2 } } \ \right] }_{ sina }^{ sinb }\)</span></p> <p><span class="math-tex">\(=[\ sinb\cdot b+\sqrt { 1-sin²b } \ ]-[ \ sina\cdot a+\sqrt { 1-sin²a } \ ]\)</span></p> <p style="margin-left: 40px;">Since, 1-sin²x = cos²x</p> <p><span class="math-tex">\(=[\ sinb\cdot b+cosb\ ]-[ \ sina\cdot a+cosa \ ]\)</span></p> <p><span class="math-tex">\(= b\cdot sinb+cosb- \ a\cdot sina-cosa \)</span></p> <p>= bsinB - a sinA + cosb - cosa</p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body"> <div class="panel-heading"> <div> <p>Volume of Revolution</p> </div> </div> <div class="panel-body"> <div> <div class="panel 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="938"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Show that <span class="math-tex">\(\large \int {x^2e^{2x}\,dx}=\frac{1}{4}e^{2x}(2x^2-2x+1)+c\)</span></p> <p>The following graph shows the function <span class="math-tex">\(\large f(x)=xe^x\)</span></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2_1.png" style="width: 400px; height: 401px;"></p> <p>b) Show that the equation of the tangent to the graph at x = 1 has the equation <span class="math-tex">\(\large y=(2e)x-e\)</span></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2_2.png" style="width: 400px; height: 356px;"></p> <p>The region bounded by <span class="math-tex">\(\large f\)</span>, the tangent <span class="math-tex">\(\large y=(2e)x-e\)</span> and y = 0 is rotated <span class="math-tex">\(\large 2\pi\)</span> around the x axis</p> <p>c) Find the volume of this solid in terms of <span class="math-tex">\(\large \pi\)</span></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) this is an integration by parts question</p> <p><content>c) the volume of the solid can be found by subtracting the volume of a cone from the volume of the curved solid </content></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2_hint.png" style="width: 400px; height: 207px;"></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)</p> <p><img alt="" src="../../files/integration/volume/esqs/esq2a_1.png" style="width: 600px; height: 219px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2a_2.png" style="width: 600px; height: 244px;"></p> <p>b)</p> <p><img alt="" src="../../files/integration/volume/esqs/esq2b.png" style="width: 600px; height: 361px;"></p> <p>c)</p> <p><img alt="" src="../../files/integration/volume/esqs/esq2c_1.png" style="width: 600px; height: 374px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2c_2.png" style="width: 600px; height: 143px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2c_3.png" style="width: 600px; height: 341px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2c_4.png" style="width: 600px; height: 145px;"></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-expandable panel-default"> <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="937"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following graph shows the curve defined by the equation <span class="math-tex">\(\large (x-1)^2+y^2=4\)</span>.</p> <p>The region bounded by the curve and the lines y = 0 and x = 0 is rotated <span class="math-tex">\(\large 2\pi\)</span> around the x axis.</p> <p>Find the volume of this solid in terms of <span class="math-tex">\(\large \pi\)</span></p> <p><img alt="" src="../../files/integration/volume/esqs/esq1.png" style="width: 450px; height: 400px;"></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>The volume of the solid is found using the formula</p> <p><content><span class="math-tex">\(\large V= \int_{0}^{3} y^2 \,dx \\ \large V= \int_{0}^{3} (4-(x-1)^2 \,)dx \)</span></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><img alt="" src="../../files/integration/volume/esqs/esq1ans_1.png" style="width: 800px; height: 445px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-expandable panel-default"> <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="939"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The graph shows <span class="math-tex">\(\large f(x) = -\arctan(1-x^2)\)</span>, the tangent to the curve at (1 , 0) and the tangent to the curve at the point <span class="math-tex">\(\large(-\frac{\pi}{4},0)\)</span>.</p> <p>The shaded region is bounded by the curve and the two tangents. This region is rotated <span class="math-tex">\(\large 2\pi\)</span> around the <strong>y axis</strong> to forma solid.</p> <p>Find the volume of this solid correct to 3 significant figures.</p> <p><img alt="" src="../../files/integration/volume/esqs/esq3.png" style="width: 400px; height: 402px;"></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>You can use your calculator for this question.</p> <p>Ensure that your calculator is in radian mode.</p> <p><content>You can find the volume gene</content><content>rated by rotating the trapezium around the y axis and subtract the volume generated by rotating the curve around the y axis</content></p> <p><img alt="" src="../../files/integration/volume/esqs/esq4_hint.png" style="width: 600px; height: 246px;"></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><img alt="" src="../../files/integration/volume/esqs/esq3_a1.png" style="width: 600px; height: 184px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq3_a2.png" style="width: 600px; height: 371px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq3_a3.png" style="width: 600px; height: 386px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq3_a4.png" style="width: 600px; height: 90px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="940"> <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 following diagram shows the graph of <span class="math-tex">\(\large x^2=\cos^3y\)</span> for <span class="math-tex">\(\large -\frac{\pi}{2}\le y\le\frac{\pi}{2}\)</span></p> <p>The shaded region <strong>R </strong>is the area bounded by the curve, the y axis and the lines <span class="math-tex">\(\large y=-\frac{\pi}{2}\)</span> and <span class="math-tex">\(\large y=\frac{\pi}{2}\)</span>.</p> <p>The rgion is rotated about the <strong>y axis</strong> through <span class="math-tex">\(\large 2\pi\)</span> to form a solid.</p> <p>Show that the volume of the solid is <span class="math-tex">\(\large \frac{4\pi}{3}\)</span></p> <p><img alt="" src="../../files/integration/volume/esqs/esq4.png" style="width: 450px; height: 401px;"></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>To perfom the integration, write</p> <p style="margin-left: 80px;"><span class="math-tex">\(\large \cos^3y=\cos y\cdot\cos^2y=\cos y\ (1-\sin^2y)\)</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><img alt="" src="../../files/integration/volume/esqs/esq4a_1.png" style="width: 600px; height: 225px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq4a_2.png" style="width: 600px; height: 251px;"></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="941"> <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 following diagram shows the graph of the function <span class="math-tex">\(\large f(x)=\frac{\sqrt{x}}{\cos x}\)</span> for <span class="math-tex">\(\large 0\le x \le\frac{\pi}{2}\)</span></p> <p>The shaded region is the area bounded by <strong><em>f</em></strong>, the x axis, x= 0 and x = <span class="math-tex">\(\large \frac{\pi}{3}\)</span></p> <p>The region is rotated about the x axis through <span class="math-tex">\(\large 2\pi\)</span> to form a solid.</p> <p>Find the volume of the solid.</p> <p><img alt="" src="../../files/integration/volume/esqs/esq5.png" style="width: 450px; height: 475px;"></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 is the integration that makes this question difficult.</p> <p><content>You will need to use both integration by parts and integration by substitution/recognition.</content></p> <p>For the integration by substitution, let</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large u=x \qquad \frac{dv}{dx}=\sec^2x\)</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><img alt="" src="../../files/integration/volume/esqs/esq5a_1.png" style="width: 600px; height: 333px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq5a_2.png" style="width: 600px; height: 273px;"></p> </section> <h4> </h4> </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-default panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Integration with Partial Fractions</p> </div> </div> <div class="panel-body"> <div> <div class="panel 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="957"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let <span class="math-tex">\(\large f(x)=\frac{-x-7}{x^2-x-6} \quad ,x\neq-2,x\neq3\)</span></p> <p>a) Express f(x) in partial fractions</p> <p>b) <strong>Hence</strong>, find the exact value of <span class="math-tex">\(\large \int_{-1}^2f(x)dx\)</span> giving your answer in the form <span class="math-tex">\(\large lnq \quad,q\in\mathbb{Q}\)</span></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"><img alt="" src="../../files/integration/partial_fractions/esq1hint.png" style="width: 650px; height: 114px;"><content> </content></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><img alt="" src="../../files/integration/partial_fractions/esq1a.png" style="width: 650px; height: 364px;"></p> <p><img alt="" src="../../files/integration/partial_fractions/esq1b.png" style="width: 650px; height: 64px;"></p> <p><img alt="" src="../../files/integration/partial_fractions/esq1c.png" style="width: 650px; height: 364px;"></p> </section> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel 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="958"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The graph shows the function <span class="math-tex">\(\large f(x)=\frac{x+10}{4-x^2} \quad ,x\neq-2,x\neq2\)</span></p> <p>By writing <strong><em>f(x)</em></strong> as the sum of two partial fractions, show that the area bounded by the curve, the x axis and the lines x = -1 and x = 1 is equal to <span class="math-tex">\(\large 5ln3\)</span></p> <p><img alt="" src="../../files/integration/partial_fractions/esq2.png" style="width: 500px; height: 337px;"></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"><img alt="" src="../../files/integration/partial_fractions/esq2hint.png" style="width: 650px; height: 60px;"><content> </content></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><img alt="" src="../../files/integration/partial_fractions/esq2a.png" style="width: 650px; height: 340px;"></p> <p><img alt="" src="../../files/integration/partial_fractions/esq2b.png" style="width: 650px; height: 303px;"></p> </section> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel 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="959"> <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/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL 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>Consider the finction <span class="math-tex">\(\large f(x)=\frac{x^2-7x+12}{x-1} \quad ,x\neq1\)</span></p> <p>Find the coordinates where the graph of <strong><em>f</em></strong> crosses the</p> <p>a i) x axis</p> <p>ii) y axis</p> <p>b) Write down the equation of the vertical asymptote of <strong><em>f</em></strong></p> <p>c) Write down the equation of the oblique asymptote of <strong><em>f</em></strong></p> <p>d) Sketch the graph of <strong><em>f</em></strong> for <span class="math-tex">\(\large -15\le x \le15\)</span> clearly indicating the points of intersection with each axis and any asymptotes</p> <p>e) Express <span class="math-tex">\(\large \frac{1}{f(x)}\)</span> in partial fractions</p> <p>f) <strong>Hence</strong>, find the exact value of <span class="math-tex">\(\large \int_1^2\frac{1}{f(x)}dx\)</span> in the form <span class="math-tex">\(\large lnq \quad,q\in\mathbb{Q}\)</span></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"><content> </content></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 i) <span class="math-tex">\(\large f(x)=\frac{x^2-7x+12}{x-1} \quad ,x\neq1\)</span></p> <p>x intercepts when f(x) = 0</p> <p><span class="math-tex">\(\large \frac{x^2-7x+12}{x-1}=0\\ \large {x^2-7x+12}=0\\ \large (x-3)(x-4)=0\\ \large x = 3, \quad x = 4\)</span></p> <hr class="hidden-separator"> <p>a ii) y intercepts when x = 0</p> <p><span class="math-tex">\(\large f(0)=\frac{12}{-1} =-12\)</span></p> <hr class="hidden-separator"> <p>b) <span class="math-tex">\(\large f(x)=\frac{x^2-7x+12}{x-1} \quad ,x\neq1\)</span></p> <p>Vertical asymptote when <span class="math-tex">\(\large x - 1 = 0\\ \large x= 1\)</span></p> <hr class="hidden-separator"> <p>c) To find oblique asymptote, divide</p> <p><img alt="" src="../../files/integration/partial_fractions/esq3a.png" style="width: 650px; height: 176px;"></p> <p>Hence, <span class="math-tex">\(\large f(x)=x-6+\frac{18}{x-1}\)</span></p> <p>Hence, oblique asymptote at <em><strong>y = x - 6</strong></em></p> <hr class="hidden-separator"> <p>d)</p> <p><img alt="" src="../../files/integration/partial_fractions/esq3.png" style="width: 600px; height: 356px;"></p> <hr class="hidden-separator"> <p>e) <span class="math-tex">\(\large \frac{1}{f(x)}=\frac{x-1}{x^2-7x+12} \quad ,x\neq3, x\neq4\)</span></p> <p><img alt="" src="../../files/integration/partial_fractions/esq3b.png" style="width: 650px; height: 331px;"></p> <p><img alt="" src="../../files/integration/partial_fractions/esq3c.png" style="width: 650px; height: 71px;"></p> <hr class="hidden-separator"> <p>f)</p> <p><img alt="" src="../../files/integration/partial_fractions/esq3d.png" style="width: 650px; height: 297px;"></p> </section> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <p> </p> <p> </p> <div class="panel panel-default panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Integration by Substitution</p> </div> </div> <div class="panel-body"> <div> <div class="panel 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="294"> <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>a) Find <span class="math-tex">\(\int { \frac { { e }^{ x } }{ 1+{ e }^{ x } } dx } \)</span></p> <p>b) Evaluate <span class="math-tex">\(\int _{ \frac { { \pi }^{ 2 } }{ 4 } }^{ \pi ^{ 2 } }{ \frac { cos\sqrt { x } }{ \sqrt { x } } } dx\)</span></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) This question requires us to use integration by substitution (or recognition)</p> <p>u =1 + e<sup>x</sup></p> <hr class="hidden-separator"> <p>b) This question requires us to use integration by substitution</p> <p>u = <span class="math-tex">\(\sqrt{x}\)</span></p> <content> </content></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/integration/by-substitution/esq-sol-intbysub1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub1.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <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/231299430"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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="296"> <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"> The following diagram shows the graph of f(x) = <span class="math-tex">\(\frac { 4x }{ \sqrt { { x }^{ 2 }+1 } }\)</span></p> <p>Let R be the region bounded by <em>f</em>, the x-axis, x = 1 and x = 2</p> <p><img alt="" src="../../files/integration/by-substitution/esq2v2.png" style="width: 300px; height: 259px;"></p> <p>Find <em><strong>R</strong></em></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>This question requires you to how to find the area under the graph: Area = <span class="math-tex">\(\int _{ a }^{ b }{ y }\ dx\)</span></p> <p>The integral requires integration by substitution</p> <p>u = x<sup>2 </sup>+1</p> <content> </content></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/integration/by-substitution/esq-sol-intbysub2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub2.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <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/232208684"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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="304"> <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-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>a) Using the fact that <span class="math-tex">\(tanx = \frac{sinx}{cosx} \)</span>, show that <span class="math-tex">\(\frac{d}{dx}(tanx) = \frac{1}{cos^2x}\)</span></p> <p>b) <strong>Hence</strong>, find <span class="math-tex">\(\int { \frac { \sqrt { tanx } }{ cos^{ 2 }x } } dx\)</span></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) This requires you to use the Quotient Rule</p> <p>b) <strong>Hence </strong>means that you should use the result from part a)</p> <p>Use Integration by substitution u = tanx</p> <content> </content></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/integration/by-substitution/esq-sol-intbysub3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub3.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <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/232255854"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="298"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find <span class="math-tex">\(\int { \frac { arcsinx+x }{ \sqrt { 1-x^{ 2 } } } } dx\)</span></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>To complete this question, we need to split the integral into two parts</p> <p><span class="math-tex">\(\int { \frac { arcsinx+x }{ \sqrt { 1-x^{ 2 } } } } dx=\int { \frac { arcsinx }{ \sqrt { 1-x^{ 2 } } } } dx+\int { \frac { x }{ \sqrt { 1-x^{ 2 } } } } dx\)</span></p> <p>Both these integrals can be completed using Integration by substitution</p> <content> </content></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/integration/by-substitution/esq-sol-intbysub4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub4.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <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/232260099"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="300"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Find <span class="math-tex">\(\int { sin^{ 5 }x\ dx } \)</span></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>This might help</p> <p><span class="math-tex">\(sin^5x = sin^4x\cdot sinx = (sin^2x)^2\cdot sinx\)</span></p> <p>You will need to use Integration by substitution</p> <p>Can you think of a substitution that will work?</p> <content> </content></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/integration/by-substitution/esq-sol-intbysub5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub5.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <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/232366286"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="302"> <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"> Find <span class="math-tex">\(\int { \sqrt { 16-9x^{ 2 } } dx } \)</span> using the substitution 3x = 4 sin<span class="math-tex">\(\theta\)</span></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>This is a tricky question that requires very careful manipulation</p> <p>The substitution gives <span class="math-tex">\(\frac{dx}{d\theta}=\frac{4}{3}cos\theta\)</span></p> <p>The integral becomes <span class="math-tex">\(\int { \sqrt { 16-16sin^{ 2 }\theta } \cdot \frac { 4 }{ 3 } cos\theta \quad d\theta } \)</span></p> <content> </content></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/integration/by-substitution/esq-sol-intbysub6.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-substitution/esq-sol-intbysub6.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <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/232389159"></iframe></div> </section> </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-default panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Integration by Parts</p> </div> </div> <div class="panel-body"> <div> <div class="panel 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="286"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> <span class="math-tex">\(\int { { e }^{ \frac { x }{ 2 } }sin(x)\quad dx } \)</span></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"><content> </content> <p>You need to use the method of integration by parts</p> <p>You should apply the method twice and the integral to find should be a multiple of the question</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/integration/by-parts/esq-sol-integration_by_parts1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-parts/esq-sol-integration_by_parts1.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <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/230121633"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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="290"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> <span class="math-tex">\(\int { arctanx \ dx } \)</span></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>The solution requires you to use Integration by Parts</p> <p>Think of the question as <span class="math-tex">\(\int { 1\cdot arctanxdx } \)</span></p> <content> </content></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/integration/by-parts/esq-sol-integration_by_parts2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-parts/esq-sol-integration_by_parts2.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <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/230247202"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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="291"> <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"> <span class="math-tex">\(\int { 2x\cdot arctanx \ dx } \)</span></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>This question requires you to use Integration by Parts</p> <p>In the second part you will need to integrate a rational function. Since the degree of the numerator is equal to the degree of the denominator, you need to divide the polynomials</p> <content> </content></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/integration/by-parts/esq-sol-integration_by_parts3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/by-parts/esq-sol-integration_by_parts3.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Video Solution</h4> <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/230247895"></iframe></div> </section> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Differential Equations - Separable Variables</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="867"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Solve the differential equation given that y(0) = 2</p> <p><span class="math-tex">\({\large \frac{dy}{dx}=\frac{e^x}{y}, \quad y>0}\)</span></p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>This is a separable differential equation.</p> <p><content>Re-arrange the equation, integrating the right-hand side with respect to y and the left-hand side with respect to x</content></p> <p><img alt="" src="../../files/integration/differential-equations/esq-de_sep1hint.png" style="width: 150px; height: 48px;"></p> </section> <h4>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><img alt="" src="../../files/integration/differential-equations/esq_des_sep1.jpg" style="width: 377px; height: 413px;"></p> </section> <h4> </h4> </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="894"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Write <span class="math-tex">\({\huge \frac{1}{4-x^2}}\)</span>as the sum of two partial fractions</p> <p>b) Hence, given that y(0) = 0, find the particular solution of the differential equation in the form</p> <h4>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) This is a separable differential equation</p> <p><content><img alt="" src="../../files/integration/differential-equations/esq-de_sep2hint.png" style="width: 300px; height: 90px;"></content></p> <p>Use the result from part a) to re-write the second integrand</p> </section> <h4>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><img alt="" src="../../files/integration/differential-equations/esq_de_sep2a.png" style="width: 600px; height: 402px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_sep2b.png" style="width: 600px; height: 839px;"></p> </section> <h4> </h4> </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="899"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>a) Express <span class="math-tex">\(\frac{3}{x(x-3)}\)</span> as the sum of two partial fractions</p> <p>The population of a species of fish can be modelled by the differential equation <span class="math-tex">\(\large \frac{\text{d}N}{\text{d}t}=\frac{2}{3}N(N-3)cos2t\)</span></p> <p>where <em><strong>N</strong></em> = population in thousands, <em><strong>t</strong></em> = time in years</p> <p>b) Given that initially the population of fish is 4000, show that <span class="math-tex">\(\large N=\frac{12}{4-e^{sin{2t}}}\)</span></p> <p>c) How many days during the first year is the population of fish above 8000?</p> <h4>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) Work out <span class="math-tex">\(\large \frac{3}{x(x-3)}\equiv\frac{A}{x}+\frac{B}{x-3}\)</span><content></content></p> <p>b) This is a variables separable differential equation</p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_sep3hinta.png" style="width: 300px; height: 49px;"></p> <p>Solve for <em><strong>N = 4 , t = 0</strong></em></p> <p>c) Use your graphical calculator to find two values in the first year where <em><strong>N > 8</strong></em></p> </section> <h4>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><img alt="" src="../../files/integration/differential-equations/esq_de_sep3a.png" style="width: 550px; height: 465px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_sep3b.png" style="width: 600px; height: 621px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_sep3c.png" style="width: 550px; height: 442px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_sep3d.png" style="width: 580px; height: 534px;"></p> </section> <h4> </h4> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Differential Equations - Integrating Factor</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="897"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Consider the following first order differential equation</p> <p><span class="math-tex">\(\large x\frac{\text{d}y}{\text{d}x}+3y=\frac{lnx}{x}\)</span></p> <p>a) Show that <em><strong>x<sup>3</sup></strong></em> is the integrating factor for this differential equation</p> <p>b) Hence, find the general solution of this differential equation in the form <em><strong>y = f(x)</strong></em></p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>a) The integrating factor, <span class="math-tex">\(I=e^{\int \frac{3}{x}dx}\)</span></p> <p>b) We should recognise the product rule for differentiation <img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor1hint.png" style="width: 250px; height: 43px;"></p> <p>We should use integration by parts to integrate xlnx</p> </section> <h4>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><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor1.png" style="width: 600px; height: 259px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor1b.png" style="width: 928px; height: 647px;"></p> </section> <h4> </h4> </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="898"> <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>Consider the following differential equation <span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}+ytanx=secx\)</span></p> <p>a) Using a suitable integrating factor show that the differential equation can be written as <span class="math-tex">\(\large \frac{y}{cosx}=\int sec^²x {dx}\)</span></p> <p>b) Given that (0 , 2) lies on the curve, show that the particular solution of the differential equation is<strong><em> <span class="math-tex">\(\large y=sinx+2cosx\)</span></em></strong></p> <h4>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) This is a differential equation in the form <span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}+P(x)y=Q(x)\)</span></p> <p>We use the integrating factor <span class="math-tex">\(I=e^{\int tanxdx }\)</span></p> <p><content> </content>You will need to use integration by substitution or recognition to integrate <em><strong>tanx</strong></em></p> <p>We should recognise the product rule for differentiation <img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor2hint.png" style="width: 300px; height: 61px;"></p> </section> <h4>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><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor2a.png" style="width: 600px; height: 256px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor2b.png" style="width: 600px; height: 553px;"></p> </section> <h4> </h4> </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="902"> <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>Consider the following differential equation <span class="math-tex">\(\large \sin x\frac{\text{d}y}{\text{d}x}+y\cos x=2\sin^2 x\)</span></p> <p>a) Show that the integrating factor of this differential equation is <span class="math-tex">\(\large \sin x\)</span></p> <p>b) Solve the differential equation, giving your answer in the form <span class="math-tex">\(y=f(x)\)</span></p> <h4>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) This is a differential equation in the form <span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}+P(x)y=Q(x)\)</span></p> <p>We use the integrating factor <span class="math-tex">\(\large I=e^{\int \frac{\cos x}{\sin x}dx }\)</span></p> <p><content> </content>You will need to use integration by substitution or recognition for this integration</p> <p>b) We should recognise the product rule for differentiation <img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactorhint.png" style="width: 400px; height: 70px;"></p> </section> <h4>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><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor3a.png" style="width: 600px; height: 348px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor3b.png" style="width: 600px; height: 310px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor3c.png" style="width: 600px; height: 118px;"></p> </section> <h4> </h4> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Differential Equations - Homogeneous</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="895"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Consider the first order differential equation</p> <p style="text-align: center;"><span class="math-tex">\( \large xy\frac{\text{d}y}{\text{d}x}+4x^2+y^2=0 \quad,\quad y\geq0\)</span></p> <p>a) Use the substitution <strong><em>y = vx </em></strong>to show that <span class="math-tex">\( \large \frac{\text{d}v}{\text{d}x}=-\frac{4+2v^2}{vx}\)</span></p> <p>b) Show that the solution to the differential equation for which <em><strong>y(1) = 0</strong></em> is <span class="math-tex">\(\large y=\frac{\sqrt{2-2x^4}}{x}\)</span></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>Divide both sides of the equations by x² to get in the recognisable form of a homogeneous differential equation</p> <p><content><img alt="" src="../../files/integration/differential-equations/esq_de_homogeneous1hint.png" style="width: 300px; height: 54px;"></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><img alt="" src="../../files/integration/differential-equations/esq_de_homogeneous1.png" style="width: 600px; height: 347px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_homogeneous1b.png" style="width: 600px; height: 291px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_homogeneous1c.png" style="width: 600px; height: 686px;"></p> </section> <h4> </h4> </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="896"> <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>Consider the homogeneous differential equation <span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}=\frac{x^2+y^2}{2xy}\)</span></p> <p>a) Using the substitution <strong><em>y = vx </em></strong>, show that <span class="math-tex">\(\large \frac{\text{d}v}{\text{d}x}=\frac{1-v^2}{2vx}\)</span></p> <p>b) Solve the differential equation and show that <strong><em>y² = x² - cx</em></strong></p> <h4>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) Divide the numerator and denominator by x² to get the differential equation in the required format to be able to substitute <span class="math-tex">\(v=\frac{y}{x}\)</span><content></content></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_homogeneous2hinta.png" style="width: 200px; height: 72px;"></p> <p>b) This is a variables separable differential equation</p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_homogeneous2hintb.png" style="width: 250px; height: 70px;"></p> </section> <h4>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><img alt="" src="../../files/integration/differential-equations/esq_de_homogeneous2a.png" style="width: 600px; height: 557px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_homogeneous2b.png" style="width: 600px; height: 545px;"></p> </section> <h4> </h4> </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="908"> <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>Consider the differential equation <span class="math-tex">\(\large xy\frac{\text{d}y}{\text{d}x}=x^2\text{cosec}\frac{y}{x}+y^2\)</span></p> <p>a) Show that this equation can be written in the form <span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}=f(\frac{y}{x})\)</span></p> <p>b) Using the substitution <strong><em>y = vx </em></strong>, show that the particular solution to the equation, given that y(1)= 0 is</p> <h4>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) Divide the numerator and denominator by x² to get the differential equation in the required format to be able to substitute <span class="math-tex">\(v=\frac{y}{x}\)</span><content></content></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_homogeneous2hinta.png" style="width: 200px; height: 72px;"></p> <p>b) This is a variables separable differential equation</p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_homogeneous2hintb.png" style="width: 250px; height: 70px;"></p> </section> <h4>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><img alt="" src="../../files/integration/differential-equations/homogeneous/esq_de_homogeneous3a.png" style="width: 650px; height: 300px;"></p> <p><img alt="" src="../../files/integration/differential-equations/homogeneous/esq_de_homogeneous3b.png" style="width: 650px; height: 286px;"></p> <p><img alt="" src="../../files/integration/differential-equations/homogeneous/esq_de_homogeneous3c.png" style="width: 650px; height: 365px;"></p> <p><img alt="" src="../../files/integration/differential-equations/homogeneous/esq_de_homogeneous3d.png" style="width: 650px; height: 231px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2912"> <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>Calculus Examination Questions HL</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer 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