Maclaurin and Power Series

2022 EJC Promo Q2 It is given that $y={{text{e}}^{2x}}cos x$. (a) Show that $frac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}=4frac{text{d}y}{text{d}x}-5y$. [3] (b) Find the Maclaurin series for $y$ up to the term in ${{x}^{2}}$. [2] (c)

2022 HCI P1 Q2 In the figure below, $ABC$ is an equilateral triangle of length $2k$ units while $ABD$ is a triangle where $AD$ is of length $k$ units and

2022 ASRJC P1 Q4 (i) Show that the first two non-zero terms of the Maclaurin series for $tan theta $ is given by $theta +frac{1}{3}{{theta }^{3}}$. You may use the

2022 TMJC P2 Q4 In the triangle $ABC$, angle $BAC=frac{pi }{4}$ radians and angle $ABC=left( frac{pi }{4}+2x right)$ radians. Show that $frac{AB}{AC}=frac{sqrt{2}cos 2x}{cos 2x+sin 2x}$. [3] Given that $x$ is

2022 RI P1 Q7 It is given that $y=sqrt{2+{{cos }^{2}}x}$. (a) Show that (i) $2yfrac{text{d}y}{text{d}x}=-sin 2x$, [1] (ii) $yfrac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}+{{left( frac{text{d}y}{text{d}x} right)}^{2}}=-cos 2x$. [1] (b) Hence find the Maclaurin series of

2022 VJC P1 Q2 The diagram shows triangle $ABC$, where angle $ACD=left( frac{pi }{4}+x right)$ radians. Point $D$ is on $BC$ such that $AD=2$ and $BD=2sqrt{3}$. Show that if $x$

Home 2006 NJC Promo Q5 Given that $y={{text{e}}^{{{sin }^{-1}}left( 2x right)}}$, show that $left( 1-4{{x}^{2}} right)frac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}-4xfrac{text{d}y}{text{d}x}=4y$. By further differentiating this result, find the Maclaurin series for $y$ in ascending powers

2022 JPJC J2 MYE P2 Q2 (a) Given that $y-2={{left( x+1 right)}^{ln left( x+1 right)}}$ , where $x>-1$, show that $left( x+1 right)frac{text{d}y}{text{d}x}=2left( y-2 right)ln left( x+1 right)$. [2] Find