Maclaurin and Power Series

Home 2022 NJC J2 CT P1 Q3 It is given that ${{tan }^{-1}}y=a{{tan }^{-1}}x+b$, for constants $a$ and $b$. (i) Show that $left( 1+{{x}^{2}} right)frac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}=2left( ay-x right)frac{text{d}y}{text{d}x}$. [2] (ii) Suppose

Home 2022 NJC J2 CT P1 Q2 In the diagram above, $AC=10$, $AD=8$, $BC=5$, angle $ADC$ is a right angle and angle $BCD=theta $ radians. By considering angle $ACD$, show

2022 ACJC J2 MYE Q1 In the triangle $ABC$, $AB=2$, angle $ABC=theta $ radians and angle $BAC=frac{pi }{4}$ radians. (i) Show that $BC=frac{2}{sin theta +cos theta }$. [3] (ii) Given

2022 HCI J2 BT Q1 Using standard series from the List of Formulae (MF26), expand ${{text{e}}^{2-ax}}$ as far as the term in ${{x}^{4}}$, where $a$ is a non-zero constant. [3]

2022 CJC J2 MYE P1 Q1 Given that $theta $ is a sufficiently small angle measured in radians, show that $frac{sin left( theta +frac{pi }{3} right)}{cos left( 2theta right)}approx a+btheta

2022 ASRJC J2 MYCT Q7 (a) If $theta $ is small such that ${{theta }^{3}}$ and higher powers of $theta $ may be neglected, express $frac{cos theta }{1+sin 2theta +cos

2018 MJC Promo Q8 (a) (i) Given that $y=cos left( {{text{e}}^{2x}}-1 right)$, show that $frac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}-2frac{text{d}y}{text{d}x}+4y{{text{e}}^{4x}}=0$. By further differentiation of this result, find the Maclaurin series of the function $cos left(

2017 ACJC Promo Q10 (a) Given that $x$ is small enough for terms involving ${{x}^{4}}$ and above to be ignored, use the Maclaurin series for $sin x$ and $cos x$

2009 HCI Promo Q3 Expand $frac{{{x}^{2}}+2x}{2{{x}^{2}}+1}$ in ascending powers of $x$ up to and including the term in ${{x}^{5}}$. State the range of values of $x$ for which this expansion

2019 RI P2 Q2 (a) The curve $y=text{f}left( x right)$ passes through the point $left( 0,81 right)$ and has gradient given by $frac{text{d}y}{text{d}x}={{left( frac{1}{3}y-15x right)}^{frac{1}{3}}}$. Find the first three non-zero