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</div><h2>HL Paper 3</h2><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\mathop {\lim }\limits_{x \to 0} \frac{{\tan x}}{{x + {x^2}}}\) ;</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\mathop {\lim }\limits_{x \to 1} \frac{{1 - {x^2} + 2{x^2}\ln x}}{{1 - \sin \frac{{\pi x}}{2}}}\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2{{\text{e}}^x} + y\tan x\) , given that <em>y</em> = 1 when <em>x</em> = 0 .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The domain of the function <em>y</em> is \(\left[ {0,\frac{\pi }{2}} \right[\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By finding the values of successive derivatives when <em>x</em> = 0 , find the Maclaurin series for <em>y</em> as far as the term in \({x^3}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Differentiate the function \({{\text{e}}^x}(\sin x + \cos x)\) and hence show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\int {{{\text{e}}^x}\cos x{\text{d}}x = \frac{1}{2}{{\text{e}}^x}(\sin x + \cos x) + c} .\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find an integrating factor for the differential equation and hence find the solution in the form \(y = f(x)\) .</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(\int_1^\infty {\frac{1}{{x(x + p)}}{\text{d}}x,{\text{ }}p \ne 0} \) is convergent if <em>p </em>> −1 and find its value in terms of <em>p</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence show that the following series is convergent.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{1}{{1 \times 0.5}} + \frac{1}{{2 \times 1.5}} + \frac{1}{{3 \times 2.5}} + ...\]</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine, for each of the following series, whether it is convergent or divergent.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(\sum\limits_{n = 1}^\infty {\sin \left( {\frac{1}{{n(n + 3)}}} \right)} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(\sqrt {\frac{1}{2}} + \sqrt {\frac{1}{6}} + \sqrt {\frac{1}{{12}}} + \sqrt {\frac{1}{{20}}} + …\)</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f </em>is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \ln \left( {\frac{1}{{1 - x}}} \right).\]</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Write down the value of the constant term in the Maclaurin series for \(f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the first three derivatives of \(f(x)\) and hence show that the Maclaurin series for \(f(x)\) up to and including the \({x^3}\) term is \(x + \frac{{{x^2}}}{2} + \frac{{{x^3}}}{3}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Use this series to find an approximate value for ln 2 .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Use the Lagrange form of the remainder to find an upper bound for the error in this approximation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) How good is this upper bound as an estimate for the actual error?</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined on the domain \(\left] { - \frac{\pi }{2},\frac{\pi }{2}} \right[{\text{ by }}f(x) = \ln (1 + \sin x)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(f''(x) = - \frac{1}{{(1 + \sin x)}}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the Maclaurin series for \(f(x)\) up to and including the term in \({x^4}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Explain briefly why your result shows that <em>f</em> is neither an even function nor an odd function.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the value of \(\mathop {\lim }\limits_{x \to 0} \frac{{\ln (1 + \sin x) - x}}{{{x^2}}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The exponential series is given by \({{\text{e}}^x} = \sum\limits_{n = 0}^\infty {\frac{{{x^n}}}{{n!}}} \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the set of values of <em>x</em> for which the series is convergent.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show, by comparison with an appropriate geometric series, that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{{\text{e}}^x} - 1 < \frac{{2x}}{{2 - x}},{\text{ for }}0 < x < 2{\text{.}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence show that \({\text{e}} < {\left( {\frac{{2n + 1}}{{2n - 1}}} \right)^n}\), for \(n \in {\mathbb{Z}^ + }\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down the first three terms of the Maclaurin series for \(1 - {{\text{e}}^{ - x}}\) and explain why you are able to state that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[1 - {{\text{e}}^{ - x}} > x - \frac{{{x^2}}}{2},{\text{ for }}0 < x < 2.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Deduce that \({\text{e}} > {\left( {\frac{{2{n^2}}}{{2{n^2} - 2n + 1}}} \right)^n}\), for \(n \in {\mathbb{Z}^ + }\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Letting <em>n</em> = 1000, use the results in parts (b) and (c) to calculate the value of e correct to as many decimal places as possible.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether or not the following series converge.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\sum\limits_{n = 0}^\infty {\left( {\sin \frac{{n\pi }}{2} - \sin \frac{{(n + 1)\pi }}{2}} \right)} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(\sum\limits_{n = 1}^\infty {\frac{{{{\text{e}}^n} - 1}}{{{\pi ^n}}}} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \(\sum\limits_{n = 2}^\infty {\frac{{\sqrt {n + 1} }}{{n(n - 1)}}} \)</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\mathop {\lim }\limits_{x \to 0} \left( {\frac{{1 - \cos {x^6}}}{{{x^{12}}}}} \right)\).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Given that \(y = \ln \cos x\) , show that the first two non-zero terms of the Maclaurin series for <em>y </em>are \( - \frac{{{x^2}}}{2} - \frac{{{x^4}}}{{12}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Use this series to find an approximation in terms of \(\pi {\text{ for }}\ln 2\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = {x^2} + {y^2}\) where <em>y</em> =1 when <em>x</em> = 0 .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use Euler’s method with step length 0.1 to find an approximate value of <em>y</em> when <em>x</em> = 0.4.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down, giving a reason, whether your approximate value for <em>y</em> is greater than or less than the actual value of <em>y</em> .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(f(x) = \sin (p\arcsin x),{\text{ }} - 1 < x < 1\) and \(p \in \mathbb{R}\).</p>
</div>
<div class="specification">
<p>The function \(f\) and its derivatives satisfy</p>
<p style="text-align: center;">\((1 - {x^2}){f^{(n + 2)}}(x) - (2n + 1)x{f^{(n + 1)}}(x) + ({p^2} - {n^2}){f^{(n)}}(x) = 0,{\text{ }}n \in \mathbb{N}\)</p>
<p>where \({f^{(n)}}(x)\) denotes the \(n\) th derivative of \(f(x)\) and \({f^{(0)}}(x)\) is \(f(x)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f’(0) = p\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({f^{(n + 2)}}(0) = ({n^2} - {p^2}){f^{(n)}}(0)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For \(p \in \mathbb{R}\backslash \{ \pm 1,{\text{ }} \pm 3\} \), show that the Maclaurin series for \(f(x)\), up to and including the \({x^5}\) term, is</p>
<p style="text-align: center;">\(px + \frac{{p(1 - {p^2})}}{{3!}}{x^3} + \frac{{p(9 - {p^2})(1 - {p^2})}}{{5!}}{x^5}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find \(\mathop {\lim }\limits_{x \to 0} \frac{{\sin (p\arcsin x)}}{x}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>If \(p\) is an odd integer, prove that the Maclaurin series for \(f(x)\) is a polynomial of degree \(p\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(n > {\text{ln}}\,n\) for \(n > 0\), use the comparison test to show that the series \(\sum\limits_{n = 0}^\infty {\frac{1}{{{\text{ln}}\left( {n + 2} \right)}}} \) is divergent.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the interval of convergence for \(\sum\limits_{n = 0}^\infty {\frac{{{{\left( {3x} \right)}^n}}}{{{\text{ln}}\left( {n + 2} \right)}}} \).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let the Maclaurin series for \(\tan x\) be</p>
<p>\[\tan x = {a_1}x + {a_3}{x^3} + {a_5}{x^5} + \ldots \]</p>
<p>where \({a_1}\), \({a_3}\) and \({a_5}\) are constants.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find series for \({\sec ^2}x\), in terms of \({a_1}\), \({a_3}\) and \({a_5}\), up to and including the \({x^4}\) term</p>
<p>by differentiating the above series for \(\tan x\);</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find series for \({\sec ^2}x\), in terms of \({a_1}\), \({a_3}\) and \({a_5}\), up to and including the \({x^4}\) term</p>
<p>by using the relationship \({\sec ^2}x = 1 + {\tan ^2}x\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, by comparing your two series, determine the values of \({a_1}\), \({a_3}\) and \({a_5}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{x}{y} - xy\) where \(y > 0\) and \(y = 2\) when \(x = 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Show that putting \(z = {y^2}\) </span>transforms the differential equation into \(\frac{{{\text{d}}z}}{{{\text{d}}x}} + 2xz = 2x\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By solving this differential equation in \(z\)<span class="s1">, obtain an expression for </span>\(y\) <span class="s1">in terms of </span>\(x\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the infinite spiral of right angle triangles as shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-01_om_17.00.25.png" alt="N16/5/MATHL/HP3/ENG/TZ0/SE/03b"></p>
<p class="p1">The \(n{\text{th}}\) triangle in the spiral has central angle \({\theta _n}\), hypotenuse of length \({a_n}\) and opposite side of length <span class="s1">1</span>, as shown in the diagram. The first right angle triangle is isosceles with the two equal sides being of length <span class="s1">1</span>.</p>
</div>
<div class="specification">
<p class="p1">Consider the series \(\sum\limits_{n = 1}^\infty {{\theta _n}} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Using l’Hôpital’s rule, find \(\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{\arcsin \left( {\frac{1}{{\sqrt {(x + 1)} }}} \right)}}{{\frac{1}{{\sqrt x }}}}} \right)\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Find \({a_1}\) and \({a_2}\) and hence write down an expression for \({a_n}\).</p>
<p class="p1">(ii) Show that \({\theta _n} = \arcsin \frac{1}{{\sqrt {(n + 1)} }}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Using a suitable test, determine whether this series converges or diverges.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">The sequence \(\{ {u_n}\} \) is defined by \({u_n} = \frac{{3n + 2}}{{2n - 1}}\), for \(n \in {\mathbb{Z}^ + }\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the sequence converges to a limit <em>L </em>, the value of which should be stated.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the least value of the integer <em>N </em>such that \(\left| {{u_n} - L} \right| < \varepsilon \)<span style="font: 12.5px Helvetica;"> </span>, for all <em>n </em>> <em>N </em>where</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(\varepsilon = 0.1\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(\varepsilon = 0.00001\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">For each of the sequences \(\left\{ {\frac{{{u_n}}}{n}} \right\},{\text{ }}\left\{ {\frac{1}{{2{u_n} - 2}}} \right\}\) and \(\left\{ {{{( - 1)}^n}{u_n}} \right\}\) , determine whether or not it converges.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that the series \(\sum\limits_{n = 1}^\infty {({u_n} - L)} \) diverges.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>The function \(f\) is defined by</p>
<p>\[f(x) = \left\{ {\begin{array}{*{20}{l}} {{x^2} - 2,}&{x < 1} \\ {ax + b,}&{x \geqslant 1} \end{array}} \right.\]</p>
<p>where \(a\) and \(b\) are real constants.</p>
<p>Given that both \(f\) and its derivative are continuous at \(x = 1\), find the value of \(a\) and the value of \(b\).</p>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">Let \(f(x)\) </span>be a function whose first and second derivatives both exist on the closed interval \([0,{\text{ }}h]\).</p>
<p class="p2">Let \(g(x) = f(h) - f(x) - (h - x)f'(x) - \frac{{{{(h - x)}^2}}}{{{h^2}}}\left( {f(h) - f(0) - hf'(0)} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the mean value theorem for a function that is continuous on the closed interval \([a,{\text{ }}b]\) and differentiable on the open interval \(]a,{\text{ }}b[\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find \(g(0)\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find \(g(h)\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Apply the mean value theorem to the function \(g(x)\) <span class="s1">on the closed interval \([0,{\text{ }}h]\)</span> to show that there exists \(c\) <span class="s1">in the open interval \(]0,{\text{ }}h[\) </span>such that \(g'(c) = 0\).</p>
<p class="p1">(iv) <span class="Apple-converted-space"> </span>Find \(g'(x)\).</p>
<p class="p2">(v) <span class="Apple-converted-space"> </span>Hence show that \( - (h - c)f''(c) + \frac{{2(h - c)}}{{{h^2}}}\left( {f(h) - f(0) - hf'(0)} \right) = 0\).</p>
<p class="p1">(vi) <span class="Apple-converted-space"> </span>Deduce that \(f(h) = f(0) + hf'(0) + \frac{{{h^2}}}{2}{\text{ }}f''(c)\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence show that, for \(h > 0\)</p>
<p class="p1">\(1 - \cos (h) \leqslant \frac{{{h^2}}}{2}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Given that \(f(x) = \ln x\)</span>, use the mean value theorem to show that, for \(0 < a < b\), \(\frac{{b - a}}{b} < \ln \frac{b}{a} < \frac{{b - a}}{a}\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence show that \(\ln (1.2)\) lies between \(\frac{1}{m}\) and \(\frac{1}{n}\), where \(m\)<span class="s1">, \(n\) </span>are consecutive positive integers to be determined.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the radius of convergence of the series \(\sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}{x^n}}}{{(n + 1){3^n}}}} \).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether the series \(\sum\limits_{n = 0}^\infty {\left( {\sqrt[3]{{{n^3} + 1}} - n} \right)} \) is convergent or divergent.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solve the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{y}{x} + \frac{{{y^2}}}{{{x^2}}}\) (where x > 0 )</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">given that <em>y</em> = 2 when <em>x</em> = 1 . Give your answer in the form \(y = f(x)\) .</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the set of values of <em>k </em>for which the improper integral \(\int_2^\infty {\frac{{{\text{d}}x}}{{x{{(\ln x)}^k}}}} \) converges.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the series \(\sum\limits_{r = 2}^\infty {\frac{{{{( - 1)}^r}}}{{r\ln r}}} \) is convergent but not absolutely convergent.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether the series \(\sum\limits_{n = 1}^\infty {\frac{{{n^{10}}}}{{{{10}^n}}}} \) is convergent or divergent.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram shows part of the graph of \(y = \frac{1}{{{x^3}}}\) together with line segments parallel to the coordinate axes.</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Using the diagram, show that \(\frac{1}{{{4^3}}} + \frac{1}{{{5^3}}} + \frac{1}{{{6^3}}} + ... < \int_3^\infty {\frac{1}{{{x^3}}}{\text{d}}x < \frac{1}{{{3^3}}} + \frac{1}{{{4^3}}} + \frac{1}{{{5^3}}} + ...} \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>Hence </strong>find upper and lower bounds for \(\sum\limits_{n = 1}^\infty {\frac{1}{{{n^3}}}} \).</span></p>
</div>
<br><hr><br><div class="specification">
<p style="text-align: center;"><img src="images/Figure_1.png" alt></p>
<p style="text-align: center;">Figure 1</p>
</div>
<div class="specification">
<p style="text-align: center;"><img src="images/Figure_2.png" alt></p>
<p style="text-align: center;">Figure 2</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Figure 1 shows part of the graph of \(y = \frac{1}{x}\) together with line segments parallel to the coordinate axes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) By considering the areas of appropriate rectangles, show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{2a + 1}}{{a(a + 1)}} < \ln \left( {\frac{{a + 1}}{{a - 1}}} \right) < \frac{{2a - 1}}{{a(a - 1)}}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence find lower and upper bounds for \(\ln (1.2)\).</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">An improved upper bound can be found by considering Figure 2 which again shows part of the graph of \(y = \frac{1}{x}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) By considering the areas of appropriate regions, show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\ln \left( {\frac{a}{{a - 1}}} \right) < \frac{{2a - 1}}{{2a(a - 1)}}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence find an upper bound for \(\ln (1.2)\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f </em>is defined by \(f(x) = \left\{ \begin{array}{r}{e^{ - x^3}}( - {x^3} + 2{x^2} + x),x \le 1\\ax + b,x > 1\end{array} \right.\), where \(a\) and \(b\) are constants.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the exact values of \(a\) and \(b\) if \(f\) is continuous and differentiable at \(x = 1\).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Use Rolle’s theorem, applied to \(f\), to prove that \(2{x^4} - 4{x^3} - 5{x^2} + 4x + 1 = 0\) has a root in the interval \(\left] { - 1,1} \right[\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence prove that \(2{x^4} - 4{x^3} - 5{x^2} + 4x + 1 = 0\) has at least two roots in the interval \(\left] { - 1,1} \right[\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Using the Maclaurin series for \({(1 + x)^n}\), write down and simplify the Maclaurin series approximation for \({(1 - {x^2})^{ - \frac{1}{2}}}\) as far as the term in \({x^4}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Use your result to show that a series approximation for arccos <em>x</em> is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\arccos x \approx \frac{\pi }{2} - x - \frac{1}{6}{x^3} - \frac{3}{{40}}{x^5}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Evaluate \(\mathop {\lim }\limits_{x \to 0} \frac{{\frac{\pi }{2} - \arccos ({x^2}) - {x^2}}}{{{x^6}}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Use the series approximation for \(\arccos x\) to find an approximate value for</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\int_0^{0.2} {\arccos \left( {\sqrt x } \right){\text{d}}x} ,\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">giving your answer to 5 decimal places. Does your answer give the actual value of the integral to 5 decimal places?</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Using l’Hopital’s Rule, show that \(\mathop {\lim }\limits_{x \to \infty } x{{\text{e}}^{ - x}} = 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine \(\int_0^a {x{{\text{e}}^{ - x}}{\text{d}}x} \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Show that the integral \(\int_0^\infty {x{{\text{e}}^{ - x}}{\text{d}}x} \) is convergent and find its value.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By successive differentiation find the first four non-zero terms in the Maclaurin series for \(f(x) = (x + 1)\ln (1 + x) - x\).</p>
<div class="marks">[11]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Deduce that, for \(n \geqslant 2\), the coefficient of \({x^n}\) <span class="s1">in this series is \({( - 1)^n}\frac{1}{{n(n - 1)}}\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By applying the ratio test, find the radius of convergence for this Maclaurin series.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\mathop {\lim }\limits_{x \to \frac{1}{2}} \left( {\frac{{\left( {\frac{1}{4} - {x^2}} \right)}}{{\cot \pi x}}} \right)\).</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = 2x + \left| x \right|\) , \(x \in \mathbb{R}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that <em>f</em> is continuous but not differentiable at the point (0, 0) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the value of \(\int_{ - a}^a {f(x){\text{d}}x} \) where \(a > 0\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve \(y = \frac{1}{x},{\text{ }}x > 0\).</p>
</div>
<div class="specification">
<p>Let \({U_n} = \sum\limits_{r = 1}^n {\frac{1}{r} - \ln n} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By drawing a diagram and considering the area of a suitable region under the curve, show that for \(r > 0\),</p>
<p>\[\frac{1}{{r + 1}} < \ln \left( {\frac{{r + 1}}{r}} \right) < \frac{1}{r}.\]</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, given that \(n\) is a positive integer greater than one, show that</p>
<p>\(\sum\limits_{r = 1}^n {\frac{1}{r} > \ln (1 + n)} \);</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, given that \(n\) is a positive integer greater than one, show that</p>
<p>\(\sum\limits_{r = 1}^n {\frac{1}{r} < 1 + \ln n} \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, given that \(n\) is a positive integer greater than one, show that</p>
<p>\({U_n} > 0\);</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, given that \(n\) is a positive integer greater than one, show that</p>
<p>\({U_{n + 1}} < {U_n}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why these two results prove that \(\{ {U_n}\} \) is a convergent sequence.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(\mathop {\lim }\limits_{x \to 1} \left( {\frac{{\ln x}}{{\sin 2\pi x}}} \right)\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">By using the series expansions for \({{\text{e}}^{{x^2}}}\) and cos <em>x</em> evaluate \(\mathop {\lim }\limits_{x \to 0} \left( {\frac{{1 - {{\text{e}}^{{x^2}}}}}{{1 - \cos x}}} \right).\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A curve that passes through the point (1, 2) is defined by the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2x(1 + {x^2} - y){\text{ }}.\]</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Use Euler’s method to get an approximate value of <em>y </em>when <em>x </em>= 1.3 , taking steps of 0.1. Show intermediate steps to four decimal places in a table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) How can a more accurate answer be obtained using Euler’s method?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Solve the differential equation giving your answer in the form </span><em style="font-family: 'times new roman', times; font-size: medium;">y </em><span style="font-family: 'times new roman', times; font-size: medium;">= </span><em style="font-family: 'times new roman', times; font-size: medium;">f</em><span style="font-family: 'times new roman', times; font-size: medium;">(</span><em style="font-family: 'times new roman', times; font-size: medium;">x</em><span style="font-family: 'times new roman', times; font-size: medium;">) .</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The variables <em>x</em> and <em>y</em> are related by \(\frac{{{\text{d}}y}}{{{\text{d}}x}} - y\tan x = \cos x\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the Maclaurin series for <em>y</em> up to and including the term in \({x^2}\) given that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = - \frac{\pi }{2}\) when <em>x</em> = 0 .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Solve the differential equation given that <em>y</em> = 0 when \(x = \pi \) . Give the solution in the form \(y = f(x)\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation \(x\frac{{{\text{d}}y}}{{{\text{d}}x}} - y = {x^p} + 1\) where \(x \in \mathbb{R},\,x \ne 0\) and \(p\) is a positive integer, \(p > 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation given that \(y = - 1\) when \(x = 1\). Give your answer in the form \(y = f\left( x \right)\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the \(x\)-coordinate(s) of the points on the curve \(y = f\left( x \right)\) where \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\) satisfy the equation \({x^{p - 1}} = \frac{1}{p}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce the set of values for \(p\) such that there are two points on the curve \(y = f\left( x \right)\) where \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\). Give a reason for your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the limit comparison test to prove that \(\sum\limits_{n = 1}^\infty {\frac{1}{{n(n + 1)}}} \) converges.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the Maclaurin series for \(\ln (1 + x)\) , show that the Maclaurin series for \(\left( {1 + x} \right)\ln \left( {1 + x} \right)\) is \(x + \sum\limits_{n = 1}^\infty {\frac{{{{( - 1)}^{n + 1}}{x^{n + 1}}}}{{n(n + 1)}}} \)</span><span style="font-family: 'times new roman', times; font-size: medium;">.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In this question you may assume that \(\arctan x\) is continuous and differentiable for \(x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the infinite geometric series</p>
<p>\[1 - {x^2} + {x^4} - {x^6} + \ldots \;\;\;\left| x \right| < 1.\]</p>
<p>Show that the sum of the series is \(\frac{1}{{1 + {x^2}}}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that an expansion of \(\arctan x\) is \(\arctan x = x - \frac{{{x^3}}}{3} + \frac{{{x^5}}}{5} - \frac{{{x^7}}}{7} + \ldots \)</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">\(f\) is a continuous function defined on \([a,{\text{ }}b]\) and differentiable on \(]a,{\text{ }}b[\) with \(f'(x) > 0\) on \(]a,{\text{ }}b[\).</p>
<p class="p1">Use the mean value theorem to prove that for any \(x,{\text{ }}y \in [a,{\text{ }}b]\), if \(y > x\) then \(f(y) > f(x)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Given \(g(x) = x - \arctan x\), prove that \(g'(x) > 0\), for \(x > 0\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Use the result from part (c) to prove that \(\arctan x < x\), for \(x > 0\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the result from part (c) to prove that \(\arctan x > x - \frac{{{x^3}}}{3}\), for \(x > 0\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that \(\frac{{16}}{{3\sqrt 3 }} < \pi < \frac{6}{{\sqrt 3 }}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} + \left( {\frac{{2x}}{{1 + {x^2}}}} \right)y = {x^2}\), <span class="s1">given that \(y = 2\) when \(x = 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(1 + {x^2}\)<span class="s2"> </span>is an integrating factor for this differential equation.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence solve this differential equation. Give the answer in the form \(y = f(x)\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the function \(f(x) = \frac{1}{{1 + {x^2}}},{\text{ }}x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Illustrate graphically the inequality, \(\frac{1}{5}\sum\limits_{r = 1}^5 {f\left( {\frac{r}{5}} \right) < \int_0^1 {f(x){\text{d}}x < \frac{1}{5}\sum\limits_{r = 0}^4 {f\left( {\frac{r}{5}} \right)} } } \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the inequality in part (a) to find a lower and upper bound for \(\pi \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\sum\limits_{r = 0}^{n - 1} {{{( - 1)}^r}{x^{2r}} = \frac{{1 + {{( - 1)}^{n - 1}}{x^{2n}}}}{{1 + {x^2}}}} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that \(\pi = 4\left( {\sum\limits_{r = 0}^{n - 1} {\frac{{{{( - 1)}^r}}}{{2r + 1}} - {{( - 1)}^{n - 1}}\int_0^1 {\frac{{{x^{2n}}}}{{1 + {x^2}}}{\text{d}}x} } } \right)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use L’Hôpital’s Rule to find \(\mathop {\lim }\limits_{x \to 0} \frac{{{{\text{e}}^x} - 1 - x\cos x}}{{{{\sin }^2}x}}\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A function \(f\) is given by \(f(x) = \int_0^x {\ln (2 + \sin t){\text{d}}t} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down \(f'(x)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">By differentiating \(f({x^2})\)</span>, obtain an expression for the derivative of \(\int_0^{{x^2}} {\ln (2 + \sin t){\text{d}}t} \) with respect to \(x\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence obtain an expression for the derivative of \(\int_x^{{x^2}} {\ln (2 + \sin t){\text{d}}t} \) <span class="s1">with respect to \(x\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The function \(f\) is defined by \(f(x) = {{\text{e}}^x}\sin x,{\text{ }}x \in \mathbb{R}\).</p>
</div>
<div class="specification">
<p class="p1">The Maclaurin series is to be used to find an approximate value for \(f(0.5)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">By finding a suitable number of derivatives of \(f\), </span>determine the Maclaurin series for \(f(x)\) as far as the term in \({x^3}\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence, or otherwise, determine the exact value of \(\mathop {\lim }\limits_{x \to 0} \frac{{{{\text{e}}^x}\sin x - x - {x^2}}}{{{x^3}}}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Use the Lagrange form of the error term to find an upper bound for the absolute value of the error in this approximation.</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Deduce from the Lagrange error term whether the approximation will be greater than or less than the actual value of \(f(0.5)\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the infinite series \(S = \sum\limits_{n = 0}^\infty {{u_n}} \) where \({u_n} = \int_{nx}^{(n + 1)\pi } {\frac{{\sin t}}{t}{\text{d}}t} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why the series is alternating.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Use the substitution \(T = t - \pi \) in the expression for \({u_{n + 1}}\) to show that \(\left| {{u_{n + 1}}} \right| < \left| {{u_n}} \right|\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Show that the series is convergent.</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(S < 1.65\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">The curves \(y = f(x)\) and \(y = g(x)\) </span>both pass through the point \((1,{\text{ }}0)\) and are defined by the differential equations \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x - {y^2}\) and \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = y - {x^2}\) <span class="s1">respectively.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the tangent to the curve \(y = f(x)\) <span class="s1">at the point \((1,{\text{ }}0)\) </span>is normal to the curve \(y = g(x)\) <span class="s1">at the point \((1,{\text{ }}0)\)</span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(g(x)\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use Euler’s method with steps of \(0.2\) <span class="s1">to estimate \(f(2)\) </span>to \(5\) <span class="s1">decimal places.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why \(y = f(x)\) cannot cross the isocline \(x - {y^2} = 0\), for \(x > 1\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Sketch the isoclines \(x - {y^2} = - 2,{\text{ }}0,{\text{ }}1\).</p>
<p>(ii) On the same set of axes, sketch the graph of \(f\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Use the integral test to determine whether the infinite series \(\sum\limits_{n = 2}^\infty {\frac{1}{{n\sqrt {\ln n} }}} \) is convergent or divergent.</p>
</div>
<br><hr><br><div class="specification">
<p>Let \(S = \sum\limits_{n = 1}^\infty {\frac{{{{(x - 3)}^n}}}{{{n^2} + 2}}} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the limit comparison test to show that the series \(\sum\limits_{n = 1}^\infty {\frac{1}{{{n^2} + 2}}} \) is convergent.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the interval of convergence for \(S\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The mean value theorem states that if \(f\) is a continuous function on \([a,{\text{ }}b]\) and differentiable on \(]a,{\text{ }}b[\) then \(f’(c) = \frac{{f(b) - f(a)}}{{b - a}}\) for some \(c \in ]a,{\text{ }}b[\).</p>
<p>The function \(g\), defined by \(g(x) = x\cos \left( {\sqrt x } \right)\), satisfies the conditions of the mean value theorem on the interval \([0,{\text{ }}5\pi ]\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For \(a = 0\) and \(b = 5\pi \), use the mean value theorem to find all possible values of \(c\) for the function \(g\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = g(x)\) on the interval \([0,{\text{ }}5\pi ]\) and hence illustrate the mean value theorem for the function \(g\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the infinite series \(\sum\limits_{n = 1}^\infty {\frac{{(n - 1){x^n}}}{{{n^2} \times {2^n}}}} \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the radius of convergence.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the interval of convergence.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{y^2}}}{{1 + x}}\), where <em>x </em>> −1 and <em>y </em>= 1 when <em>x </em>= 0 .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use Euler’s method, with a step length of 0.1, to find an approximate value of <em>y </em>when <em>x </em>= 0.5.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = \frac{{2{y^3} - {y^2}}}{{{{(1 + x)}^2}}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence find the Maclaurin series for <em>y</em>, up to and including the term in \({x^2}\)<span style="font: 7.0px Helvetica;"> </span>.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Solve the differential equation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the value of <em>a </em>for which \(y \to \infty \) as \(x \to a\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that the solution of the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{{\text{d}}y}}{{{\text{d}}x}} = \cos x{\cos ^2}y{\text{,}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">given that \(y = \frac{\pi }{4}{\text{ when }}x = \pi {\text{, is }}y = \arctan (1 + \sin x){\text{.}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine the value of the constant <em>a </em>for which the following limit exists</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{{\arctan (1 + \sin x) - a}}{{{{\left( {x - \frac{\pi }{2}} \right)}^2}}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and evaluate that limit<em>.</em></span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the radius of convergence of the infinite series</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{1}{2}x + \frac{{1 \times 3}}{{2 \times 5}}{x^2} + \frac{{1 \times 3 \times 5}}{{2 \times 5 \times 8}}{x^3} + \frac{{1 \times 3 \times 5 \times 7}}{{2 \times 5 \times 8 \times 11}}{x^4} + \ldots {\text{ .}}\]</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether the series \(\sum\limits_{n = 1}^\infty {\sin \left( {\frac{1}{n} + n\pi } \right)} \) is convergent or divergent.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the functions \(f\) and \(g\) given by \(f(x) = \frac{{{{\text{e}}^x} + {{\text{e}}^{ - x}}}}{2}{\text{ and }}g(x) = \frac{{{{\text{e}}^x} - {{\text{e}}^{ - x}}}}{2}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(f'(x) = g(x)\) and \(g'(x) = f(x)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the first three non-zero terms in the Maclaurin expansion of \(f(x)\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the value of \(\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{1 - f(x)}}{{{x^2}}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of the improper integral \(\int_0^\infty {\frac{{g(x)}}{{{{\left[ {f(x)} \right]}^2}}}{\text{d}}x} \).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A differential equation is given by \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{y}{x}\) , where <em>x </em>> 0 and <em>y </em>> 0.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Solve this differential equation by separating the variables, giving your answer in the form <em>y </em>= <em>f </em>(<em>x</em>) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Solve the same differential equation by using the standard homogeneous substitution <em>y </em>= <em>vx </em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Solve the same differential equation by the use of an integrating factor.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">If <em>y </em>= 20 when <em>x </em>= 2 , find <em>y </em>when <em>x </em>= 5 .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>The function \(f\) is defined by</p>
<p>\[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}<br> {\left| {x - 2} \right| + 1}&{x < 2} \\ <br> {a{x^2} + bx}&{x \geqslant 2} <br>\end{array}} \right.\]</p>
<p>where \(a\) and \(b\) are real constants</p>
<p>Given that both \(f\) and its derivative are continuous at \(x = 2\), find the value of \(a\) and the value of \(b\).</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The mean value theorem states that if \(f\) is a continuous function on \([a,{\text{ }}b]\) and differentiable on \(]a,{\text{ }}b[\) <span class="s1">then \(f'(c) = \frac{{f(b) - f(a)}}{{b - a}}\) </span>for some \(c \in ]a,{\text{ }}b[\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the two possible values of \(c\) for the function defined by \(f(x) = {x^3} + 3{x^2} - 2\) on the interval \([ - 3,{\text{ }}1]\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Illustrate this result graphically.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>The function \(f\) is continuous on \([a,{\text{ }}b]\), differentiable on \(]a,{\text{ }}b[\) and \(f'(x) = 0\) for all \(x \in ]a,{\text{ }}b[\). Show that \(f(x)\) is constant on \([a,{\text{ }}b]\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence, prove that for \(x \in [0,{\text{ }}1],{\text{ }}2\arccos x + \arccos (1 - 2{x^2}) = \pi \).</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Use l’Hôpital’s rule to determine the value of</p>
<p>\[\mathop {\lim }\limits_{x \to 0} \frac{{{{\sin }^2}x}}{{x\ln (1 + x)}}.\]</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solve the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[(x - 1)\frac{{{\text{d}}y}}{{{\text{d}}x}} + xy = (x - 1){{\text{e}}^{ - x}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">given that <em>y</em> = 1 when <em>x</em> = 0. Give your answer in the form \(y = f(x)\).</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove by induction that \(n! > {3^n}\), for \(n \ge 7,{\text{ }}n \in \mathbb{Z}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence use the comparison test to prove that the series \(\sum\limits_{r = 1}^\infty {\frac{{{2^r}}}{{r!}}} \) converges.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} + \frac{x}{{{x^2} + 1}}y = x\) where \(y = 1\) when \(x = 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\sqrt {{x^2} + 1} \) is an integrating factor for this differential equation.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation giving your answer in the form \(y = f(x)\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(n! \geqslant {2^{n - 1}}\), for \(n \geqslant 1\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence use the comparison test to determine whether the series \(\sum\limits_{n = 1}^\infty {\frac{1}{{n!}}} \) converges or diverges.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the series \(\sum\limits_{n = 2}^\infty {\frac{1}{{{n^2}\ln n}}} \) converges.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Show that \(\ln (n) + \ln \left( {1 + \frac{1}{n}} \right) = \ln (n + 1)\).</p>
<p>(ii) Using this result, show that an application of the ratio test fails to determine whether or not \(\sum\limits_{n = 2}^\infty {\frac{1}{{n\ln n}}} \) converges.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) State why the integral test can be used to determine the convergence or divergence of \(\sum\limits_{n = 2}^\infty {\frac{1}{{n\ln n}}} \).</p>
<p>(ii) Hence determine the convergence or divergence of \(\sum\limits_{n = 2}^\infty {\frac{1}{{n\ln n}}} \).</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The Taylor series of \(\sqrt x \) about <em>x</em> = 1 is given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{a_0} + {a_1}(x - 1) + {a_2}{(x - 1)^2} + {a_3}{(x - 1)^3} + \ldots \]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the values of \({a_0},{\text{ }}{a_1},{\text{ }}{a_2}\) and \({a_3}\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, or otherwise, find the value of \(\mathop {\lim }\limits_{x \to 1} \frac{{\sqrt x - 1}}{{x - 1}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use an integrating factor to show that the general solution for \(\frac{{{\text{d}}x}}{{{\text{d}}t}} - \frac{x}{t} = - \frac{2}{t},{\text{ }}t > 0\) is \(x = 2 + ct\), where <em>\(c\) </em>is a constant.</p>
<p class="p1">The weight in kilograms of a dog, <em>\(t\) </em>weeks after being bought from a pet shop, can be modelled by the following function:</p>
<p class="p1">\[w(t) = \left\{ {\begin{array}{*{20}{c}} {2 + ct}&{0 \le t \le 5} \\ {16 - \frac{{35}}{t}}&{t > 5} \end{array}.} \right.\]</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(w(t)\) is continuous, find the value of \(c\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>the weight of the dog when bought from the pet shop;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>an upper bound for the weight of the dog.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove from first principles that \(w(t)\) is differentiable at \(t = 5\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the power series \(\sum\limits_{k = 1}^\infty {k{{\left( {\frac{x}{2}} \right)}^k}} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the radius of convergence.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the interval of convergence.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the infinite series \(\sum\limits_{k = 1}^\infty {{{( - 1)}^{k + 1}} \times \frac{k}{{2{k^2} + 1}}} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that the series is convergent.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that the sum to infinity of the series is less than 0.25.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that the solution of the homogeneous differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{y}{x} + 1,{\text{ }}x > 0,\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">given that \(y = 0{\text{ when }}x = {\text{e, is }}y = x(\ln x - 1)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Determine the first three derivatives of the function \(f(x) = x(\ln x - 1)\).</span></p>
<p style="margin: 0px 0px 0px 30px; font: 21px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Hence find the first three non-zero terms of the Taylor series for <em>f</em>(<em>x</em>) about <em>x </em>= 1.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="text-align: left;">The function \(f\) is defined by \(f(x){\text{ }}={\text{ }}{(\arcsin{\text{ }}x)^2},{\text{ }} - 1 \leqslant x \leqslant 1\).</p>
<p> </p>
</div>
<div class="specification">
<p>The function \(f\) satisfies the equation \(\left( {1 - {x^2}} \right)f''\left( x \right) - xf'\left( x \right) - 2 = 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f'\left( 0 \right) = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating the above equation twice, show that</p>
<p>\[\left( {1 - {x^2}} \right){f^{\left( 4 \right)}}\left( x \right) - 5x{f^{\left( 3 \right)}}\left( x \right) - 4f''\left( x \right) = 0\]</p>
<p>where \({f^{\left( 3 \right)}}\left( x \right)\) and \({f^{\left( 4 \right)}}\left( x \right)\) denote the 3rd and 4th derivative of \(f\left( x \right)\) respectively.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the Maclaurin series for \(f\left( x \right)\) up to and including the term in \({x^4}\) is \({x^2} + \frac{1}{3}{x^4}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use this series approximation for \(f\left( x \right)\) with \(x = \frac{1}{2}\) to find an approximate value for \({\pi ^2}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A function \(f\) is defined in the interval \(\left] { - k,{\text{ }}k} \right[\), where \(k > 0\). The gradient function \({f'}\) exists at each point of the domain of \(f\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following diagram shows the graph of \(y = f(x)\), its asymptotes and its vertical symmetry axis.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-17_om_14.31.15.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Sketch the graph of \(y = f'(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(p(x) = a + bx + c{x^2} + d{x^3} + \ldots \) be the Maclaurin expansion of \(f(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Justify that \(a > 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Write down a condition for the largest set of possible values for each of the parameters \(b\), \(c\) and \(d\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) State, with a reason, an upper bound for the radius of convergence.</span></p>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[x\frac{{{\text{d}}y}}{{{\text{d}}x}} - 2y = \frac{{{x^3}}}{{{x^2} + 1}}.\]</span></p>
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<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find an integrating factor for this differential equation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Solve the differential equation given that \(y = 1\) when \(x = 1\) , giving your answer in the forms \(y = f(x)\) .</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The real and imaginary parts of a complex number \(x + {\text{i}}y\) are related by the differential equation \((x + y)\frac{{{\text{d}}y}}{{{\text{d}}x}} + (x - y) = 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By solving the differential equation, given that \(y = \sqrt 3 \) when <em>x</em> =1, show that the relationship between the modulus <em>r</em> and the argument \(\theta \) of the complex number is \(r = 2{{\text{e}}^{\frac{\pi }{3} - \theta }}\).</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the differential equation</p>
<p>\[\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {\frac{y}{x}} \right),{\text{ }}x > 0.\]</p>
<p>Use the substitution \(y = vx\) to show that the general solution of this differential equation is</p>
<p>\[\int {\frac{{{\text{d}}v}}{{f(v) - v}} = \ln x + } {\text{ Constant.}}\]</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, solve the differential equation</p>
<p>\[\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{x^2} + 3xy + {y^2}}}{{{x^2}}},{\text{ }}x > 0,\]</p>
<p>given that \(y = 1\) when \(x = 1\). Give your answer in the form \(y = g(x)\).</p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solve the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({x^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} = {y^2} + 3xy + 2{x^2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">given that <em>y</em> = −1 when <em>x</em> =1. Give your answer in the form \(y = f(x)\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {{\text{e}}^x}\sin x\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(f''(x) = 2\left( {f'(x) - f(x)} \right)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By further differentiation of the result in part (a) , find the Maclaurin expansion of \(f(x)\), as far as the term in \({x^5}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the exact value of \(\int_0^\infty {\frac{{{\text{d}}x}}{{(x + 2)(2x + 1)}}} \).</span></p>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} + y\tan x = {\cos ^2}x\), given that <em>y</em> = 2 when <em>x</em> = 0.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use Euler’s method with a step length of 0.1 to find an approximation to the value of <em>y</em> when <em>x</em> = 0.3.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 35.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that the integrating factor for solving the differential equation is \(\sec x\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 35.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence solve the differential equation, giving your answer in the form \(y = f(x)\).</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(g(x) = \sin {x^2}\), where \(x \in \mathbb{R}\).</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Using the result \(\mathop {{\text{lim}}}\limits_{t \to 0} \frac{{\sin t}}{t} = 1\), or otherwise, calculate \(\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{g(2x) - g(3x)}}{{4{x^2}}}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the Maclaurin series of \(\sin x\) to show that \(g(x) = \sum\limits_{n = 0}^\infty {{{( - 1)}^n}\frac{{{x^{4n + 2}}}}{{(2n + 1)!}}} \)</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine the minimum number of terms of the expansion of \(g(x)\) required to approximate the value of \(\int_0^1 {g(x){\text{d}}x} \) to four decimal places.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Using the Maclaurin series for the function \({{\text{e}}^x}\), write down the first four terms of the Maclaurin series for \({{\text{e}}^{ - \frac{{{x^2}}}{2}}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence find the first four terms of the series for \(\int_0^x {{{\text{e}}^{ - \frac{{{u^2}}}{2}}}} {\text{d}}u\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Use the result from part (b) to find an approximate value for \(\frac{1}{{\sqrt {2\pi } }}\int_0^1 {{{\text{e}}^{ - \frac{{{x^2}}}{2}}}{\text{d}}x} \).</span></p>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the infinite series \(\sum\limits_{n = 1}^\infty {\frac{{{n^2}}}{{{2^n}}}{x^n}} \).</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the radius of convergence.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the interval of convergence.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that <em>x</em> = – 0.1, find the sum of the series correct to three significant figures.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{y}{{x + \sqrt {xy} }}\), for \(x,{\text{ }}y > 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Use Euler’s method starting at the point \((x,{\text{ }}y) = (1,{\text{ }}2)\), with interval \(h = 0.2\), to find an approximate value of <em>y </em>when \(x = 1.6\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Use the substitution \(y = vx\) to show that \(x\frac{{{\text{d}}v}}{{{\text{d}}x}} = \frac{v}{{1 + \sqrt v }} - v\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Hence find the solution of the differential equation in the form \(f(x,{\text{ }}y) = 0\), given that \(y = 2\) when \(x = 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the value of \(y\) when \(x = 1.6\).</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(y = \frac{1}{x}\int {f(x){\text{d}}x} \) is a solution of the differential equation</p>
<p class="p1">\(x\frac{{{\text{d}}y}}{{{\text{d}}x}} + y = f(x),{\text{ }}x > 0\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence solve \(x\frac{{{\text{d}}y}}{{{\text{d}}x}} + y = {x^{ - \frac{1}{2}}},{\text{ }}x > 0\), given that \(y = 2\) when \(x = 4\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[x\frac{{{\text{d}}y}}{{{\text{d}}x}} = y + \sqrt {{x^2} - {y^2}} ,{\text{ }}x > 0,{\text{ }}{x^2} > {y^2}.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that this is a homogeneous differential equation.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the general solution, giving your answer in the form \(y = f(x)\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f(x) = \frac{{1 + ax}}{{1 + bx}}\) can be expanded as a power series in <em style="font-family: 'times new roman', times; font-size: medium;">x</em>, within its radius of convergence R, in the form \(f(x) \equiv 1 + \sum\limits_{n = 1}^\infty {{c_n}{x^n}} \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Show that \({c_n} = {( - b)^{n - 1}}(a - b)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) State the value of R.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine the values of <em>a</em> and <em>b</em> for which the expansion of <em>f</em>(<em>x</em>) agrees with that of \({{\text{e}}^x}\) up to and including the term in \({x^2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Hence find a rational approximation to \({{\text{e}}^{\frac{1}{3}}}\) .</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the first three terms of the Maclaurin series for \(\ln (1 + {{\text{e}}^x})\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, or otherwise, determine the value of \(\mathop {\lim }\limits_{x \to 0} \frac{{2\ln (1 + {{\text{e}}^x}) - x - \ln 4}}{{{x^2}}}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined by \(f(x) = {{\text{e}}^{({{\text{e}}^x} - 1)}}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Assuming the Maclaurin series for \({{\text{e}}^x}\) , show that the Maclaurin series for \(f(x)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">is \(1 + x + {x^2} + \frac{5}{6}{x^3} + \ldots {\text{ .}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence or otherwise find the value of \(\mathop {\lim }\limits_{x \to 0} \frac{{f(x) - 1}}{{f'(x) - 1}}\) .</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether the series \(\sum\limits_{n = 1}^\infty {\sin \frac{1}{n}} \) is convergent or divergent.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the series \(\sum\limits_{n = 2}^\infty {\frac{1}{{n{{(\ln n)}^2}}}} \) is convergent.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the general solution of the differential equation \(t\frac{{{\text{d}}y}}{{{\text{d}}t}} = \cos t - 2y\) , for <em>t </em>> 0 .</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the infinite series</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{1}{{2\ln 2}} - \frac{1}{{3\ln 3}} + \frac{1}{{4\ln 4}} - \frac{1}{{5\ln 5}} + \ldots {\text{ .}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that the series converges.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine if the series converges absolutely or conditionally.</span></p>
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<p>Find the value of \(\int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<p>Illustrate graphically the inequality \(\sum\limits_{n = 5}^\infty {\frac{1}{{{n^3}}}} < \int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} < \sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<p>Hence write down a lower bound for \(\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} \).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
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<p>Find an upper bound for \(\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the infinite series \(\sum\limits_{n = 1}^\infty {\frac{2}{{{n^2} + 3n}}} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Use a comparison test to show that the series converges.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The general term of a sequence \(\{ {a_n}\} \) is given by the formula \({a_n} = \frac{{{{\text{e}}^n} + {2^n}}}{{2{{\text{e}}^n}}},{\text{ }}n \in {\mathbb{Z}^ + }\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Determine whether the sequence \(\{ {a_n}\} \) is decreasing or increasing.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that the sequence \(\{ {a_n}\} \) is convergent and find the limit <em>L</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the smallest value of \(N \in {\mathbb{Z}^ + }\) such that \(\left| {{a_n} - L} \right| < 0.001\), for all \(n \geqslant N\).</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 37.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the series \(\sum\limits_{n = 1}^\infty {{{( - 1)}^n}\frac{{{x^n}}}{{n \times {2^n}}}} \).</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the radius of convergence of the series.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence deduce the interval of convergence.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solve the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{x^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} = {y^2} + xy + 4{x^2},\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">given that <em>y</em> = 2 when <em>x</em> =1. Give your answer in the form \(y = f(x)\).</span></p>
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<p class="p1">The function \(f\) is defined by \(f(x) = {{\text{e}}^{ - x}}\cos x + x - 1\).</p>
<p class="p1">By finding a suitable number of derivatives of \(f\), determine the first non-zero term in its Maclaurin series.</p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{y^2} + {x^2}}}{{2{x^2}}}\) for which <em>y</em> = −1 when <em>x</em> = 1.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Use Euler’s method with a step length of 0.25 to find an estimate for the value of <em>y</em> when <em>x</em> = 2 .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Solve the differential equation giving your answer in the form \(y = f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Find the value of <em>y</em> when <em>x</em> = 2 .</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(y = \ln \left( {\frac{{1 + {{\text{e}}^{ - x}}}}{2}} \right)\), show that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{{\text{e}}^{ - y}}}}{2} - 1\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, by repeated differentiation of the above differential equation, find the Maclaurin series for <em>y</em> as far as the term in \({x^3}\), showing that two of the terms are zero.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Each term of the power series \(\frac{1}{{1 \times 2}} + \frac{1}{{4 \times 5}}x + \frac{1}{{7 \times 8}}{x^2} + \frac{1}{{10 \times 11}}{x^3} + \ldots \) has the form \(\frac{1}{{b(n) \times c(n)}}{x^n}\), where \(b(n)\) and \(c(n)\) are linear functions of \(n\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the functions \(b(n)\) and \(c(n)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the radius of convergence.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the interval of convergence.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} - 2{y^2} = {{\text{e}}^x}\) and <em>y</em> = 1 when <em>x</em> = 0, use Euler’s method with a step length of 0.1 to find an approximation for the value of <em>y</em> when <em>x</em> = 0.4. Give all intermediate values with maximum possible accuracy.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the functions \(f(x) = {(\ln x)^2},{\text{ }}x > 1\) and \(g(x) = \ln \left( {f(x)} \right),{\text{ }}x > 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find \(f'(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find \(g'(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Hence, show that \(g(x)\) is increasing on \(\left] {1,{\text{ }}\infty } \right[\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\[(\ln x)\frac{{{\text{d}}y}}{{{\text{d}}x}} + \frac{2}{x}y = \frac{{2x - 1}}{{(\ln x)}},{\text{ }}x > 1.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the general solution of the differential equation in the form \(y = h(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that the particular solution passing through the point with coordinates \(\left( {{\text{e, }}{{\text{e}}^2}} \right)\) is given by \(y = \frac{{{x^2} - x + {\text{e}}}}{{{{(\ln x)}^2}}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Sketch the graph of your solution for \(x > 1\), clearly indicating any asymptotes and any maximum or minimum points.</span></p>
<div class="marks">[12]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the integral test, show that \(\sum\limits_{n = 1}^\infty {\frac{1}{{4{n^2} + 1}}} \) is convergent.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show, by means of a diagram, that \(\sum\limits_{n = 1}^\infty {\frac{1}{{4{n^2} + 1}}} < \frac{1}{{4 \times {1^2} + 1}} + \int_1^\infty {\frac{1}{{4{x^2} + 1}}{\text{d}}x} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence find an upper bound for \(\sum\limits_{n = 1}^\infty {\frac{1}{{4{n^2} + 1}}} \)</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that \(\mathop {\lim }\limits_{H \to \infty } \int_a^H {\frac{1}{{{x^2}}}{\text{d}}x} \) exists and find its value in terms of \(a{\text{ (where }}a \in {\mathbb{R}^ + })\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the integral test to prove that \(\sum\limits_{n = 1}^\infty {\frac{1}{{{n^2}}}} \) converges.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\sum\limits_{n = 1}^\infty {\frac{1}{{{n^2}}}} = L\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows the graph of \(y = \frac{1}{{{x^2}}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Times;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Shade suitable regions on a copy of the diagram above and show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sum\limits_{n = 1}^k {\frac{1}{{{n^2}}}} + \int_{k + 1}^\infty {\frac{1}{{{x^2}}}} {\text{d}}x < L\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Similarly shade suitable regions on another copy of the diagram above and</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">show that \(L < \sum\limits_{n = 1}^k {\frac{1}{{{n^2}}}} + \int_k^\infty {\frac{1}{{{x^2}}}} {\text{d}}x\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence show that \(\sum\limits_{n = 1}^k {\frac{1}{{{n^2}}}} + \frac{1}{{k + 1}} < L < \sum\limits_{n = 1}^k {\frac{1}{{{n^2}}}} + \frac{1}{k}\)</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">You are given that \(L = \frac{{{\pi ^2}}}{6}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By taking <em>k </em>= 4 , use the upper bound and lower bound for <em>L </em>to find an upper bound and lower bound for \(\pi \) . Give your bounds to three significant figures.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = f(x,{\text{ }}y)\) where \(f(x,{\text{ }}y) = y - 2x\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch, on one diagram, the four isoclines corresponding to \(f(x,{\text{ }}y) = k\) where <em>\(k\) </em>takes the values \(-1\), \(-0.5\), \(0\) and \(1\). Indicate clearly where each isocline crosses the <em>\(y\) </em>axis.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A curve, \(C\), passes through the point \((0,1)\) and satisfies the differential equation above.</p>
<p>Sketch \(C\) on your diagram.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A curve, \(C\), passes through the point \((0,1)\) and satisfies the differential equation above.</p>
<p>State a particular relationship between the isocline \(f(x,{\text{ }}y) = - 0.5\) and the curve \(C\), at their point of intersection.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A curve, \(C\), passes through the point \((0,1)\) and satisfies the differential equation above.</p>
<p class="p1">Use Euler’s method with a step interval of \(0.1\) to find an approximate value for <em>\(y\) </em>on \(C\), when \(x = 0{\text{.}}5\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \sqrt {x + y} ,{\text{ }}(x + y \geqslant 0)\) satisfying the initial conditions <em>y </em>= 1 when <em>x </em>= 1. Also let <em>y </em>= <em>c </em>when <em>x </em>= 2 .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use Euler’s method to find an approximation for the value of <em>c </em>, using a step length of <em>h </em>= 0.1 . Give your answer to four decimal places.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">You are told that if Euler’s method is used with <em>h </em>= 0.05 then \(c \simeq 2.7921\) , if it is used with <em>h </em>= 0.01 then \(c \simeq 2.8099\) and if it is used with <em>h </em>= 0.005 then \(c \simeq 2.8121\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Plot on graph paper, with <em>h </em>on the horizontal axis and the approximation for <em>c</em> on the vertical axis, the four points (one of which you have calculated and three of which have been given). Use a scale of 1 cm = 0.01 on both axes. Take the horizontal axis from 0 to 0.12 and the vertical axis from 2.76 to 2.82.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Draw, by eye, the straight line that best fits these four points, using a ruler.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Use your graph to give the best possible estimate for <em>c </em>, giving your answer to three decimal places.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br>