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H2 Math

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N(A) E Math

PSLE

PSLE Foundation Mathematics

**Do not use a calculator in answering this question.**

Small Angle Approximations

Approximations Using Power Series

Maclaurin Series from Differential Equations

Higher Derivatives by Implicit Differentiation

2026 RI P2 Q4

Techniques of Differentiation

Maclaurin Series and Power Series

<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

Substitution and Composition of Power Series

Derivatives of Reciprocal Trigonometric Functions

Accuracy of Power Series Approximations

2026 TJC P1 Q5

<p>It is given that&nbsp;$y=\sqrt{1+\ln(1+2x)}$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-0" data-list-item-path="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-0"><div class="list-item-paragraph"><p>Show that&nbsp;$y\left(\dfrac{ \mathrm{d}y}{ \mathrm{d}x}\right)=\dfrac{1}{1+2x}$.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-1" data-list-item-path="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-1"><div class="list-item-paragraph">By further differentiation, obtain the Maclaurin series of $y$, up to and including the term in $x^2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-2" data-list-item-path="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-2"><div class="list-item-paragraph"><p>Hence, find the Maclaurin series of $\dfrac{1}{\left(1+\ln(1+2x)\right)^{\frac{3}{2}}},$ up to and including the term in $x^2$.</p><span class="inline-marks">[3]</span></div></li></ol>

It is given that $y=\sqrt{1+\ln(1+2x)}$.

By further differentiation, obtain the Maclaurin series of $y$, up to and including the term in $x^2$.

Derivatives of Natural Logarithmic Functions

Chain Rule

2025 RVHS Promo Q8

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

Derivatives of Inverse Trigonometric Functions

Maclaurin Series by Successive Differentiation

Problems involving Calculus

Deriving Series by Differentiation and Integration

N2025 P1 Q9

<p>It is given that&nbsp;$y = \mathrm{e}^{\tan x}$, for&nbsp;$-\frac{\pi}{2} &lt; x &lt; \frac{\pi}{2}$, and $y_r = \frac{\mathrm{d}^r y}{\mathrm{d}x^r}$, for&nbsp;$r = 1, 2, 3$.</p>
<ol class="abc-list"><li data-list-item-id="part-a" data-list-item-path="part-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{y_1}{y} = 1 + \tan^2 x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="part-b" data-list-item-path="part-b"><div class="list-item-paragraph">By expressing $\ln y_1$ in terms of $\tan x$, show that&nbsp;$\frac{y_2}{y_1} = (1 + \tan x)^2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="part-c" data-list-item-path="part-c"><div class="list-item-paragraph">Find $\frac{y_3}{y_2}$ in terms of $\tan x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="part-d" data-list-item-path="part-d"><div class="list-item-paragraph">Hence obtain the Maclaurin series for&nbsp;$\mathrm{e}^{\tan x}$ in ascending powers of $x$ up to and including the term in $x^3$.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $y = \mathrm{e}^{\tan x}$, for $-\frac{\pi}{2} < x < \frac{\pi}{2}$, and $y_r = \frac{\mathrm{d}^r y}{\mathrm{d}x^r}$, for $r = 1, 2, 3$.

Derivatives of Trigonometric Functions

Logarithmic Differentiation

Derivatives of Exponential Functions

<p>It is given that&nbsp;$\mathrm{e}^y = 1 + \sin 3x$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that</div><div class="list-item-paragraph" style="text-align: center">$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 + 9 = 9\mathrm{e}^{-y}.$ </div><div class="list-item-paragraph">Hence find the series expansion of $y$ in ascending powers of $x$, up to and including the term in $x^3$, simplifying your answer.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Using standard series from the List of Formulae, verify that the series expansion obtained in <strong>part (a)</strong> is correct.<span class="inline-marks">[3]</span></div></li></ol>

It is given that $\mathrm{e}^y = 1 + \sin 3x$.

Implicit Differentiation

Maclaurin Series by Implicit Differentiation

2024 RI Promo Q6

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Given that&nbsp;$y = \sqrt{\ln(x + \mathrm{e})}$ for&nbsp;$x &gt; 1 - \mathrm{e}$, show that</div><div class="list-item-paragraph" style="text-align: center">\[2y\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{e}^{-y^2}.\]</div><div class="list-item-paragraph">Hence find the first three terms of the Maclaurin expansion of $y$. Give the coefficients as exact fractions in their simplest form.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Using standard series from the List of Formulae (MF27), expand $\ln(x + \mathrm{e})$ as far as the term in $x^2$.</div><div class="list-item-paragraph">Hence, verify the correctness of your expansion in <strong>(a)</strong>, showing your working clearly.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Use your expansion in part <strong>(a)</strong> to find an approximation to $\sqrt{\ln\left(\frac{1 + 10\mathrm{e}}{10}\right)}$, leaving your answer in terms of $\mathrm{e}$.<span class="inline-marks">[2]</span></div></li></ol>

Maclaurin Series Using Standard Series

2024 NYJC Promo Q8

<p>It is given that&nbsp;$y = \tan[\ln(1+2x)]$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$(1+2x)\frac{\mathrm{d}y}{\mathrm{d}x} = 2(1+y^2)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">By differentiating the result in part <strong>(a)</strong>, find the Maclaurin expansion of $\tan[\ln(1+2x)]$ up to and including the term in $x^2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Hence, find the Maclaurin expansion of $\mathrm{e}^{\frac{1}{2}y}$, up to and including the term in $x^2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Using your answer in part <strong>(c)</strong>, evaluate $\lim \limits_{x \to 0} \frac{\mathrm{e}^{\frac{1}{2}y}-1}{3x}$.<span class="inline-marks">[2]</span></div></li></ol>

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

2024 EJC Promo Q8

<p>It is given that&nbsp;$e^y = 1 + e^x$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 = \frac{\mathrm{d}y}{\mathrm{d}x}.$<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Hence find the first four non-zero terms of the Maclaurin series for&nbsp;$y$.</span><span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">Hence find the Maclaurin series for&nbsp;$\frac{e^{2x}}{1 + e^{2x}}$, up to and including the term in $x^3$.</span><span class="inline-marks">[3]</span></div></li></ol>

Show that \[\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 = \frac{\mathrm{d}y}{\mathrm{d}x}.\]

Hence find the first four non-zero terms of the Maclaurin series for $y$.

Hence find the Maclaurin series for $\frac{e^{2x}}{1 + e^{2x}}$, up to and including the term in $x^3$.

2025 YIJC P2 Q5

<p>It is given that&nbsp;$y = \ln \sqrt{1 + \mathrm{e}^{3x}}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$2\mathrm{e}^{2y} \frac{\mathrm{d}y}{\mathrm{d}x} = 3\mathrm{e}^{3x}$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><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 $x^2$, giving the coefficients in exact form.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">Deduce the Maclaurin series for&nbsp;$\ln \left( \frac{\sqrt{1 + \mathrm{e}^{-3x}}}{2} \right)$, up to and including the term in $x^2$.</span><span class="inline-marks">[2]</span></div></li></ol>

It is given that $y = \ln \sqrt{1 + \mathrm{e}^{3x}}$.

Show that $2\mathrm{e}^{2y} \frac{\mathrm{d}y}{\mathrm{d}x} = 3\mathrm{e}^{3x}$.

Deduce the Maclaurin series for $\ln \left( \frac{\sqrt{1 + \mathrm{e}^{-3x}}}{2} \right)$, up to and including the term in $x^2$.

2025 TJC P2 Q1

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">In a triangle $ABC$, $AB = 2$, angle $CAB = x$ radians and angle $CBA = \frac{\pi}{6}$ radians.</div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">Show that&nbsp;$AC = \frac{2}{\cos x + \sqrt{3} \sin x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">Given that&nbsp;$x$ is a sufficiently small angle, show that</div><div class="list-item-paragraph" style="text-align: center">\[AC \approx a + bx + cx^2,\]</div><div class="list-item-paragraph">where&nbsp;$a$, $b$ and $c$ are constants to be determined.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">It is given that&nbsp;$2xy + \ln y = \ln 3$. Show that</div><div class="list-item-paragraph" style="text-align: center">\[(2xy^2 + y)\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 4y^2\frac{\mathrm{d}y}{\mathrm{d}x} - \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 = 0.\]</div><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">[5]</span></div></li></ol>

2025 RI P2 Q2

<p>It is given that&nbsp;$y = \mathrm{e}^{2\sin^{-1} x}$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Show that&nbsp;$\sqrt{1 - x^2} \frac{\mathrm{d}y}{\mathrm{d}x} = 2y$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph"><p>By further differentiation of the result in part <strong>(i)</strong>, find the Maclaurin series for&nbsp;$y$ up to and including the term in $x^2$.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Using your result from part <strong>(ii)</strong>, find an approximate value for&nbsp;$\displaystyle \displaystyle\int_{0}^{0.1} x\mathrm{e}^{2\sin^{-1} x} \, \mathrm{d}x$, giving your answer to $4$ significant figures.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $y = \mathrm{e}^{2\sin^{-1} x}$.

Show that $\sqrt{1 - x^2} \frac{\mathrm{d}y}{\mathrm{d}x} = 2y$.

2025 MI P1 Q8

<p>It is given that&nbsp;$y = \ln \left[ \sin \left( x + \frac{\pi}{4} \right) \right]$, where&nbsp;$-\frac{\pi}{4} &lt; x &lt; \frac{3\pi}{4}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)^2 + 1 = 0$. Hence find the first four non-zero terms of the Maclaurin expansion of $y$, leaving your answer in exact form.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph"><p>Verify the result obtained in part <strong>(a)</strong> using standard series from the List of Formulae (MF27).</p><span class="inline-marks">[5]</span></div></li></ol>

It is given that $y = \ln \left[ \sin \left( x + \frac{\pi}{4} \right) \right]$, where $-\frac{\pi}{4} < x < \frac{3\pi}{4}$.

Show that $\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)^2 + 1 = 0$. Hence find the first four non-zero terms of the Maclaurin expansion of $y$, leaving your answer in exact form.

Second and Higher Derivatives

2025 HCI P2 Q2

<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 $b$.

2025 DHS Promo Q6

<p>Given that&nbsp;$y={{({{\sin }^{-1}}x)}^{2}}$, show that&nbsp;$(1-{{x}^{2}}){{\left(\frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}=4y$ and $(1-{{x}^{2}})\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-x\frac{\mathrm{d}y}{\mathrm{d}x}=2$.</p>

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

<p>Given that&nbsp;$y = \sin^{-1}x$, show that&nbsp;$(\cos y)\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} - (\sin y)\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2} = 0$.</p>

Given that $y = \sin^{-1}x$, show that $(\cos y)\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} - (\sin y)\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2} = 0$.

<p>Differentiate the following equations with respect to $x$:</p>
<ol class="i-list"><li><div class="list-item-paragraph">$x\frac{\mathrm{d}y}{\mathrm{d}x} = \cos(xy) + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}$</div></li><li><div class="list-item-paragraph">$y\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2} = \cos(xy) + \tan z$</div></li></ol>

Differentiate the following equations with respect to $x$:

<p>The curve $C$ has equation $\ln {{y}^{2}}={{x}^{2}}y$, where&nbsp;$x\in \mathbb{R}$, $-1\le y&lt;0$.</p>
<p>It is given that&nbsp;$C$ has only one turning point.</p>
<ol class="abc-list"><li data-list-item-id="item-1761023606800-n7ncdbt60" data-list-item-path="item-1761023606800-n7ncdbt60"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}\left( 2-{{x}^{2}}y \right)=2x{{y}^{2}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761023607242-sn64qwfph" data-list-item-path="item-1761023607242-sn64qwfph"><div class="list-item-paragraph">Find the coordinates of the turning point.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761023607962-itvendbfa" data-list-item-path="item-1761023607962-itvendbfa"><div class="list-item-paragraph">Calculate the value of $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}$ at the turning point and determine whether the turning point is a maximum or a minimum.<span class="inline-marks">[3]</span></div></li></ol>

The curve $C$ has equation $\ln {{y}^{2}}={{x}^{2}}y$, where $x\in \mathbb{R}$, $-1\le y<0$.

It is given that $C$ has only one turning point.

Stationary Points and Their Nature

2024 HCI Promo Q6

Applications of Differentiation

<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 $y=\ln \left( 2-{{\mathrm{e}}^{-2x}} \right)$.

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>

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