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Examination Questions SL</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>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> <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 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> <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="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> <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="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" 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