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 curve $C$ has equation $y=x^{-3}\ln x$.</p>
<ol class="abc-list"><li data-list-item-id="p1q7-q-b2-i1" data-list-item-path="p1q7-q-b2-i1"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1-3\ln x}{x^4}$ and hence find the coordinates of the turning point of $C$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="p1q7-q-b2-i2" data-list-item-path="p1q7-q-b2-i2"><div class="list-item-paragraph">Find the exact area enclosed by $C$, the $x$-axis and the line $x=3$.<span class="inline-marks">[5]</span></div></li></ol>

A curve $C$ has equation $y=x^{-3}\ln x$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1-3\ln x}{x^4}$ and hence find the coordinates of the turning point of $C$.

Find the exact area enclosed by $C$, the $x$-axis and the line $x=3$.

Derivatives of Natural Logarithmic Functions

Product Rule

Area Bounded by Curves

Stationary Points and Their Nature

N2022 P1 Q7

Techniques of Differentiation

Applications of Integration

Applications of Differentiation

Techniques of Integration

<p>The variables $x$ and $y$ satisfy the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=y^3-\frac{y}{x}$, where&nbsp;$x,y&gt;0$. It is given that&nbsp;$y=2$ when&nbsp;$x=1$.</p>
<p>Euler's method is a numerical method for estimating a solution of a differential equation. Over a small horizontal step of length $h$, it uses a straight line with the gradient given by the differential equation at the current estimated point.</p>
<p>The estimated change in $y$ is therefore&nbsp;$h\times\text{current gradient}$. Let&nbsp;$y_n$ be the estimate of $y$ at $x=x_n$, with $x_0=1$ and $y_0=2$. For this differential equation, the rule is</p>
<p>\[x_{n+1}=x_n+h,\qquad y_{n+1}=y_n+h\left(y_n^3-\frac{y_n}{x_n}\right).\]</p>
<p>At each step, calculate the gradient using the current values of $x$ and $y$, then add $h$ times this gradient to the current estimate of $y$. Recalculate the gradient at the new estimated point before repeating. Keep unrounded values in subsequent calculations.</p>
<ol class="abc-list"><li data-list-item-id="p2-tzc-q12-item01" data-list-item-path="p2-tzc-q12-item01"><div class="list-item-paragraph">Take $h=0.01$. First calculate the gradient at $(1,2)$. Use this gradient in the Euler formula to show that the first estimate, at $x_1=1.01$, is $y_1=2.06$.</div><div class="list-item-paragraph">Then calculate $y_2$, showing your substitution and giving its value to 5 decimal places. Continue the process to estimate $y$ when&nbsp;$x=1.05$, giving this estimate to 3 significant figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p2-tzc-q12-item02" data-list-item-path="p2-tzc-q12-item02"><div class="list-item-paragraph">Using the substitution $u=xy$, show that the differential equation can be written as&nbsp;$\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{u^3}{x^2}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p2-tzc-q12-item03" data-list-item-path="p2-tzc-q12-item03"><div class="list-item-paragraph">Hence, by separating the variables and using the given initial condition, solve the differential equation, giving $y$ as a function of $x$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="p2-tzc-q12-item04" data-list-item-path="p2-tzc-q12-item04"><div class="list-item-paragraph">State the largest possible domain of this solution containing $x=1$, and its corresponding range.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p2-tzc-q12-item05" data-list-item-path="p2-tzc-q12-item05"><div class="list-item-paragraph">By considering the shape of the solution curve for&nbsp;$1\leq x\leq1.05$ and the straight-line steps in Euler's method, state whether the approximation in part (a) is an overestimate or an underestimate. Give a reason.<span class="inline-marks">[1]</span></div></li></ol>

The variables $x$ and $y$ satisfy the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=y^3-\frac{y}{x}$, where $x,y>0$. It is given that $y=2$ when $x=1$.

Euler's method is a numerical method for estimating a solution of a differential equation. Over a small horizontal step of length $h$, it uses a straight line with the gradient given by the differential equation at the current estimated point.

The estimated change in $y$ is therefore $h\times\text{current gradient}$. Let $y_n$ be the estimate of $y$ at $x=x_n$, with $x_0=1$ and $y_0=2$. For this differential equation, the rule is

\[x_{n+1}=x_n+h,\qquad y_{n+1}=y_n+h\left(y_n^3-\frac{y_n}{x_n}\right).\]

At each step, calculate the gradient using the current values of $x$ and $y$, then add $h$ times this gradient to the current estimate of $y$. Recalculate the gradient at the new estimated point before repeating. Keep unrounded values in subsequent calculations.

Euler's Method

Concavity of Graphs

Solving Differential Equations by Separation of Variables

Function Notation, Domain and Range

Solving Differential Equations by Substitution

Range of a Function

Domain of a Function

Sketching Solution Curves of Differential Equations

IB AA HL May 2026 P2 TZC Q12 [modified]

Functions

Differential Equations

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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 to Parametric Curves

Derivatives of Trigonometric Functions

Differentiation of Parametric Equations

Tangents and Normals Parallel to the Axes

Parametric Equations

N2025 P1 Q11

Graphing Techniques

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x$.<span class="inline-marks">[2]</span></div></li></ol>
<p>A curve $C$ is given by the equation $y = \ln x$. The region $R$ is bounded by $C$, the $x$-axis and the line $x = \mathrm{e}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph"><p>Using the result in part <strong>(a)</strong>, find the exact volume generated when&nbsp;$R$ is rotated $360^\circ$ about the $x$-axis.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">Find the volume generated when&nbsp;$R$ is rotated $360^\circ$ about the $y$-axis, giving your answer correct to $2$ decimal places.<span class="inline-marks">[3]</span></div></li></ol>

Show that $\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x$.

A curve $C$ is given by the equation $y = \ln x$. The region $R$ is bounded by $C$, the $x$-axis and the line $x = \mathrm{e}$.

Find the volume generated when $R$ is rotated $360^\circ$ about the $y$-axis, giving your answer correct to $2$ decimal places.

Volume of Revolution about the y-axis

Volume of Revolution about the x-axis

N2025 P1 Q7

<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 $\displaystyle\int_0^{\pi/6}x\sin3x\,\mathrm{d}x$.

Definite Integration as the Reverse of Differentiation

Product derivative with a triple angle

Integration

Differentiation

<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-q16-question-1" data-list-item-path="amint-q16-question-1"><div class="list-item-paragraph">Differentiate $(2x+1)\mathrm{e}^{-x}$ with respect to $x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q16-question-2" data-list-item-path="amint-q16-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of $\displaystyle\int_0^1x\mathrm{e}^{-x}\,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol>

Differentiate $(2x+1)\mathrm{e}^{-x}$ with respect to $x$.

Hence evaluate the exact value of $\displaystyle\int_0^1x\mathrm{e}^{-x}\,\mathrm{d}x$.

Linear factor with a negative exponential

<ol class="i-list"><li data-list-item-id="amint-q07-question-1" data-list-item-path="amint-q07-question-1"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(x^2-2x+3)\sqrt{2x+1}\right]=\dfrac{5x^2-4x+1}{\sqrt{2x+1}}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="amint-q07-question-2" data-list-item-path="amint-q07-question-2"><div class="list-item-paragraph">Hence evaluate $\displaystyle\int_0^4\dfrac{5x^2-4x+1}{\sqrt{2x+1}}\,\mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(x^2-2x+3)\sqrt{2x+1}\right]=\dfrac{5x^2-4x+1}{\sqrt{2x+1}}$.

Hence evaluate $\displaystyle\int_0^4\dfrac{5x^2-4x+1}{\sqrt{2x+1}}\,\mathrm{d}x$.

Chain Rule

Quadratic factor with a square root

<ol class="i-list"><li data-list-item-id="amint-q19-question-1" data-list-item-path="amint-q19-question-1"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{\mathrm{d}}{\mathrm{d}x}\left[-(x^2+2x+2)\mathrm{e}^{-x}\right]=x^2\mathrm{e}^{-x}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="amint-q19-question-2" data-list-item-path="amint-q19-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of $\displaystyle\int_0^2x^2\mathrm{e}^{-x}\,\mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

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

Hence evaluate the exact value of $\displaystyle\int_0^2x^2\mathrm{e}^{-x}\,\mathrm{d}x$.

Quadratic factor with a negative exponential

<ol class="i-list"><li data-list-item-id="amint-q06-question-1" data-list-item-path="amint-q06-question-1"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(2x+5)\sqrt{3x-1}\right]=\dfrac{18x+11}{2\sqrt{3x-1}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q06-question-2" data-list-item-path="amint-q06-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of $\displaystyle\int_1^3\dfrac{18x+11}{2\sqrt{3x-1}}\,\mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(2x+5)\sqrt{3x-1}\right]=\dfrac{18x+11}{2\sqrt{3x-1}}$.

Hence evaluate the exact value of $\displaystyle\int_1^3\dfrac{18x+11}{2\sqrt{3x-1}}\,\mathrm{d}x$.

Scaled radical product

<ol class="i-list"><li data-list-item-id="amint-q17-question-1" data-list-item-path="amint-q17-question-1"><div class="list-item-paragraph">Differentiate $\mathrm{e}^{2x}(3-2x)$ with respect to $x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q17-question-2" data-list-item-path="amint-q17-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of $\displaystyle\int_0^1x\mathrm{e}^{2x}\,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol>

Differentiate $\mathrm{e}^{2x}(3-2x)$ with respect to $x$.

Hence evaluate the exact value of $\displaystyle\int_0^1x\mathrm{e}^{2x}\,\mathrm{d}x$.

Linear factor with a positive exponential

<ol class="i-list"><li data-list-item-id="amint-q21-question-1" data-list-item-path="amint-q21-question-1"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{\mathrm{d}}{\mathrm{d}x}[(2x+1)\ln x]=2\ln x+2+\dfrac1x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q21-question-2" data-list-item-path="amint-q21-question-2"><div class="list-item-paragraph">Hence evaluate $\displaystyle\int_1^{\mathrm{e}}\ln x\,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol>

Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}[(2x+1)\ln x]=2\ln x+2+\dfrac1x$.

Hence evaluate $\displaystyle\int_1^{\mathrm{e}}\ln x\,\mathrm{d}x$.

Logarithmic integral between one and e

<ol class="i-list"><li data-list-item-id="amint-q08-question-1" data-list-item-path="amint-q08-question-1"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(x^2+x+1)\sqrt[3]{x+1}\right]=\dfrac{7x^2+10x+4}{3(x+1)^{2/3}}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="amint-q08-question-2" data-list-item-path="amint-q08-question-2"><div class="list-item-paragraph">Hence evaluate $\displaystyle\int_0^7\dfrac{7x^2+10x+4}{3(x+1)^{2/3}}\,\mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(x^2+x+1)\sqrt[3]{x+1}\right]=\dfrac{7x^2+10x+4}{3(x+1)^{2/3}}$.

Hence evaluate $\displaystyle\int_0^7\dfrac{7x^2+10x+4}{3(x+1)^{2/3}}\,\mathrm{d}x$.

Polynomial factor with a cube root

<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-q05-question-1" data-list-item-path="amint-q05-question-1"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(3x-1)\sqrt{x+2}\right]=\dfrac{9x+11}{2\sqrt{x+2}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q05-question-2" data-list-item-path="amint-q05-question-2"><div class="list-item-paragraph">Hence evaluate $\displaystyle\int_{-1}^{2}\dfrac{9x+11}{2\sqrt{x+2}}\,\mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(3x-1)\sqrt{x+2}\right]=\dfrac{9x+11}{2\sqrt{x+2}}$.

Hence evaluate $\displaystyle\int_{-1}^{2}\dfrac{9x+11}{2\sqrt{x+2}}\,\mathrm{d}x$.

Linear factor with a square root

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

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

Product of exponential and trigonometric functions

<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}}{\mathrm{d}x}\left[(2x)(3 - 5x)^{\frac{6}{5}}\right] = (6 - 22x)(3 - 5x)^{\frac{1}{5}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Hence, evaluate $\int x(3 - 5x)^{\frac{1}{5}} dx$.<span class="inline-marks">[3]</span></div></li></ol>

Integration as an Inverse Process of Differentiation

2024 GESS PRELIMS P2 Q11

<p>The equation of a curve is $y = \frac{x}{e^{2x}}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Find an expression for&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Given that&nbsp;$x$ is changing at a constant rate of $0.18$ units per second, find the rate of change of $y$ when&nbsp;$x = 1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Explain if the curve has more than one stationary point.<span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve is $y = \frac{x}{e^{2x}}$.

Connected Rates of Change

2024 GESS PRELIMS P2 Q8

<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;$\frac{\mathrm{d}}{\mathrm{d}x}\left[\left(x^2+1\right)e^{2x-1}\right]=2e^{2x-1}\left(x^2+x+1\right)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence, show that&nbsp;$\int -2e^{2x-1}\left(x^2+x\right) \mathrm{d}x = -x^2e^{2x-1}+c$.<span class="inline-marks">[4]</span></div></li></ol>

Show that $\frac{\mathrm{d}}{\mathrm{d}x}\left[\left(x^2+1\right)e^{2x-1}\right]=2e^{2x-1}\left(x^2+x+1\right)$.

Hence, show that $\int -2e^{2x-1}\left(x^2+x\right) \mathrm{d}x = -x^2e^{2x-1}+c$.

TKSS 4AM 2022 Prelims P2 Q8

<p>The polynomial $\mathrm{f}(x)$ has the property $\mathrm{f}(a)=0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down a factor of $\mathrm{f}(x)$.</span></div></li></ol>
<p>It is further given that&nbsp;$\mathrm{f}'(a)=0$.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">What can be deduced about the factorised form of $\mathrm{f}(x)$?</span></div></li></ol>

The polynomial $\mathrm{f}(x)$ has the property $\mathrm{f}(a)=0$.

Write down a factor of $\mathrm{f}(x)$.

It is further given that $\mathrm{f}'(a)=0$.

What can be deduced about the factorised form of $\mathrm{f}(x)$?

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