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<ol class="abc-list"><li data-list-item-id="item-1759818545285-ebbtrex2a" data-list-item-path="item-1759818545285-ebbtrex2a"><ol class="i-list"><li data-list-item-id="subitem-1759818547640-sfehueh22" data-list-item-path="item-1759818545285-ebbtrex2a/subitem-1759818547640-sfehueh22"><div class="list-item-paragraph">Given that&nbsp;$y=\cos \left( {{\mathrm{e}}^{2x}}-1 \right)$, show that&nbsp;$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-2\frac{\mathrm{d}y}{\mathrm{d}x}+4y{{\mathrm{e}}^{4x}}=0$. By further differentiation of this result, find the Maclaurin series of the function $\cos \left( {{\mathrm{e}}^{2x}}-1 \right)$ up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[7]</span></div></li><li data-list-item-id="subitem-1759818549013-mjvmt0crz" data-list-item-path="item-1759818545285-ebbtrex2a/subitem-1759818549013-mjvmt0crz"><div class="list-item-paragraph">Verify the correctness of your answer in part (ai) by using the standard series for&nbsp;${{\mathrm{e}}^{x}}$ and $\cos x$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1759818546725-5curbb5da" data-list-item-path="item-1759818546725-5curbb5da"><ol class="i-list"><li data-list-item-id="subitem-1759818566953-2rhdpmtbr" data-list-item-path="item-1759818546725-5curbb5da/subitem-1759818566953-2rhdpmtbr"><div class="list-item-paragraph">Explain why it is not possible to obtain a Maclaurin series for&nbsp;$\ln x$ .<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1759818567830-spjcun9uk" data-list-item-path="item-1759818546725-5curbb5da/subitem-1759818567830-spjcun9uk"><div class="list-item-paragraph">A Taylor series is an expansion of a real function $\mathrm{f}\left( x \right)$ about a point $x=a$ and it is defined by<br>\[\mathrm{f}\left( x \right)=\mathrm{f}\left( a \right)+\mathrm{f}'\left( a \right)\left( x-a \right)+\frac{\mathrm{f}''\left( a \right)}{2!}{{\left( x-a \right)}^{2}}+...+\frac{{\mathrm{f}^{\left( n \right)}}\left( a \right)}{n!}{{\left( x-a \right)}^{n}}+... \]</div><div class="list-item-paragraph">where&nbsp;${{\mathrm{f}}^{\left( n \right)}}\left( a \right)$ is the value of the $n$th derivative of $\mathrm{f}\left( x \right)$ when&nbsp;$x=a$.</div><div class="list-item-paragraph">Given that&nbsp;$\mathrm{f}\left( x \right)=\ln x$, find $\mathrm{f}\left( 1 \right)$, $\mathrm{f}'\left( 1 \right)$ and $\mathrm{f}''\mathrm{ }\left( 1 \right)$. Hence write down the Taylor series for&nbsp;$\mathrm{f}\left( x \right)$ about the point $x=1$, giving your answer in the form of $\ln x=b+c\left( x-a \right)+d{{\left( x-a \right)}^{2}}+...$, where&nbsp;$a$, $b$, $c$ and $d$ are real constants.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Maclaurin Series Using Standard Series

Maclaurin Series from Differential Equations

Derivatives of Trigonometric Functions

Substitution and Composition of Power Series

Derivatives of Natural Logarithmic Functions

Existence of a Maclaurin Expansion

Maclaurin Series by Successive Differentiation

Higher Derivatives by Implicit Differentiation

Second and Higher Derivatives

Derivatives of Exponential Functions

Chain Rule

Taylor Expansions about a Nonzero Point

2018 MJC Promo Q8

Maclaurin Series and Power Series

Techniques of Differentiation

<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.

Series from Products, Substitution, Integration and Differentiation

Approximations Using Power Series

Accuracy of Power Series Approximations

Deriving Series by Differentiation and Integration

N2019 P2 Q4

<ol class="abc-list"><li data-list-item-id="item-1759215319532-l5rfxu6ty" data-list-item-path="item-1759215319532-l5rfxu6ty"><div class="list-item-paragraph">Expand ${{\left( b-\frac{x}{2} \right)}^{n}}$ in ascending powers of $x$, where&nbsp;$b$ is a positive constant and $n$ is a negative constant, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">It is given that the coefficient of $x$ is four times the coefficient of ${{x}^{2}}$ and the constant term in the expansion is $\frac{1}{2}$. Find the integer values of $b$ and $n$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759215320139-999u6fsuj" data-list-item-path="item-1759215320139-999u6fsuj"><ol class="i-list"><li data-list-item-id="subitem-1759215344342-kg0sbhw0v" data-list-item-path="item-1759215320139-999u6fsuj/subitem-1759215344342-kg0sbhw0v"><div class="list-item-paragraph">Explain why it is not possible to obtain a Maclaurin series for&nbsp;$\ln \left( 2{{x}^{2}} \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1759215362042-cxfoul2s3" data-list-item-path="item-1759215320139-999u6fsuj/subitem-1759215362042-cxfoul2s3"><div class="list-item-content"><span class="list-item-text">A Taylor series is an expansion of a real function $\mathrm{f}\left( x \right)$ about a point $x=a$ and it is defined by</span></div><div class="list-item-paragraph" style="text-align: center">$\mathrm{f}\left( x \right)=\mathrm{f}\left( a \right)+\mathrm{f}'\left( a \right)\left( x-a \right)+\frac{\mathrm{f}''\left( a \right)}{2!}{{\left( x-a \right)}^{2}}+...+\frac{{{\mathrm{f}}^{\left( n \right)}}\left( a \right)}{n!}{{\left( x-a \right)}^{n}}+...$</div><div class="list-item-paragraph">where&nbsp;${{\mathrm{f}}^{\left( n \right)}}\left( a \right)$ is the value of the $n$th derivative of $\mathrm{f}\left( x \right)$ when&nbsp;$x=a$.</div><div class="list-item-paragraph">Find the first three exact non-zero terms of the Taylor series for&nbsp;$\ln \left( 2{{x}^{2}} \right)$ about the point $x=2$.</div><div class="list-item-paragraph">[You need not simplify your answer.]<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Expand ${{\left( b-\frac{x}{2} \right)}^{n}}$ in ascending powers of $x$, where $b$ is a positive constant and $n$ is a negative constant, up to and including the term in ${{x}^{2}}$.

Finding Parameters from Power Series Coefficients

Binomial Expansion as a Maclaurin Series

2021 TMJC P2 Q4

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