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H2 Math

IB AA HL Math

IB AA SL Math

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IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>It is given that&nbsp;$y=\mathrm{f}(x)$ is such that \[\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=\frac{\mathrm{d}y}{\mathrm{d}x}\left(2\tan x+p\sec^2x\right),\] where&nbsp;$p$ is a constant. The Maclaurin series for&nbsp;$\mathrm{f}(x)$ is given by \[y=1+x+\frac12x^2+qx^3+\cdots,\] where&nbsp;$q$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="tjc-q05-item01" data-list-item-path="tjc-q05-item01"><div class="list-item-paragraph">State the values of $\mathrm{f}(0)$ and $\mathrm{f}'(0)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="tjc-q05-item02" data-list-item-path="tjc-q05-item02"><div class="list-item-paragraph">Find the values of $p$ and $q$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="tjc-q05-item03" data-list-item-path="tjc-q05-item03"><div class="list-item-paragraph">Using the given series for&nbsp;$\mathrm{f}(x)$ and standard series from the List of Formulae (MF27), find the first three non-zero terms of the series expansion of $\sqrt{\mathrm{f}(x)}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="tjc-q05-item04" data-list-item-path="tjc-q05-item04"><div class="list-item-paragraph">The given Maclaurin series for&nbsp;$\mathrm{f}(x)$ is used to estimate $\displaystyle \int_0^{0.2}\mathrm{f}(x)\,\mathrm{d}x$ and $\displaystyle \int_{0.2}^{0.4}\mathrm{f}(x)\,\mathrm{d}x$. Without any further computations, determine which estimate has a smaller percentage error. Give an explanation for your choice.<span class="inline-marks">[1]</span></div></li></ol>

It is given that $y=\mathrm{f}(x)$ is such that \[\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=\frac{\mathrm{d}y}{\mathrm{d}x}\left(2\tan x+p\sec^2x\right),\] where $p$ is a constant. The Maclaurin series for $\mathrm{f}(x)$ is given by \[y=1+x+\frac12x^2+qx^3+\cdots,\] where $q$ is a constant.

State the values of $\mathrm{f}(0)$ and $\mathrm{f}'(0)$.

Using the given series for $\mathrm{f}(x)$ and standard series from the List of Formulae (MF27), find the first three non-zero terms of the series expansion of $\sqrt{\mathrm{f}(x)}$.

The given Maclaurin series for $\mathrm{f}(x)$ is used to estimate $\displaystyle \int_0^{0.2}\mathrm{f}(x)\,\mathrm{d}x$ and $\displaystyle \int_{0.2}^{0.4}\mathrm{f}(x)\,\mathrm{d}x$. Without any further computations, determine which estimate has a smaller percentage error. Give an explanation for your choice.

Finding Parameters from Power Series Coefficients

Maclaurin Series from Differential Equations

Substitution and Composition of Power Series

Derivatives of Reciprocal Trigonometric Functions

Accuracy of Power Series Approximations

Higher Derivatives by Implicit Differentiation

2026 TJC P1 Q5

Maclaurin Series and Power Series

Techniques of Differentiation

<p>It is given that&nbsp;$\mathrm{f}(x) = \ln(a + x)$, $x \in \mathbb{R}$, $x &gt; -a$, where&nbsp;$a$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-4-a" data-list-item-path="item-4-a"><div class="list-item-paragraph">Using the standard series from the List of Formulae (MF27), find the series expansion for&nbsp;$\mathrm{f}(x)$, up to and including the term in $x^3$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that&nbsp;$a=1$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-4-b" data-list-item-path="item-4-b"><div class="list-item-paragraph">Hence, or otherwise, show that the series expansion of $\sin[\mathrm{f}(x)]$, up to and including the term in $x^3$ is given by $x - \frac{1}{2}x^2 + \frac{1}{6}x^3$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-4-c" data-list-item-path="item-4-c"><div class="list-item-paragraph">Deduce the Maclaurin series for&nbsp;$\cos[\mathrm{f}(x)]$ up to and including the term in $x^2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-4-d" data-list-item-path="item-4-d"><div class="list-item-paragraph">Find $\displaystyle \displaystyle\int_{1}^{3} (x - \frac{1}{2}x^2 + \frac{1}{6}x^3) \, \mathrm{d}x$. Without the use of a calculator or any further calculation, explain, whether this value is a good approximation to the value of $\displaystyle \displaystyle\int_{1}^{3} \sin[\mathrm{f}(x)] \, \mathrm{d}x$.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $\mathrm{f}(x) = \ln(a + x)$, $x \in \mathbb{R}$, $x > -a$, where $a$ is a constant.

Using the standard series from the List of Formulae (MF27), find the series expansion for $\mathrm{f}(x)$, up to and including the term in $x^3$.

Hence, or otherwise, show that the series expansion of $\sin[\mathrm{f}(x)]$, up to and including the term in $x^3$ is given by $x - \frac{1}{2}x^2 + \frac{1}{6}x^3$.

Deduce the Maclaurin series for $\cos[\mathrm{f}(x)]$ up to and including the term in $x^2$.

Find $\displaystyle \displaystyle\int_{1}^{3} (x - \frac{1}{2}x^2 + \frac{1}{6}x^3) \, \mathrm{d}x$. Without the use of a calculator or any further calculation, explain, whether this value is a good approximation to the value of $\displaystyle \displaystyle\int_{1}^{3} \sin[\mathrm{f}(x)] \, \mathrm{d}x$.

Maclaurin Series Using Standard Series

Approximations Using Power Series

Deriving Series by Differentiation and Integration

2025 VJC P2 Q1

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the expansion of $(8 - x)^{\frac{1}{3}}$ in ascending powers of $x$ up to and including the term in $x^2$. State the set of values of $x$ for which this expansion is valid.</span></div></li><li><div class="list-item-content"><span class="list-item-text">By substituting $x=-1$, find an approximation of $\sqrt[3]{3}$, leaving your answer in the form $\frac{m}{n}$, where $m$ and $n$ are integers to be determined.</span></div></li><li><div class="list-item-content"><span class="list-item-text">The substitution $x=5$ can also be used to find an approximation for $\sqrt[3]{3}$. Explain whether this will give a better approximation than using $x=-1$.</span></div></li></ol>

Find the expansion of $(8 - x)^{\frac{1}{3}}$ in ascending powers of $x$ up to and including the term in $x^2$. State the set of values of $x$ for which this expansion is valid.

By substituting $x=-1$, find an approximation of $\sqrt[3]{3}$, leaving your answer in the form $\frac{m}{n}$, where $m$ and $n$ are integers to be determined.

The substitution $x=5$ can also be used to find an approximation for $\sqrt[3]{3}$. Explain whether this will give a better approximation than using $x=-1$.

Convergence and Validity of Power Series

Binomial Expansion as a Maclaurin Series

<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>

Series from Products, Substitution, Integration and Differentiation

Small Angle Approximations

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.

Derivatives of Inverse Trigonometric Functions

Maclaurin Series by Successive Differentiation

Derivatives of Exponential Functions

Chain Rule

2024 JPJC P2 Q3

<ol class="abc-list"><li data-list-item-id="item-1760706771310-0lbd2knc3" data-list-item-path="item-1760706771310-0lbd2knc3"><div class="list-item-paragraph">Using standard series from the List of Formulae (MF26), find the Maclaurin expansion of $\frac{1}{{{\left( 1+\cos x \right)}^{2}}}$ in ascending powers of $x$ up to and including the term in ${{x}^{4}}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760706771895-x94ud5bna" data-list-item-path="item-1760706771895-x94ud5bna"><div class="list-item-paragraph">Find the set of values of $x$ for which $\frac{1}{{{\left( 1+\cos x \right)}^{2}}}$ is within $\pm \,0.5$ of the polynomial found in part <strong>(a)</strong>, where&nbsp;$0\le x &lt; \pi $.<span class="inline-marks">[3]</span></div></li></ol>

2024 HCI P2 Q5

<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

<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>In triangle $PQR$, $\angle PRQ=\frac{2\pi }{3}$, $\angle RPQ=\frac{\pi }{6}+\theta $, and $PR=a$ units, where&nbsp;$a$ is a positive real constant.</p>
<ol class="i-list"><li data-list-item-id="item-1760423747573-c4mvj1pjc" data-list-item-path="item-1760423747573-c4mvj1pjc"><div class="list-item-paragraph">Show that&nbsp;$PQ=\frac{\sqrt{3}a}{\cos \theta -\sqrt{3}\sin \theta }$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760423748158-9fegtmk7c" data-list-item-path="item-1760423748158-9fegtmk7c"><div class="list-item-paragraph">Given that&nbsp;$\theta $ is sufficiently small, show that&nbsp;$PQ\approx \sqrt{3}a\left( 1+b\theta +c{{\theta }^{2}} \right)$, where&nbsp;$b$ and $c$ are real constants.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760423749013-a9nf00s5i" data-list-item-path="item-1760423749013-a9nf00s5i"><div class="list-item-paragraph">Given that&nbsp;$\theta &gt;0$, find the range of values of $\theta $ such that the percentage error of the approximation in <strong>(ii)</strong> is less than $5\%$.<span class="inline-marks">[2]</span></div></li></ol>

In triangle $PQR$, $\angle PRQ=\frac{2\pi }{3}$, $\angle RPQ=\frac{\pi }{6}+\theta $, and $PR=a$ units, where $a$ is a positive real constant.

2023 SAJC P2 Q2

<p>The function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}\left( x \right)=\frac{x}{\sqrt{4-x}}$, $x\in \mathbb{R}$, $0 < x < 4$.</p>
<p>It is given that $\mathrm{f}\left( x \right)=ax+b{{x}^{2}}+c{{x}^{3}}$ is the binomial expansion of $\mathrm{g}\left( x \right)=\frac{x}{\sqrt{4-x}}$ in ascending powers of $x$ up to and including terms in ${{x}^{3}}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the exact value of $a$, $b$ and $c$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the range of values of $x$ such that the percentage error in using $\mathrm{f}\left( x \right)$ to approximate $\mathrm{g}\left( x \right)$ is less than $4\%$.</div><p style="text-align: right;">[2]</p></li></ol>

$\mathrm{g}\left( x \right)=\frac{x}{\sqrt{4-x}}$, $x\in \mathbb{R}$, $0 < x < 4$.

It is given that $\mathrm{f}\left( x \right)=ax+b{{x}^{2}}+c{{x}^{3}}$ is the binomial expansion of $\mathrm{g}\left( x \right)=\frac{x}{\sqrt{4-x}}$ in ascending powers of $x$ up to and including terms in ${{x}^{3}}$.

2023 TJC P1 Q3

<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 Trigonometric Functions

Derivatives of Natural Logarithmic Functions

2020 TJC P2 Q3

<p>Given that&nbsp;$y={{({{\sin }^{-1}}x)}^{2}}$, show that</p>
<p style="text-align: center">\[\left( 1-{{x}^{2}} \right){{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}=4y\]</p>
<p style="text-align: center">and $\left( 1-{{x}^{2}} \right)\frac{{{\mathrm{d}}^{\text{2}}}y}{\mathrm{d}{{x}^{2}}}-x\frac{\mathrm{d}y}{\mathrm{d}x}=2$.<span class="inline-marks">[3]</span></p>
<p>By further differentiation of these results, find the Maclaurin series of $y$ up to including the term in ${{x}^{4}}$.<span class="inline-marks">[3]</span></p>
<p>The series found above is to be used to estimate a value for&nbsp;${{\pi }^{2}}$. Explain with the aid of a sketch why using $x=\frac{1}{2}$ in the series gives a better approximation than $x=1$, and hence find an approximation to ${{\pi }^{2}}$, giving your answer in fraction form. <span class="inline-marks">[4]</span></p>

Given that $y={{({{\sin }^{-1}}x)}^{2}}$, show that

\[\left( 1-{{x}^{2}} \right){{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}=4y\]

and $\left( 1-{{x}^{2}} \right)\frac{{{\mathrm{d}}^{\text{2}}}y}{\mathrm{d}{{x}^{2}}}-x\frac{\mathrm{d}y}{\mathrm{d}x}=2$.

By further differentiation of these results, find the Maclaurin series of $y$ up to including the term in ${{x}^{4}}$.

The series found above is to be used to estimate a value for ${{\pi }^{2}}$. Explain with the aid of a sketch why using $x=\frac{1}{2}$ in the series gives a better approximation than $x=1$, and hence find an approximation to ${{\pi }^{2}}$, giving your answer in fraction form. 

2013 ACJC P1 Q12

<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;$x$ is small enough for terms involving ${{x}^{4}}$ and above to be ignored, use the Maclaurin series for&nbsp;$\sin x$ and $\cos x$ to show that</div><div class="list-item-paragraph" style="text-align: center">$\tan x\approx x+\frac{1}{3}{{x}^{3}}$.<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">A straight pole, $AB$, leans against a wall as shown in the diagram below. $AC$ is horizontal where&nbsp;$C$ is the base of the wall and $AC=2$ metres. $B$ is vertically above $C$ and angle $BAC={{85}^{\circ }}$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760155508834-e2kr9cc98n.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAxNTU1MDg4MzQtZTJrcjljYzk4bi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQyNjEyLCJleHAiOjE3OTM1NDI5MTJ9.TgRY2UYCTQt3qwt8VdZX7Gm66QXP25IimAhw-W94vq4" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div><div class="list-item-paragraph">Use the Maclaurin series for&nbsp;$\tan x$ to find an approximate value for&nbsp;$BC$, giving your answer correct to $5$ decimal places.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">Use a calculator to find the value of $BC$, giving your answer correct to $5$ decimal places, and hence show that the error in your approximation is about $0.0008\%$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;$y=\tan \left( \frac{1}{x+a} \right)$, where&nbsp;$a$ is a positive constant, show that</div><div class="list-item-paragraph" style="text-align: center">${{\left( x+a \right)}^{2}}\frac{\mathrm{d}y}{\mathrm{d}x}+{{y}^{2}}+1=0$.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">Using $a=\frac{4}{\pi }$, find the Maclaurin series for&nbsp;$\tan \left( \frac{1}{x+a} \right)$, up to and including the term in ${{x}^{2}}$, giving all coefficients in exact form.<span class="inline-marks">[4]</span></div></li></ol>

2017 ACJC Promo Q10

<p>Given that $y={{\mathrm{e}}^{{{\sin }^{-1}}\left( 2x \right)}}$, show that $\left( 1-4{{x}^{2}} \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-4x\frac{\mathrm{d}y}{\mathrm{d}x}=4y$.</p>
<p>By further differentiating this result, find the Maclaurin series for $y$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.</p>
<p>Hence, find an approximation value of ${{\mathrm{e}}^{\frac{\pi }{2}}}$, by substituting a suitable value of $x$ in the Maclaurin series $y$.</p>
<p>Suggest one way to improve the accuracy of the approximated value obtained.</p>

Given that $y={{\mathrm{e}}^{{{\sin }^{-1}}\left( 2x \right)}}$, show that $\left( 1-4{{x}^{2}} \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-4x\frac{\mathrm{d}y}{\mathrm{d}x}=4y$.

By further differentiating this result, find the Maclaurin series for $y$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.

Hence, find an approximation value of ${{\mathrm{e}}^{\frac{\pi }{2}}}$, by substituting a suitable value of $x$ in the Maclaurin series $y$.

Suggest one way to improve the accuracy of the approximated value obtained.

2006 NJC Promo Q5

<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

<div class="graph-block" data-graph-config="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" data-width="230" data-height="147" style="width:230px;height:147px;max-width:100%;display:block;margin:0 auto;"></div>
<p>$ABC$ is a sector of a circle with radius $AB$ and $AC$ and $\measuredangle BAC=\theta $. $D$ is a point on $AC$ such that&nbsp;$AD=x$cm, $CD=y$m and $\measuredangle ADB=\frac{\pi }{2}$.</p>
<ol class="i-list"><li data-list-item-id="item-1759149348798-sjfyiamge" data-list-item-path="item-1759149348798-sjfyiamge"><div class="list-item-paragraph">Show that&nbsp;$BD=\sqrt{{{y}^{2}}+2xy}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759149349502-4fmhiz3ah" data-list-item-path="item-1759149349502-4fmhiz3ah"><div class="list-item-paragraph">Hence show that&nbsp;$\sin \theta \approx {{\left( 2\alpha \right)}^{\frac{1}{2}}}\left( 1-\frac{3}{4}\alpha \right)$, where&nbsp;$\alpha =\frac{y}{x}$, and $y$ is small compared to $x$.<span class="inline-marks">[4]</span></div></li></ol>
<p>You are now given that&nbsp;$\theta =10{}^\circ $ and $x=100$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1759149367750-73wyuzc9u" data-list-item-path="item-1759149367750-73wyuzc9u"><div class="list-item-paragraph">Use part <strong>(ii)</strong> to find estimates for&nbsp;$\alpha $ and $y$ correct to $2$ decimal places.</div></li><li data-list-item-id="item-1759149369738-a68qfclh7" data-list-item-path="item-1759149369738-a68qfclh7"><div class="list-item-paragraph">Find the actual value of $y$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759149387904-8w9wly4hh" data-list-item-path="item-1759149387904-8w9wly4hh"><div class="list-item-paragraph">Comparing the values of $y$ obtained in parts <strong>(iii)</strong> and <strong>(iv)</strong>, comment on the accuracy of your approximations and explain your answer.<span class="inline-marks">[1]</span></div></li></ol>

$ABC$ is a sector of a circle with radius $AB$ and $AC$ and $\measuredangle BAC=\theta $. $D$ is a point on $AC$ such that $AD=x$cm, $CD=y$m and $\measuredangle ADB=\frac{\pi }{2}$.

Hence show that $\sin \theta \approx {{\left( 2\alpha \right)}^{\frac{1}{2}}}\left( 1-\frac{3}{4}\alpha \right)$, where $\alpha =\frac{y}{x}$, and $y$ is small compared to $x$.

You are now given that $\theta =10{}^\circ $ and $x=100$.

2021 SAJC 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

<ol class="abc-list"><li data-list-item-id="item-1759127994505-k8dwirnfr" data-list-item-path="item-1759127994505-k8dwirnfr"><div class="list-item-content"><span class="list-item-text">In the triangle $ABC$, $AC=BC=1$ and angle $ACB=\left( \frac{1}{2}\pi +\theta \right)$ radians. Given that&nbsp;$\theta $ is sufficiently small angle, show that</span></div><div class="list-item-paragraph" style="text-align: center">$AB\approx \sqrt{2}+\frac{\sqrt{2}}{2}\theta -\frac{\sqrt{2}}{8}{{\theta }^{2}}$<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1759127995375-tmm8n1own" data-list-item-path="item-1759127995375-tmm8n1own"><div class="list-item-paragraph">Let&nbsp;$\mathrm{f}(x)=\ln (2-x)$.</div><ol class="i-list"><li data-list-item-id="subitem-1759128015183-514zvhi2t" data-list-item-path="item-1759127995375-tmm8n1own/subitem-1759128015183-514zvhi2t"><div class="list-item-paragraph">Write down the first three non-zero terms of the Maclaurin series for&nbsp;$\mathrm{f}(x)$, where&nbsp;$-2\le x&lt;2$.<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">Denote the answer to part <strong>(i)</strong> by $\mathrm{g}(x)$.</div></li><li data-list-item-id="subitem-1759128015393-d1c3ksmlt" data-list-item-path="item-1759127995375-tmm8n1own/subitem-1759128015393-d1c3ksmlt"><div class="list-item-paragraph">Find the range of values of $x$ for which the percentage error between the value of $\mathrm{g}(x)$ and the value of $\mathrm{f}(x)$ is within $10\%$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

In the triangle $ABC$, $AC=BC=1$ and angle $ACB=\left( \frac{1}{2}\pi +\theta \right)$ radians. Given that $\theta $ is sufficiently small angle, show that

2021 TJC P2 Q3

<p>Let&nbsp;$\mathrm{f}\left( x \right)={{\mathrm{e}}^{x}}\sin x$.</p>
<ol class="i-list"><li data-list-item-id="item-1758939701469-ommzt9cx7" data-list-item-path="item-1758939701469-ommzt9cx7"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ for&nbsp;$-3\le x\le 3$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1758939702235-vegro0btv" data-list-item-path="item-1758939702235-vegro0btv"><div class="list-item-paragraph">Find the series expansion of $\mathrm{f}\left( x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>Denote the answer to part (ii) by $\mathrm{g}\left( x \right)$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1758939717602-tgzee98iw" data-list-item-path="item-1758939717602-tgzee98iw"><div class="list-item-paragraph">On the same diagram as in part (i), sketch the graph of $y=\mathrm{g}\left( x \right)$. Label the two graphs clearly.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1758939718532-w51svjaao" data-list-item-path="item-1758939718532-w51svjaao"><div class="list-item-paragraph">Find, for&nbsp;$-3\le x\le 3$, the set of values of $x$ for which the value of $\mathrm{g}\left( x \right)$ is within $\pm 0.5$ of the value of $\mathrm{f}\left( x \right)$.<span class="inline-marks">[3]</span></div></li></ol>

Let $\mathrm{f}\left( x \right)={{\mathrm{e}}^{x}}\sin x$.

Sketch the graph of $y=\mathrm{f}\left( x \right)$ for $-3\le x\le 3$.

Find the series expansion of $\mathrm{f}\left( x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.

Denote the answer to part (ii) by $\mathrm{g}\left( x \right)$.

On the same diagram as in part (i), sketch the graph of $y=\mathrm{g}\left( x \right)$. Label the two graphs clearly.

Find, for $-3\le x\le 3$, the set of values of $x$ for which the value of $\mathrm{g}\left( x \right)$ is within $\pm 0.5$ of the value of $\mathrm{f}\left( x \right)$.

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