A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<ol class="i-list"><li><div class="list-item-paragraph">Using standard result given in the List of Formulae (MF26), find the Maclaurin’s series for&nbsp;$\ln (1+x+{{x}^{2}})$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Determine the set of values of $x$ for which the expansion in part <strong>(i)</strong> is valid.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence, or otherwise, obtain the series expansion of $\frac{1+2x}{1+x+{{x}^{2}}}$ up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[2]</span></div></li></ol>

Maclaurin Series Using Standard Series

Series from Products, Substitution, Integration and Differentiation

Convergence and Validity of Power Series

Substitution and Composition of Power Series

Deriving Series by Differentiation and Integration

Maclaurin Series and Power Series

<ol class="i-list"><li><div class="list-item-paragraph">Using standard result given in the List of Formulae (MF26), find the Maclaurin’s series for&nbsp;$\ln (1-2x+{{x}^{2}})$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Determine the set of values of $x$ for which the expansion in part <strong>(i)</strong> is valid.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence, or otherwise, obtain the series expansion of $\frac{x-1}{1-2x+{{x}^{2}}}$ up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[2]</span></div></li></ol>

<ol class="abc-list"><li data-list-item-id="item-1760748073985-6zkh4y147" data-list-item-path="item-1760748073985-6zkh4y147"><div class="list-item-paragraph">Find the series expansion for&nbsp;$\frac{\cos 2x}{{{\left( 1+\tan x \right)}^{2}}}$ up to and including the term in ${{x}^{2}}$, given that&nbsp;$x$ is a sufficiently small angle.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760748074735-fzd6yxb7p" data-list-item-path="item-1760748074735-fzd6yxb7p"><div class="list-item-paragraph">It is given that&nbsp;$\mathrm{f}(x)={{\left( 1-x \right)}^{\frac{1}{2}}}\ln \sqrt{1+x}$.</div><ol class="i-list"><li data-list-item-id="subitem-1760748088122-r4auxec5s" data-list-item-path="item-1760748074735-fzd6yxb7p/subitem-1760748088122-r4auxec5s"><div class="list-item-paragraph">Find the numerical value of $\mathrm{f}'(x)$ when&nbsp;$x=0.01$, leaving the answer in $5$ decimal places.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1760748095885-ni5ilr8hq" data-list-item-path="item-1760748074735-fzd6yxb7p/subitem-1760748095885-ni5ilr8hq"><div class="list-item-paragraph">Using the standard series from the List of Formulae (MF27), expand $\mathrm{f}\left( x \right)$ as far as the term in ${{x}^{3}}$. State the range of values of $x$ for which the expansion is valid.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="subitem-1760748103077-oufpzpbxf" data-list-item-path="item-1760748074735-fzd6yxb7p/subitem-1760748103077-oufpzpbxf"><div class="list-item-paragraph">Denote the answer in part <strong>(b)(ii)</strong> as&nbsp;$\mathrm{g}\left( x \right)$. Hence find $\mathrm{g }'\mathrm{ }\left( x \right)$ when&nbsp;$x=0.01$, leaving the answer in $5$ decimal places.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760748118528-hw1tpiuno" data-list-item-path="item-1760748074735-fzd6yxb7p/subitem-1760748118528-hw1tpiuno"><div class="list-item-paragraph">Comparing your answer in part <strong>(b)(i)</strong> against <strong>(b)(iii)</strong>, comment on the validity of this answer.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

Small Angle Approximations

Approximations Using Power Series

Accuracy of Power Series Approximations

Binomial Expansion as a Maclaurin Series

Products and Quotients of Power Series

2024 MI P2 Q5

<p>It is given that&nbsp;$\mathrm{f}\left( x \right)={{\mathrm{e}}^{{{\tan }^{-1}}x}}$, where&nbsp;${{\tan }^{-1}}x$ denotes the principal values.</p>
<ol class="i-list"><li data-list-item-id="item-1760747941731-9vyv1phos" data-list-item-path="item-1760747941731-9vyv1phos"><div class="list-item-paragraph">Show that&nbsp;$\left( 1+{{x}^{2}} \right)\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)=\mathrm{f}\left( x \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760747942428-8ls2y4o9j" data-list-item-path="item-1760747942428-8ls2y4o9j"><div class="list-item-paragraph">By further differentiation of the above result, find the Maclaurin series for&nbsp;$\mathrm{f}(x)$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760747943185-0r00ysebb" data-list-item-path="item-1760747943185-0r00ysebb"><div class="list-item-paragraph">Using the series in part <strong>(ii)</strong> find an approximate value of $\displaystyle\displaystyle \displaystyle\int_{0}^{0.5}{\mathrm{f}\left( x \right)\,}\mathrm{d}x$, giving your answer to $4$ significant figures.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760747944566-nz3f4hhzo" data-list-item-path="item-1760747944566-nz3f4hhzo"><div class="list-item-paragraph">Comment on the suitability of substituting $x=1$ into the series in part <strong>(ii)</strong> to estimate the value of ${{\mathrm{e}}^{\frac{\pi }{4}}}$.<span class="inline-marks">[1]</span></div></li></ol>

It is given that $\mathrm{f}\left( x \right)={{\mathrm{e}}^{{{\tan }^{-1}}x}}$, where ${{\tan }^{-1}}x$ denotes the principal values.

Maclaurin Series from Differential Equations

Derivatives of Inverse Trigonometric Functions

Maclaurin Series by Successive Differentiation

Higher Derivatives by Implicit Differentiation

Derivatives of Exponential Functions

Chain Rule

2024 JPJC P2 Q3

Techniques of Differentiation

<p>For a right-angled triangle $ABC$ with $BC=a$ and $AC=b$, $\measuredangle CAB=x$ and $\measuredangle ABC=\frac{\pi }{2}$ as shown below, a sector with radius $a$ is extended from the vertex $C$. This sector meets the side $AC$ at the point $Y$.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760707177047-d4so0iwxo4a.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA3MDcxNzcwNDctZDRzbzBpd3hvNGEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzOTMwOCwiZXhwIjoxNzkzNTM5NjA4fQ.HCL-DVxpBAqUR5utTG2KNyzw_aFg9ebzqkJvCh_T2rI" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1760707176669-hpgk7x371" data-list-item-path="item-1760707176669-hpgk7x371"><div class="list-item-paragraph">Show that&nbsp;${{\left( \frac{AC}{AY} \right)}^{2}}=\frac{1}{{{\left( 1-\sin x \right)}^{2}}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760707177209-c5ixvvjxc" data-list-item-path="item-1760707177209-c5ixvvjxc"><div class="list-item-paragraph">Using appropriate expansions from the List of Formulae (MF26), find the Maclaurin series for&nbsp;$\mathrm{f}(x)={{\left( 1-\sin x \right)}^{-2}}$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760707178147-p1fikf4k6" data-list-item-path="item-1760707178147-p1fikf4k6"><div class="list-item-paragraph">Deduce the series expansion for&nbsp;$\frac{2\cos x}{{{\left( 1-\sin x \right)}^{3}}}$, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760707178831-sau53jtmy" data-list-item-path="item-1760707178831-sau53jtmy"><div class="list-item-paragraph">Using your answer in part <strong>(b)</strong> or otherwise, show that&nbsp;$\frac{AC}{AY}\approx 1+x+{{x}^{2}}$, when&nbsp;$x$ is sufficiently small.<span class="inline-marks">[3]</span></div></li></ol>

For a right-angled triangle $ABC$ with $BC=a$ and $AC=b$, $\measuredangle CAB=x$ and $\measuredangle ABC=\frac{\pi }{2}$ as shown below, a sector with radius $a$ is extended from the vertex $C$. This sector meets the side $AC$ at the point $Y$.

2024 EJC P1 Q7

<p>It is given that&nbsp;$y={{3}^{\left( {{x}^{2}}+2x \right)}}$</p>
<ol class="i-list"><li data-list-item-id="item-1760700301254-i57tkd6d0" data-list-item-path="item-1760700301254-i57tkd6d0"><div class="list-item-paragraph">Show that&nbsp;$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\frac{1}{y}{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}+2y\ln 3$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760700301801-0spy3h3cg" data-list-item-path="item-1760700301801-0spy3h3cg"><div class="list-item-paragraph">Find Maclaurin’s series for&nbsp;$y$, up to and including the term in ${{x}^{2}}$, leaving the coefficients in exact form.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760700303129-d83sph6t1" data-list-item-path="item-1760700303129-d83sph6t1"><div class="list-item-paragraph">Hence estimate the value of $\displaystyle\displaystyle \displaystyle\int_{0}^{1}{{{3}^{\left( {{x}^{2}}+2x \right)}}\,}\mathrm{d}x$, giving your answer in exact form. Explain how the accuracy of the estimation can be improved.<span class="inline-marks">[3]</span></div></li></ol>

It is given that $y={{3}^{\left( {{x}^{2}}+2x \right)}}$

Second and Higher Derivatives

2013 MI P1 Q2

<p>It is given that&nbsp;$y=\frac{1}{3+\sin 2x}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760690577236-g5ojqpjt3" data-list-item-path="item-1760690577236-g5ojqpjt3"><div class="list-item-paragraph">Show that&nbsp;$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-4{{y}^{2}}\sin 2x+4y\frac{\mathrm{d}y}{\mathrm{d}x}\cos 2x=0$. Hence find the Maclaurin series for&nbsp;$y$, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760690578376-mdfu6pvwh" data-list-item-path="item-1760690578376-mdfu6pvwh"><div class="list-item-paragraph"><p>Given further that&nbsp;$x$ is small so that&nbsp;${{x}^{3}}$ and higher powers of $x$ can be neglected, use appropriate expansions from the List of Formulae <strong>(MF26)</strong> to verify the correctness of the series of $y$ in part <strong>(a)</strong>.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760690578909-9polo9nl0" data-list-item-path="item-1760690578909-9polo9nl0"><div class="list-item-paragraph"><p>Use the series expansion found in part <strong>(a)</strong> and integration to find an approximate value for&nbsp;$\int_0^{1.5} \frac{1}{3+\sin 2x}\mathrm{d}x.$. </p><span class="inline-marks">[2]</span></div></li></ol>

It is given that $y=\frac{1}{3+\sin 2x}$.

Derivatives of Trigonometric Functions

2023 ASRJC P1 Q8

<ol class="abc-list"><li><div class="list-item-paragraph">Using standard series, find the Maclaurin series expansion of $\frac{1}{\sqrt{1-{{a}^{2}}{{x}^{2}}}}$ up to and including the term in ${{x}^{4}}$, where $a$ is a positive constant. Find also the range of values of $x$, in terms of $a$, for which the expansion is valid.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Hence, by using a suitable value for $a$, show that ${{\cos }^{-1}}x\approx \frac{\pi }{2}-x-\frac{1}{6}{{x}^{3}}-\frac{3}{40}{{x}^{5}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Using the result in part <strong>(b)</strong>, estimate $\displaystyle\int_{0}^{\frac{1}{2}}{{{\cos }^{-1}}x\,\,\mathrm{d}x}$ to $~5$ significant figures.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Comparing your estimate in <strong>(c)</strong> with the actual value of $\displaystyle\int_{0}^{\frac{1}{2}}{{{\cos }^{-1}}x\,\,\mathrm{d}x}$, comment on the accuracy of your estimate and suggest how it can be further improved.</div><p style="text-align: right;">[2]</p></li></ol>

2023 EJC P1 Q9

<p>It is given that&nbsp;$\mathrm{f}(x)=\sin \left[ \ln \left( 1+x \right) \right]$, $x\in \mathbb{R}$, $x&gt;-1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;${{(1+x)}^{2}}\mathrm{f}''(x)+(1+x)\,\mathrm{f}'(x)+\mathrm{f}(x)=0$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">By further differentiation of the result in <strong>(i)</strong>, find the first three non-zero terms of the Maclaurin series for&nbsp;$\mathrm{f}(x)$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Verify the correctness of the series found in part <strong>(ii)</strong> using the standard series from the List of Formulae (MF26).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Use the series in part <strong>(ii)</strong> to approximate the value of  $\displaystyle\int_{0}^{2}{\mathrm{f}(x)}\,\mathrm{d}x$.</div><div class="list-item-paragraph">Use your calculator to find an accurate value of $\displaystyle\int_{0}^{2}{\mathrm{f}(x)}\,\mathrm{d}x$. Why is the approximated value not very good?<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Using the series obtained in part <strong>(ii)</strong>, deduce the Maclaurin series for&nbsp;$\cos \left[ \ln (1+x) \right]$ in ascending powers of $x$, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $\mathrm{f}(x)=\sin \left[ \ln \left( 1+x \right) \right]$, $x\in \mathbb{R}$, $x>-1$.

Derivatives of Natural Logarithmic Functions

2020 TJC P2 Q3

<p>It is given that $\mathrm{f}\left( x \right)=\ln \left( 1+{{\mathrm{e}}^{-x}} \right)$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Using standard series from the List of Formulae (MF26) or otherwise, show that the first three non- zero terms in the Maclaurin series for $\mathrm{f}\left( x \right)$ is $\ln a+bx+c{{x}^{2}}$, where $a$, $b$ and $c$ are constants to be determined.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence find an expansion for $\frac{1}{1+{{\mathrm{e}}^{x}}}$, up to and including the term in $x$.</div><p style="text-align: right;">[2]</p></li></ol>

It is given that $\mathrm{f}\left( x \right)=\ln \left( 1+{{\mathrm{e}}^{-x}} \right)$.

2023 JPJC P2 Q1

<p>It is given that&nbsp;$y=\ln \left( 1+{{\mathrm{e}}^{x}} \right)$.</p>
<ol class="i-list"><li data-list-item-id="item-1759645673988-or8g0q0zf" data-list-item-path="item-1759645673988-or8g0q0zf"><div class="list-item-paragraph">Show that&nbsp;$\left( 1+{{\mathrm{e}}^{x}} \right)\frac{\mathrm{d}y}{\mathrm{d}x}-{{\mathrm{e}}^{x}}=0$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759645674520-8dztg7dpo" data-list-item-path="item-1759645674520-8dztg7dpo"><div class="list-item-paragraph">By further differentiation of the result in <strong>(i)</strong>, find the Maclaurin series for&nbsp;$y$, up to and including the term in ${{x}^{2}}$. Hence find the series for&nbsp;$\frac{{{\mathrm{e}}^{x}}}{1+{{\mathrm{e}}^{x}}}$ up to and including $x$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759645675240-7u40kdbhv" data-list-item-path="item-1759645675240-7u40kdbhv"><div class="list-item-paragraph">Using appropriate expansion from the List of Formulae (MF26), verify the correctness of the series $\frac{{{\mathrm{e}}^{x}}}{1+{{\mathrm{e}}^{x}}}$ found in <strong>(ii)</strong>.<span class="inline-marks">[3]</span></div></li></ol>

It is given that $y=\ln \left( 1+{{\mathrm{e}}^{x}} \right)$.

2021 NYJC P1 Q7

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Use the first three non-zero terms of the Maclaurin series for $\cos x$ to find the Maclaurin series for $\mathrm{g}\left( x \right)$, where $\mathrm{g}\left( x \right)={{\cos }^{6}}x$, up to and including the term in ${{x}^{4}}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use your answer to part (i) to give an approximation for $\displaystyle\int_{0}^{a}{\mathrm{g}\left( x \right)}\mathrm{ d}x$ in terms of $a$, and evaluate this approximation in the case where $a=\frac{1}{4}\pi $.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Use your calculator to find an accurate value for $\displaystyle\int_{0}^{\frac{1}{4}\pi }{\mathrm{g}\left( x \right)}\mathrm{ d}x$. Why is the approximation in part (ii)(a) not very good?</span></div><p style="text-align: right;">[2]</p></li></ol></li></ol>

Use the first three non-zero terms of the Maclaurin series for $\cos x$ to find the Maclaurin series for $\mathrm{g}\left( x \right)$, where $\mathrm{g}\left( x \right)={{\cos }^{6}}x$, up to and including the term in ${{x}^{4}}$.

N2011 P1 Q4

<ol class="i-list"><li data-list-item-id="item-1759220108455-ufm8z5o2x" data-list-item-path="item-1759220108455-ufm8z5o2x"><div class="list-item-paragraph">Given that&nbsp;$\mathrm{f}\left( x \right)=\sec 2x$, find $\mathrm{f}'\left( x \right)$ and $\mathrm{f}''\left( x \right)$. Hence, or otherwise, find the Maclaurin series for&nbsp;$\mathrm{f}\left( x \right)$, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1759220109040-jud8p62xc" data-list-item-path="item-1759220109040-jud8p62xc"><div class="list-item-paragraph"><p>Use your series from part <strong>(i)</strong> to estimate $\int_{0}^{0.02} \sec 2x \mathrm{d}x$, correct to $5$ decimal places.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759220110225-knqdovwj6" data-list-item-path="item-1759220110225-knqdovwj6"><div class="list-item-paragraph"><p>Use your calculator to find $\int_{0}^{0.02} \sec 2x \mathrm{d}x$, correct to $5$ decimal places.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759220110728-s4lvixscp" data-list-item-path="item-1759220110728-s4lvixscp"><div class="list-item-paragraph"><p>Comparing your answers to parts <strong>(ii)</strong> and <strong>(iii)</strong>, and with reference to the value of $x$, comment on the accuracy of your approximations.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759220111365-hyesi83zo" data-list-item-path="item-1759220111365-hyesi83zo"><div class="list-item-paragraph">Explain why a Maclaurin series for&nbsp;$\mathrm{g}\left( x \right)=\operatorname{cosec}2x$ cannot be found.<span class="inline-marks">[1]</span></div></li></ol>
<p style="text-align: center"></p>

Given that $\mathrm{f}\left( x \right)=\sec 2x$, find $\mathrm{f}'\left( x \right)$ and $\mathrm{f}''\left( x \right)$. Hence, or otherwise, find the Maclaurin series for $\mathrm{f}\left( x \right)$, up to and including the term in ${{x}^{2}}$.

Explain why a Maclaurin series for $\mathrm{g}\left( x \right)=\operatorname{cosec}2x$ cannot be found.

Existence of a Maclaurin Expansion

N2019 P2 Q4

<p>In this question you may use expansions from the List of Formulae (MF26).</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the Maclurin expansion of $\ln (\cos 2x)$ in ascending powers of $x$, up to and including the term ${{x}^{6}}$. State any value(s) of $x$ in the domain $0\le x\le \frac{1}{4}\pi $ for which the expansion is not valid.</span></div><p style="text-align: right;">[6]</p></li><li><div class="list-item-content"><span class="list-item-text">Use your expansion from part (i) and integration to find an approximate expression for $\displaystyle\int{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}$. Hence find an approximate value for $\displaystyle\int_{0}^{0.5}{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}$, giving your answer to $4$ decimal places.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Use your graphing calculator to find a second approximate value for $\displaystyle\int_{0}^{0.5}{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}$, giving your answer to $4$decimal places.</span></div><p style="text-align: right;">[1]</p></li></ol>

In this question you may use expansions from the List of Formulae (MF26).

Find the Maclurin expansion of $\ln (\cos 2x)$ in ascending powers of $x$, up to and including the term ${{x}^{6}}$. State any value(s) of $x$ in the domain $0\le x\le \frac{1}{4}\pi $ for which the expansion is not valid.

Use your expansion from part (i) and integration to find an approximate expression for $\displaystyle\displaystyle \displaystyle\int{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}$. Hence find an approximate value for $\displaystyle\displaystyle \displaystyle\int_{0}^{0.5}{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}$, giving your answer to $4$ decimal places.

Use your graphing calculator to find a second approximate value for $\displaystyle\displaystyle \displaystyle\int_{0}^{0.5}{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}$, giving your answer to $4$decimal places.

N2018 P2 Q4

<p>In this question you may use expansions from the List of Formulae <strong>(MF26).</strong></p>
<ol class="i-list"><li data-list-item-id="item-1759131988922-d2el9fs7v" data-list-item-path="item-1759131988922-d2el9fs7v"><div class="list-item-paragraph">Find the Maclaurin expansion of $\ln \left( 1+\cos 3x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{4}}$, for&nbsp;$0\le x&lt;\frac{\pi }{3}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759131989732-evy9e0yb5" data-list-item-path="item-1759131989732-evy9e0yb5"><div class="list-item-paragraph"><p>Use your expansion from part <strong>(i)</strong> and to find an approximate value for&nbsp;$\int_{0}^{0.5} x \ln (1 + \cos 3x) \mathrm{d}x$, giving your answer to $5$ decimal places.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759131990504-5i5mv58ef" data-list-item-path="item-1759131990504-5i5mv58ef"><div class="list-item-paragraph"><p>Use your calculator to find the value of $\int_{0}^{0.5} x \ln (1 + \cos 3x) \mathrm{d}x$ up to $5$ decimal places.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759131991712-bljzrw6nx" data-list-item-path="item-1759131991712-bljzrw6nx"><div class="list-item-paragraph"><p>With the aid of a suitable diagram, comparing your answers in <strong>(ii)</strong> and <strong>(iii)</strong>, comment on the accuracy of your approximations.</p><span class="inline-marks">[2]</span></div></li></ol>
<p style="text-align: center"></p>

In this question you may use expansions from the List of Formulae **(MF26).**

Find the Maclaurin expansion of $\ln \left( 1+\cos 3x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{4}}$, for $0\le x<\frac{\pi }{3}$.

2022 SAJC P2 Q3

<p>It is given that&nbsp;$\mathrm{f}\left( x \right)=\ln \left( 2+2\sin x \right)$.</p>
<ol class="i-list"><li data-list-item-id="item-1759129600239-hucpyo9hc" data-list-item-path="item-1759129600239-hucpyo9hc"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{f}''\left( x \right)=\frac{k}{1+\sin x}$, where&nbsp;$k$ is a constant to be found.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759129600855-ihgp8f1cc" data-list-item-path="item-1759129600855-ihgp8f1cc"><div class="list-item-paragraph">Hence find the Maclaurin series for&nbsp;$\mathrm{f}\left( x \right)$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759129602002-2ltailw89" data-list-item-path="item-1759129602002-2ltailw89"><div class="list-item-paragraph">Use the series in part <strong>(ii)</strong> to approximate the value of $\displaystyle\displaystyle \displaystyle\int_{0}^{2}{\mathrm{f}\left( x \right)\,\,\mathrm{d}x}$.<span class="inline-marks">[1]</span></div></li></ol>

It is given that $\mathrm{f}\left( x \right)=\ln \left( 2+2\sin x \right)$.

Show that $\mathrm{f}''\left( x \right)=\frac{k}{1+\sin x}$, where $k$ is a constant to be found.

Hence find the Maclaurin series for $\mathrm{f}\left( x \right)$, up to and including the term in ${{x}^{3}}$.

2021 MI P2 Q3

<ol class="i-list"><li><div class="list-item-paragraph">Find the series expansion of ${{e}^{4x}}\sin x$, up to and including the term in ${{x}^{4}}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Using your answer in <strong>(i)</strong>, find the series expansion of ${{e}^{4x}}\cos x$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[2]</span></div></li></ol>

Find the series expansion of ${{e}^{4x}}\sin x$, up to and including the term in ${{x}^{4}}$.

YJC P2 Q3

<p>It is given that $y=\mathrm{f}\left( x \right)$, where $\mathrm{f}\left( x \right)={{\mathrm{e}}^{3x}}\sec 2x$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that $\mathrm{{f}'}\left( x \right)=y\left( 3+2\tan 2x \right)$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the Maclaurin series for $\mathrm{f}\left( x \right)$, up to and including the term in ${{x}^{2}}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence find the first two non-zero terms in the Maclaurin series for $\left( {{\mathrm{e}}^{3x}}\sec 2x \right)\left( 3+2\tan 2x \right)$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Using the standard series from the List of Formulae (MF26), verify the result found in part (ii).</span></div><p style="text-align: right;">[2]</p></li></ol>

It is given that $y=\mathrm{f}\left( x \right)$, where $\mathrm{f}\left( x \right)={{\mathrm{e}}^{3x}}\sec 2x$.

Show that $\mathrm{{f}'}\left( x \right)=y\left( 3+2\tan 2x \right)$.

Find the Maclaurin series for $\mathrm{f}\left( x \right)$, up to and including the term in ${{x}^{2}}$.

Hence find the first two non-zero terms in the Maclaurin series for $\left( {{\mathrm{e}}^{3x}}\sec 2x \right)\left( 3+2\tan 2x \right)$.

Using the standard series from the List of Formulae (MF26), verify the result found in part (ii).

2020 ACJC P1 Q3

<p>It is given that $\frac{\mathrm{d}y}{\mathrm{d}x}+xy={{\mathrm{e}}^{-x}}$ and $y=1$ when $x=0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the Maclaurin series for $y$, up to and including the term in ${{x}^{3}}$.</span></div><p style="text-align: right;">[5]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence, or otherwise, state the first three non-zero terms of the Maclaurin series for $\frac{\mathrm{d}y}{\mathrm{d}x}$.</span></div><p style="text-align: right;">[1]</p></li></ol>

It is given that $\frac{\mathrm{d}y}{\mathrm{d}x}+xy={{\mathrm{e}}^{-x}}$ and $y=1$ when $x=0$.

Find the Maclaurin series for $y$, up to and including the term in ${{x}^{3}}$.

Hence, or otherwise, state the first three non-zero terms of the Maclaurin series for $\frac{\mathrm{d}y}{\mathrm{d}x}$.

2020 VJC Promo Q1

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that ${y = \ln{(\sec{x})}}$, show that $\frac{\mathrm{d}^3 y}{\mathrm{d}x^3} = 2 \left( \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} \right) \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence or otherwise, obtain the Maclaurin expansion of ${y}$ in terms of ${x}$ up to and including the term in ${x^4}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">By putting ${x = \frac{\pi}{4}}$, find an approximation for ${\ln{2}}$ in terms of ${\pi}$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Using your answer to part (b), find an approximation to $\displaystyle\int_{0}^{\frac{\pi }{10}}{\ln }(\sec x)\mathrm{d}x$. Give your answer correct to 4 significant figures.</span></div><p style="text-align: right;">[1]</p></li></ol>

Given that ${y = \ln{(\sec{x})}}$, show that $\frac{\mathrm{d}^3 y}{\mathrm{d}x^3} = 2 \left( \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} \right) \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)$.

Hence or otherwise, obtain the Maclaurin expansion of ${y}$ in terms of ${x}$ up to and including the term in ${x^4}$.

By putting ${x = \frac{\pi}{4}}$, find an approximation for ${\ln{2}}$ in terms of ${\pi}$.

Using your answer to part (b), find an approximation to $\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{10}}{\ln }(\sec x)\mathrm{d}x$. Give your answer correct to 4 significant figures.

Math Question Bank