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PSLE Foundation Mathematics

<p>It is given that&nbsp;$y = \sin^{-1} px$, where&nbsp;$p$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = p \sec y$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph"><p>By further differentiation of the result in part <strong>(a)</strong> find, in terms of $p$, the Maclaurin series for&nbsp;$y$ up to and including the term in $x^3$.</p><span class="inline-marks">[5]</span></div></li><li data-list-item-id="imported-item-block-1-item-2" data-list-item-path="imported-item-block-1-item-2"><div class="list-item-paragraph"><p>Use your answer to part <strong>(b)</strong> to find the series expansion of $\frac{1}{\sqrt{1 - 9x^2}}$, up to and including the term in $x^2$.</p><span class="inline-marks">[3]</span></div></li></ol>

It is given that $y = \sin^{-1} px$, where $p$ is a constant.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = p \sec y$.

Maclaurin Series from Differential Equations

Derivatives of Inverse Trigonometric Functions

Derivatives of Reciprocal Trigonometric Functions

Maclaurin Series by Successive Differentiation

Higher Derivatives by Implicit Differentiation

Problems involving Calculus

Deriving Series by Differentiation and Integration

N2025 P1 Q9

Maclaurin Series and Power Series

Techniques of Differentiation

<p>Prove that ${{\left( 1+x \right)}^{n}}=1+nx+\frac{n\left( n-1 \right)}{2!}{{x}^{2}}+\frac{n\left( n-1 \right)\left( n-2 \right)}{3!}{{x}^{3}}+...$.</p>

Prove that ${{\left( 1+x \right)}^{n}}=1+nx+\frac{n\left( n-1 \right)}{2!}{{x}^{2}}+\frac{n\left( n-1 \right)\left( n-2 \right)}{3!}{{x}^{3}}+...$.

Worked Example 1

<p>It is given that $y={{e}^{{{\tan }^{-1}}4x}}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that $\left( 1+16{{x}^{2}} \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+k\frac{\mathrm{d}y}{\mathrm{d}x}\left( 8x-1 \right)=0$, where $k$ is a constant to be found.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">By further differentiation of the result in part <strong>(a)</strong>, find the Maclaurin series for $y$ up to and including the term in ${{x}^{3}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">The first three non-zero terms of the Maclaurin expansion of ${{e}^{{{\tan }^{-1}}4x}}$ are equal to the first three non-zero terms of the series expansion of $\ln \left( \frac{a}{1-bx} \right)$, where $a$ and $b$ are constants. Using standard series from the List of Formulae (MF27), find the exact values of $a$ and $b$.</div><p style="text-align: right;">[2]</p></li></ol>

Show that $(1+16x^2)\frac{d^2y}{dx^2}+k\frac{dy}{dx}(8x-1)=0$, where $k$ is a constant to be found.

By further differentiation of the result in part (a), find the Maclaurin series for $y$ up to and including the term in $x^3$.

The first three non-zero terms of the Maclaurin expansion of $e^{\tan^{-1}4x}$ are equal to the first three non-zero terms of the series expansion of $\ln\left(\frac{a}{1-bx}\right)$, where $a$ and $b$ are constants. Using standard series from the List of Formulae (MF27), find the exact values of $a$ and $b$.

Finding Parameters from Power Series Coefficients

Maclaurin Series Using Standard Series

Second and Higher Derivatives

Derivatives of Exponential Functions

Chain Rule

2025 DHS Promo Q6

<p>It is given that&nbsp;$y=\ln \left( 2-{{\mathrm{e}}^{-2x}} \right)$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760870612493-0lhphq5um" data-list-item-path="item-1760870612493-0lhphq5um"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}=4{{\mathrm{e}}^{-y}}-2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760870613475-t4r53w0xb" data-list-item-path="item-1760870613475-t4r53w0xb"><div class="list-item-paragraph">Hence find the Maclaurin series for&nbsp;$y$, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760870614360-lu2pah4wx" data-list-item-path="item-1760870614360-lu2pah4wx"><div class="list-item-paragraph"><p>Using standard series from the List of Formulae (MF26), expand $\ln \left( 2-{{\mathrm{e}}^{-2x}} \right)$ as far as the term in ${{x}^{2}}$, and use this expansion as a check on the correctness of the series found in part <strong>(b)</strong>.</p><span class="inline-marks">[4]</span></div></li></ol>

It is given that $y=\ln \left( 2-{{\mathrm{e}}^{-2x}} \right)$.

Substitution and Composition of Power Series

Derivatives of Natural Logarithmic Functions

2023 EJC Promo Q9

<p>It is given that&nbsp;$y=\sqrt{1+\tan x}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760850707670-du9gekgw8" data-list-item-path="item-1760850707670-du9gekgw8"><div class="list-item-paragraph">Show that&nbsp;$2y\frac{\mathrm{d}y}{\mathrm{d}x}=1+{{\left( {{y}^{2}}-1 \right)}^{2}}$ and $y\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}=2y\left( {{y}^{2}}-1 \right)\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760850708179-wypfpif5n" data-list-item-path="item-1760850708179-wypfpif5n"><div class="list-item-paragraph">Hence find the first three terms in the Maclaurin expansion of $y$.<span class="inline-marks">[2]</span></div></li></ol>

Implicit Differentiation

Derivatives of Trigonometric Functions

2024 ACJC Promo Q2

<p>By successive differentiation, find the Maclaurin’s series for&nbsp;$\sin 3x$ up to the term ${{x}^{3}}$.</p>

By successive differentiation, find the Maclaurin’s series for $\sin 3x$ up to the term ${{x}^{3}}$.

<p>By successive differentiation, find the Maclaurin’s series for&nbsp;$\ln (1+2x)$ up to the term ${{x}^{3}}$.</p>

By successive differentiation, find the Maclaurin’s series for $\ln (1+2x)$ up to the term ${{x}^{3}}$.

The Maclaurin series is the Taylor series of a sufficiently differentiable function about $x=0$:

\[\mathrm{f}(x)=\mathrm{f}(0)+x\mathrm{f}'(0)+\frac{x^2}{2!}\mathrm{f}''(0)+\cdots+\frac{x^n}{n!}\mathrm{f}^{(n)}(0)+\cdots\]

Differentiate $\mathrm{f}(x)$ successively to obtain $\mathrm{f}'(x),\mathrm{f}''(x),\mathrm{f}'''(x),\ldots$.

Evaluate the function and derivatives at $x=0$.

Substitute these values into the Maclaurin formula, retaining the terms required by the question and checking the interval of validity.

<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.

Series from Products, Substitution, Integration and Differentiation

Approximations Using Power Series

Accuracy of Power Series Approximations

2024 JPJC P2 Q3

<p>It is given that $y=\sin \left[ \ln \left( 1+2x \right) \right]$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that ${{\left( 1+2x \right)}^{2}}\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+2\left( 1+2x \right)\frac{\mathrm{d}y}{\mathrm{d}x}+ky=0$, where $k$ is a constant to be found.</div><div class="list-item-paragraph">Hence, find the first three non-zero terms of the Maclaurin expansion for $y$.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Using standard series from the List of Formulae, verify the correctness of your result from part <strong>(i)</strong> up to and including the term in ${{x}^{3}}$.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">Explain why the expansion is not valid when $x=-\frac{1}{2}$.</div><p style="text-align: right;">[1]</p></li></ol>

It is given that $y=\sin \left[ \ln \left( 1+2x \right) \right]$.

Convergence and Validity of Power Series

RI Maclaurin Series Tutorial Q3

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760706907296-yet4o3lzty.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA3MDY5MDcyOTYteWV0NG8zbHp0eS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTM5MzEyLCJleHAiOjE3OTM1Mzk2MTJ9.Ag3DrGCQ2mxe-ApGAemWMlhIHA4VrS0FRKdVLbfZwnk" width="400" class="aligncenter" /></figure>
<p>In the triangle $ABC$, $AB=x$, $BC=y$, $AC=\frac{1}{6}$, angle $ABC=\frac{\pi }{6}$ radians and angle $ACB=\theta $ radians (see diagram).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Show that $\frac{x}{y}=\frac{2\sin \theta }{\cos \theta +\sqrt{3}\sin \theta }$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $\theta $ is sufficiently small, express $\frac{x}{y}$ as a cubic polynomial in $\theta $.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Show that $\theta ={{\sin }^{-1}}\left( 3x \right)$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the Maclaurin series for $\theta $, up to and including the term in ${{x}^{3}}$.</div><p style="text-align: right;">[5]</p></li></ol></li></ol>

In the triangle $ABC$, $AB=x$, $BC=y$, $AC=\frac{1}{6}$, angle $ABC=\frac{\pi }{6}$ radians and angle $ACB=\theta $ radians (see diagram).

Small Angle Approximations

2016 HCI P1 Q10

<p>It is given that&nbsp;$y=\mathrm{f}\left( x \right)$, where&nbsp;${{y}^{3}}+8=3xy$.</p>
<p>Find the Maclaurin series for&nbsp;$\mathrm{f}\left( x \right)$ up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[6]</span></p>
<p>Hence write down the equation of the normal at the point on the curve of $y=\mathrm{f}\left( x \right)$ at $x=0$.<span class="inline-marks">[1]</span></p>

It is given that $y=\mathrm{f}\left( x \right)$, where ${{y}^{3}}+8=3xy$.

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

Hence write down the equation of the normal at the point on the curve of $y=\mathrm{f}\left( x \right)$ at $x=0$.

Tangents and Normals to Implicit Curves

Maclaurin Series by Implicit Differentiation

2024 DHS P2 Q2

Applications of Differentiation

<p>It is given that&nbsp;$\ln \left( 1+y \right)=\tan x$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760706591850-i302r80vw" data-list-item-path="item-1760706591850-i302r80vw"><div class="list-item-paragraph">Show that&nbsp;$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\frac{\mathrm{d}y}{\mathrm{d}x}\left( 2\tan x+{{\sec }^{2}}x \right)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760706592383-z6af7y2id" data-list-item-path="item-1760706592383-z6af7y2id"><div class="list-item-paragraph">By further differentiation of the result in part <strong>(a)</strong>, find the Maclaurin series for&nbsp;$y$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760706593619-3s8feuomy" data-list-item-path="item-1760706593619-3s8feuomy"><div class="list-item-paragraph">Hence find an estimate for the value of $\mathrm{e}$.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $\ln \left( 1+y \right)=\tan x$.

2024 CJC P1 Q6

<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 $y={{3}^{\left( {{x}^{2}}+2x \right)}}$

2013 MI P1 Q2

<p>It is given that $y={{\mathrm{e}}^{-x}}\sin x+x-1$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=k{{\mathrm{e}}^{-x}}\cos x$, where $k$ is a constant to be determined.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">By further differentiation of this result, find the Maclaurin series for $y$, up to and including the term in ${{x}^{3}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">By using the result in part <strong>(b)</strong> and standard series from the List of Formulae (MF26), find the expansion of $\frac{{{\mathrm{e}}^{-x}}\sin x+x-1}{\cos 2x}$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$, giving the coefficients in exact form.</div><p style="text-align: right;">[3]</p></li></ol>

It is given that $y={{\mathrm{e}}^{-x}}\sin x+x-1$.

Product Rule

Products and Quotients of Power Series

2023 CJC P2 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 $y=\frac{1}{3+\sin 2x}$.

2023 ASRJC P1 Q8

<p>It is given that $\mathrm{f}\left( x \right)=\ln \left( \sin x+\cos x \right)$ for $-\frac{\pi }{4} < x\le \frac{\pi }{4}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)+{{\left[ \mathrm{{f}'}(x) \right]}^{2}}+1=0$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">By further differentiation of the result in part <strong>(i)</strong>, find the Maclaurin series for $\mathrm{f}\left( x \right)$ up to and including the term in ${{x}^{3}}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Using the expansion obtained in part <strong>(ii)</strong> and the standard series from the List of Formulae (MF26), find the series expansion of $\mathrm{g}\left( x \right)=\ln \left( 1+\tan x \right)$ for $-\frac{\pi }{4} < x\le \frac{\pi }{4}$ as far as the term in ${{x}^{3}}$.</div><p style="text-align: right;">[4]</p></li></ol>

It is given that $\mathrm{f}\left( x \right)=\ln \left( \sin x+\cos x \right)$ for $-\frac{\pi }{4} < x\le \frac{\pi }{4}$.

2018 AJC P1 Q5

<p>It is given that&nbsp;$y={{\sin }^{-1}}x$, $-1\le x\le 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760430006158-aony6z2na" data-list-item-path="item-1760430006158-aony6z2na"><div class="list-item-paragraph">Show that&nbsp;$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=x{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{3}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760430006939-x6lgv1h1z" data-list-item-path="item-1760430006939-x6lgv1h1z"><div class="list-item-paragraph">Using the result in part <strong>(a)</strong>, find the Maclaurin expansion for&nbsp;$y$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760430007540-7hyj4gagf" data-list-item-path="item-1760430007540-7hyj4gagf"><div class="list-item-paragraph">By substituting $x=0.5$ into the expansion found in part <strong>(b)</strong>, find an approximate value of $\pi $.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $y={{\sin }^{-1}}x$, $-1\le x\le 1$.

2023 CJC Promo Q5

<ol class="i-list"><li><div class="list-item-paragraph">Given that $y=\tan \left( {{\mathrm{e}}^{2x}}-1 \right)$, show that $\frac{\mathrm{d}y}{\mathrm{d}x}=k{{\mathrm{e}}^{2x}}\left( 1+{{y}^{2}} \right)$, where $k$ is to be found. Hence find the values of $\frac{\mathrm{d}y}{\mathrm{d}x}$, $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}$ and $\frac{{{\mathrm{d}}^{3}}y}{\mathrm{d}{{x}^{3}}}$ when $x=0$.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Write down the first three non-zero terms in the Maclaurin series for $\tan \left( {{\mathrm{e}}^{2x}}-1 \right)$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">The first two non-zero terms in the Maclaurin series for $\tan \left( {{\mathrm{e}}^{2x}}-1 \right)$ are equal to the first two non-zero terms in the series expansion of ${{\mathrm{e}}^{ax}}\ln \left( 1+nx \right)$. By using appropriate expansions from the List of Formulae (MF26), find the constants $a$ and $n$. Hence find the third non-zero term of the series expansion of ${{\mathrm{e}}^{ax}}\ln \left( 1+nx \right)$ for these values of $a$ and $n$.</div><p style="text-align: right;">[4]</p></li></ol>

RI Maclaurin Series Tutorial Q4

<ol class="abc-list"><li><div class="list-item-paragraph">Logarithmic functions are useful as measurement units for phenomena with large scale of values, such as decibels (for sound) or Richter scale (for earthquakes). Logarithmic functions make the measurement units smaller and easier to work with. An example of a logarithmic function is given as</div><div class="list-item-paragraph" style="text-align: center">$y=\ln \left( 1+2x+3{{x}^{2}} \right)$.</div><ol class="i-list"><li><div class="list-item-paragraph">By using differentiation, find the Maclaurin series for $y$, up to and including the term in ${{x}^{2}}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">By using the standard series in MF26, verify that the answer obtained in part <strong>(i)</strong> is correct.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-paragraph">Find the series expansion of $\frac{x}{\sqrt{4+x}}$, up to and including the term in ${{x}^{3}}$.</div><p style="text-align: right;">[4]</p></li></ol>

Math Question Bank