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

<p>A particle moves back and forth in a horizontal line between two end points. A fixed origin \(O\) is the midpoint between these end points. Its displacement \(x\) metres from \(O\) at time \(t\) seconds is \(x=4\sin(2t+\frac\pi3)\), where&nbsp;\(0\leq t\leq\pi\).</p>
<ol class="abc-list"><li data-list-item-id="p3-tzc-q02-item01" data-list-item-path="p3-tzc-q02-item01"><div class="list-item-paragraph">Determine</div><ol class="i-list"><li data-list-item-id="p3-tzc-q02-sub01-item01" data-list-item-path="p3-tzc-q02-item01/p3-tzc-q02-sub01-item01"><div class="list-item-paragraph">the amplitude;<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p3-tzc-q02-sub01-item02" data-list-item-path="p3-tzc-q02-item01/p3-tzc-q02-sub01-item02"><div class="list-item-paragraph">the initial displacement;<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p3-tzc-q02-sub01-item03" data-list-item-path="p3-tzc-q02-item01/p3-tzc-q02-sub01-item03"><div class="list-item-paragraph">the value of \(t\) when the particle first passes through \(O\).<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="p3-tzc-q02-item02" data-list-item-path="p3-tzc-q02-item02"><div class="list-item-paragraph">The acceleration \(a\) at displacement \(s\) satisfies \(a=-n^2s\), where&nbsp;\(n&gt;0\).</div><div class="list-item-paragraph">The displacement is \(s=A\sin(nt+b)\), where&nbsp;\(t\geq0\), \(A,n&gt;0\) and \(-\pi\leq b\leq\pi\).</div><div class="list-item-paragraph">By finding expressions for&nbsp;\(\frac{\mathrm{d}s}{\mathrm{d}t}\) and \(\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\), verify that&nbsp;\(s=A\sin(nt+b)\) satisfies \(a=-n^2s\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p3-tzc-q02-item03" data-list-item-path="p3-tzc-q02-item03"><ol class="i-list"><li data-list-item-id="p3-tzc-q02-sub03-item01" data-list-item-path="p3-tzc-q02-item03/p3-tzc-q02-sub03-item01"><div class="list-item-paragraph">Use the chain rule to show that&nbsp;\(a=v\frac{\mathrm{d}v}{\mathrm{d}s}\), where&nbsp;\(v\) is velocity.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p3-tzc-q02-sub03-item02" data-list-item-path="p3-tzc-q02-item03/p3-tzc-q02-sub03-item02"><div class="list-item-paragraph">By solving \(v\frac{\mathrm{d}v}{\mathrm{d}s}=-n^2s\), show that&nbsp;\(v^2=n^2(A^2-s^2)\).<span class="inline-marks">[5]</span></div></li><li data-list-item-id="p3-tzc-q02-sub03-item03" data-list-item-path="p3-tzc-q02-item03/p3-tzc-q02-sub03-item03"><div class="list-item-paragraph">Hence, or otherwise, find the particle’s maximum speed.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="p3-tzc-q02-item04" data-list-item-path="p3-tzc-q02-item04"><div class="list-item-paragraph">The continuous random variable \(S\) denotes the particle’s displacement \(s\) from \(O\) at time \(t\).</div><div class="list-item-paragraph">\[f(s)=\begin{cases}\frac1{\pi\sqrt{A^2-s^2}},&-A&lt;s&lt;A,\\0,&\text{otherwise}.\end{cases}\]</div><div class="list-item-paragraph">Show that&nbsp;\(P(-\frac A2\leq S\leq\frac A2)=\frac13\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="p3-tzc-q02-item05" data-list-item-path="p3-tzc-q02-item05"><div class="list-item-paragraph">For \(-A&lt;s&lt;A\), \(f(s)\) can be expressed as \(\frac{k}{|v(s)|}\), where \(k&gt;0\) and \(v(s)\) is the particle’s velocity at displacement \(s\) from \(O\).</div><div class="list-item-paragraph">Find the value of \(k\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p3-tzc-q02-item06" data-list-item-path="p3-tzc-q02-item06"><ol class="i-list"><li data-list-item-id="p3-tzc-q02-sub06-item01" data-list-item-path="p3-tzc-q02-item06/p3-tzc-q02-sub06-item01"><div class="list-item-paragraph">Determine \(E(S)\), justifying your answer.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p3-tzc-q02-sub06-item02" data-list-item-path="p3-tzc-q02-item06/p3-tzc-q02-sub06-item02"><div class="list-item-paragraph">Interpret the result in the context of the particle’s motion.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

A particle moves back and forth in a horizontal line between two end points. A fixed origin \(O\) is the midpoint between these end points. Its displacement \(x\) metres from \(O\) at time \(t\) seconds is \(x=4\sin(2t+\frac\pi3)\), where \(0\leq t\leq\pi\).

Solving Differential Equations by Separation of Variables

Given the Displacement Function

Derivatives of Trigonometric Functions

Probability Density Functions

Expectation and Variance of Continuous Random Variables

Second and Higher Derivatives

Product, Quotient and Chain Rules

Probabilities by Integrating a Density Function

IB AA HL May 2026 P3 TZC Q2

Techniques of Differentiation

Continuous Random Variables

Differential Equations

Kinematics

<p>A particle moves back and forth in a horizontal line between two end points. A fixed origin \(O\) is the midpoint between these end points. Its displacement \(x\) metres from \(O\) at time \(t\) seconds is \(x=4\sin(2t+\frac\pi3)\), where&nbsp;\(0\leq t\leq\pi\).</p>
<ol class="abc-list"><li data-list-item-id="h2-motion-3" data-list-item-path="h2-motion-3"><div class="list-item-paragraph">Determine</div><ol class="i-list"><li data-list-item-id="h2-motion-4" data-list-item-path="h2-motion-3/h2-motion-4"><div class="list-item-paragraph">the amplitude;<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-motion-6" data-list-item-path="h2-motion-3/h2-motion-6"><div class="list-item-paragraph">the initial displacement;<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-motion-8" data-list-item-path="h2-motion-3/h2-motion-8"><div class="list-item-paragraph">the value of \(t\) when the particle first passes through \(O\).<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="h2-motion-11" data-list-item-path="h2-motion-11"><div class="list-item-paragraph">More generally, a particle moves along a straight line with displacement \(s=A\sin(nt+b)\) metres at time \(t\) seconds, where&nbsp;\(t\geq0\), \(A,n&gt;0\) and \(-\pi\leq b\leq\pi\).</div><div class="list-item-paragraph">Its velocity and acceleration are \(v=\frac{\mathrm{d}s}{\mathrm{d}t}\) and \(a=\frac{\mathrm{d}v}{\mathrm{d}t}\), respectively.</div><div class="list-item-paragraph">Find expressions for&nbsp;\(\frac{\mathrm{d}s}{\mathrm{d}t}\) and \(\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\), and hence show that&nbsp;\(a=-n^2s\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-motion-15" data-list-item-path="h2-motion-15"><ol class="i-list"><li data-list-item-id="h2-motion-16" data-list-item-path="h2-motion-15/h2-motion-16"><div class="list-item-paragraph">Use the chain rule to show that&nbsp;\(a=v\frac{\mathrm{d}v}{\mathrm{d}s}\), where&nbsp;\(v\) is velocity.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-motion-18" data-list-item-path="h2-motion-15/h2-motion-18"><div class="list-item-paragraph">The particle is instantaneously at rest when&nbsp;\(s=A\). By separating the variables in \(v\frac{\mathrm{d}v}{\mathrm{d}s}=-n^2s\), show that&nbsp;\(v^2=n^2(A^2-s^2)\).</div></li><li data-list-item-id="h2-motion-20" data-list-item-path="h2-motion-15/h2-motion-20"><div class="list-item-paragraph">Hence, or otherwise, find the particle’s maximum speed.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Verifying Solutions of Differential Equations

Chain Rule

IB AA HL May 2026 P3 TZC Q2 (H2 adaptation)

<p>The asteroid, a curve \(C\) which is used to characterize various properties of energy and magnetism, has parametric equations \[x = a \cos^3 t, y = a \sin^3 t,\] where&nbsp;\(0 \le t \le \frac{1}{2}\pi\) and \(a\) is a positive constant.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find the equation of the tangent to \(C\) at the point \(P\) with parameter \(p\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><div class="list-item-paragraph">The tangent at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\). Show that the length \(AB\) depends only on \(a\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-2" data-list-item-path="imported-item-block-3-item-2"><div class="list-item-paragraph">Find a cartesian equation of \(C\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-3-item-3" data-list-item-path="imported-item-block-3-item-3"><div class="list-item-paragraph">The region bounded by \(C\) and the \(x\)- and \(y\)-axes is rotated through \(360^\circ\) about the \(y\)-axis. Find the exact value of the volume of revolution of the solid formed.<span class="inline-marks">[4]</span></div></li></ol>

The asteroid, a curve \(C\) which is used to characterize various properties of energy and magnetism, has parametric equations \[x = a \cos^3 t, y = a \sin^3 t,\] where \(0 \le t \le \frac{1}{2}\pi\) and \(a\) is a positive constant.

Find the equation of the tangent to \(C\) at the point \(P\) with parameter \(p\).

The tangent at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\). Show that the length \(AB\) depends only on \(a\).

The region bounded by \(C\) and the \(x\)- and \(y\)-axes is rotated through \(360^\circ\) about the \(y\)-axis. Find the exact value of the volume of revolution of the solid formed.

Tangents and Normals to Parametric Curves

Volume of Revolution about the y-axis

Differentiation of Parametric Equations

Converting Parametric Equations to Cartesian Form

Graphing Techniques

Applications of Integration

Applications of Differentiation

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<p>Engineers are designing a roller coaster. The diagram shows the side view of a section of track with a loop. This section is modelled by the curve with parametric equations</p>
<p style="text-align: center">\(x = 28(\theta \cos \theta + 1), \quad y = 20(2 - \theta \sin \theta),\)</p>
<p>for&nbsp;\(-\frac{2}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\), where&nbsp;\(x\) and \(y\) are measured in metres. The \(x\)-axis models the horizontal ground.</p>
<p>The maximum point on the curve is at \(A\), and the line \(AB\) is vertical. Points \(C\) and \(D\) are where the tangent to the curve is parallel to the \(y\)-axis. Point \(E\) is the intersection of the lines \(AB\) and \(CD\).</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(\theta\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><ol class="i-list"><li data-list-item-id="imported-item-block-3-item-1-sub-0" data-list-item-path="imported-item-block-3-item-1/imported-item-block-3-item-1-sub-0"><div class="list-item-paragraph">Find the values of \(\theta\) at points \(C\) and \(D\), giving your answers correct to \(2\) significant figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1-sub-1" data-list-item-path="imported-item-block-3-item-1/imported-item-block-3-item-1-sub-1"><div class="list-item-paragraph">Hence find the greatest width, \(CD\), of the loop.<span class="inline-marks">[2]</span></div></li></ol></li></ol>
<p>The \(x\)-coordinate of point \(A\) is \(28\).</p>
<ol class="abc-list" start="3"><li data-list-item-id="imported-item-block-5-item-0" data-list-item-path="imported-item-block-5-item-0"><ol class="i-list"><li data-list-item-id="imported-item-block-5-item-0-sub-0" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-0"><div class="list-item-paragraph">Find the value of \(\theta\) at point \(A\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-5-item-0-sub-1" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-1"><div class="list-item-paragraph">Find the \(2\) possible values of \(\theta\) at point \(B\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-5-item-0-sub-2" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-2"><div class="list-item-paragraph">Hence find the distance \(AB\).<span class="inline-marks">[1]</span></div></li></ol></li></ol>
<p>Engineers know that, for safety reasons, the distance \(AE\) should be within \(10\%\) of the distance \(\frac{1}{2}CD\).</p>
<ol class="abc-list" start="4"><li data-list-item-id="imported-item-block-7-item-0" data-list-item-path="imported-item-block-7-item-0"><div class="list-item-paragraph">Determine whether the design of this section of track satisfies this condition.<span class="inline-marks">[2]</span></div></li></ol>

Engineers are designing a roller coaster. The diagram shows the side view of a section of track with a loop. This section is modelled by the curve with parametric equations

\(x = 28(\theta \cos \theta + 1), \quad y = 20(2 - \theta \sin \theta),\)

for \(-\frac{2}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\), where \(x\) and \(y\) are measured in metres. The \(x\)-axis models the horizontal ground.

The maximum point on the curve is at \(A\), and the line \(AB\) is vertical. Points \(C\) and \(D\) are where the tangent to the curve is parallel to the \(y\)-axis. Point \(E\) is the intersection of the lines \(AB\) and \(CD\).

Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(\theta\).

The \(x\)-coordinate of point \(A\) is \(28\).

Engineers know that, for safety reasons, the distance \(AE\) should be within \(10\%\) of the distance \(\frac{1}{2}CD\).

Determine whether the design of this section of track satisfies this condition.

Tangents and Normals Parallel to the Axes

Product Rule

Parametric Equations

N2025 P1 Q11

<ol class="i-list"><li data-list-item-id="amint-q14-question-1" data-list-item-path="amint-q14-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin4x-4x)=-8\sin^22x\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q14-question-2" data-list-item-path="amint-q14-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/8}\sin^22x\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin4x-4x)=-8\sin^22x\).

Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/8}\sin^22x\,\mathrm{d}x\).

Definite Integration as the Reverse of Differentiation

Squared-sine integral from a derivative identity

Integration

Differentiation

<ol class="i-list"><li data-list-item-id="amint-q23-question-1" data-list-item-path="amint-q23-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\sin3x)=3\cot3x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q23-question-2" data-list-item-path="amint-q23-question-2"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\cos3x)=-3\tan3x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q23-question-3" data-list-item-path="amint-q23-question-3"><div class="list-item-paragraph">Show that&nbsp;\(\cot6x=\dfrac12(\cot3x-\tan3x)\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q23-question-4" data-list-item-path="amint-q23-question-4"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_{\pi/18}^{\pi/12}\cot6x\,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\sin3x)=3\cot3x\).

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\cos3x)=-3\tan3x\).

Show that \(\cot6x=\dfrac12(\cot3x-\tan3x)\).

Hence evaluate the exact value of \(\displaystyle\int_{\pi/18}^{\pi/12}\cot6x\,\mathrm{d}x\).

Derivatives of Natural Logarithmic Functions

Proving and Simplifying Trigonometric Identities

Cotangent integral from logarithmic derivatives

Trigonometry

<ol class="i-list"><li data-list-item-id="amint-q15-question-1" data-list-item-path="amint-q15-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin3x-3x)=-6\sin^2\!\left(\dfrac{3x}{2}\right)\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q15-question-2" data-list-item-path="amint-q15-question-2"><div class="list-item-paragraph">Without using a calculator, find the values of the constants \(a\), \(b\) and \(c\) for which \(\displaystyle\int_0^{\pi/9}\left[\sin^2\!\left(\dfrac{3x}{2}\right)-\sin3x\right]\,\mathrm{d}x=a\sqrt3+b\pi+c\).<span class="inline-marks">[5]</span></div></li></ol>
<p>The coefficients in the requested exact form are rational numbers.</p>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin3x-3x)=-6\sin^2\!\left(\dfrac{3x}{2}\right)\).

Without using a calculator, find the values of the constants \(a\), \(b\) and \(c\) for which \(\displaystyle\int_0^{\pi/9}\left[\sin^2\!\left(\dfrac{3x}{2}\right)-\sin3x\right]\,\mathrm{d}x=a\sqrt3+b\pi+c\).

The coefficients in the requested exact form are rational numbers.

Problems involving Unknowns

Exact constants from a trigonometric integral

<ol class="i-list"><li data-list-item-id="amint-q13-question-1" data-list-item-path="amint-q13-question-1"><div class="list-item-paragraph">Differentiate \(x\tan x\) with respect to \(x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q13-question-2" data-list-item-path="amint-q13-question-2"><div class="list-item-paragraph">Differentiate \(\ln(\cos x)\) with respect to \(x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q13-question-3" data-list-item-path="amint-q13-question-3"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/4}x\sec^2x\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Differentiate \(x\tan x\) with respect to \(x\).

Differentiate \(\ln(\cos x)\) with respect to \(x\).

Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/4}x\sec^2x\,\mathrm{d}x\).

Tangent product and logarithmic derivative

<ol class="i-list"><li data-list-item-id="amint-q10-question-1" data-list-item-path="amint-q10-question-1"><div class="list-item-paragraph">Differentiate \(x\cos3x\) with respect to \(x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q10-question-2" data-list-item-path="amint-q10-question-2"><div class="list-item-paragraph">Hence evaluate \(\displaystyle\int_0^{\pi/6}x\sin3x\,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Differentiate \(x\cos3x\) with respect to \(x\).

Hence evaluate \(\displaystyle\int_0^{\pi/6}x\sin3x\,\mathrm{d}x\).

Product derivative with a triple angle

<ol class="i-list"><li data-list-item-id="amint-q18-question-1" data-list-item-path="amint-q18-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[\mathrm{e}^{-2x}(\cos x+2\sin x)\right]=-5\mathrm{e}^{-2x}\sin x\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q18-question-2" data-list-item-path="amint-q18-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/4}\mathrm{e}^{-2x}\sin x\,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[\mathrm{e}^{-2x}(\cos x+2\sin x)\right]=-5\mathrm{e}^{-2x}\sin x\).

Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/4}\mathrm{e}^{-2x}\sin x\,\mathrm{d}x\).

Derivatives of Exponential Functions

Exponential and trigonometric product

<ol class="i-list"><li data-list-item-id="amint-q11-question-1" data-list-item-path="amint-q11-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\sin2x}{1+\cos2x}\right)=\dfrac{2}{1+\cos2x}\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q11-question-2" data-list-item-path="amint-q11-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/6}\dfrac{1}{1+\cos2x}\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\sin2x}{1+\cos2x}\right)=\dfrac{2}{1+\cos2x}\).

Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/6}\dfrac{1}{1+\cos2x}\,\mathrm{d}x\).

Quotient Rule

Sine quotient with a double angle

<ol class="i-list"><li data-list-item-id="amint-q12-question-1" data-list-item-path="amint-q12-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\sin3x}{1-\cos3x}\right)=-\dfrac{3}{1-\cos3x}\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q12-question-2" data-list-item-path="amint-q12-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_{\pi/12}^{\pi/6}\dfrac{1}{1-\cos3x}\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\sin3x}{1-\cos3x}\right)=-\dfrac{3}{1-\cos3x}\).

Hence evaluate the exact value of \(\displaystyle\int_{\pi/12}^{\pi/6}\dfrac{1}{1-\cos3x}\,\mathrm{d}x\).

Sine quotient with a triple angle

<p>Differentiate \(y=\mathrm{e}^{3x}\cos 2x\).</p>

Differentiate \(y=\mathrm{e}^{3x}\cos 2x\).

Product of exponential and trigonometric functions

<p>The diagram shows part of the graph \(y = x^{\cos 2x}\), for&nbsp;\(0 \le x \le 5\), which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by \(x\) units and its vertical distance travelled is denoted by \(y\) units.</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/pasted/1783573222728-q8fnjduy6xq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODM1NzMyMjI3MjgtcThmbmpkdXk2eHEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTE2NCwiZXhwIjoxNzkyNzg5NDY0fQ.BeSntq9ri816CbD69DrN6b1jFc3UDKYCz8dDqt-n-Ig" style="max-width: min(280px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<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;\(\frac{\mathrm{d}y}{\mathrm{d}x} = x^{\cos 2x} \left[ \frac{\cos 2x}{x} - (2 \sin 2x) \ln x \right]\).<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">At the point on the graph where&nbsp;\(x = \pi\), find the rate at which the roller coaster is moving vertically when it is moving horizontally at a rate of \(8\) units per hour.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the acute angle that the tangent to the graph where&nbsp;\(x = \pi\) makes with the horizontal.<span class="inline-marks">[1]</span></div></li></ol>

The diagram shows part of the graph \(y = x^{\cos 2x}\), for \(0 \le x \le 5\), which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by \(x\) units and its vertical distance travelled is denoted by \(y\) units.

Logarithmic Differentiation

Derivative as the Gradient of a Curve

Connected Rates of Change

2024 ACJC P2 Q1

<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 \(r = 1, 2, 3\).

Maclaurin Series from Differential Equations

Higher Derivatives by Implicit Differentiation

Maclaurin Series and Power Series

<p>Evaluate the following: </p>
<ol class="abc-list"><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(4^{2x}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(2^{x^2}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(5^{\ln(2x)}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(\pi^{\sin 2x^3}\right)\)</div></li></ol>

<p>Given that</p>
<p style="text-align: center">\[y = A \cos [\ln(x+1)] + B \sin [\ln(x+1)],\]</p>
<p>where&nbsp;\(A\) and \(B\) are constants with \(x &gt; -1\).</p>
<p>Show that&nbsp;\((x+1)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (x+1) \frac{\mathrm{d}y}{\mathrm{d}x} + y = 0\).<span class="inline-marks">[5]</span></p>

\[y = A \cos [\ln(x+1)] + B \sin [\ln(x+1)],\]

where \(A\) and \(B\) are constants with \(x > -1\).

Show that \((x+1)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (x+1) \frac{\mathrm{d}y}{\mathrm{d}x} + y = 0\).

2018 AJC C1BT1 Q3

<p>Find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(5^{\tan x}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(2^{x^2}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(3^{\ln x}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(\pi^{\sin 2x^3}\right)\)</div></li></ol>

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><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">State the derivative of \(\mathrm{e}^{\tan x}\).<span class="inline-marks">[1]</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">Hence, find \(\int \mathrm{e}^{\tan x} (\sec^3 x)(\sin x) \,\mathrm{d}x\).<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Write down constants \(A\) and \(B\) such that, for all values of \(x\), \(1 - 4x = A(3 - 8x) + B\). Hence, find </div><div class="list-item-paragraph"><p>\(\int \frac{1 - 4x}{\sqrt{1 - 4x^2 + 3x}} \,\mathrm{d}x.\)</p><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-paragraph">Use the substitution \(u = \sqrt{1 + x}\) to find the exact value of \(\int_{1}^{3} \frac{x^2}{\left(\sqrt{1 + x}\right)^3} \,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Integration by Parts

Integration as an Inverse Process of Differentiation

Integration by Substitution

2024 VJC Promo Q10

Techniques of Integration

<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>

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