Maclaurin and Power Series

2021 NYJC P1 Q7 It is given that $y=ln left( 1+{{text{e}}^{x}} right)$. (i) Show that $left( 1+{{text{e}}^{x}} right)frac{text{d}y}{text{d}x}-{{text{e}}^{x}}=0$. [1] (ii) By further differentiation of the result in (i), find the

2021 TMJC P2 Q4 (a) Expand ${{left( b-frac{x}{2} right)}^{n}}$ in ascending powers of $x$, where $b$ is a positive constant and $n$ is a negative constant, up to and including

2021 MI P2 Q3 It is given that $text{f}left( x right)=ln left( 2+2sin x right)$. (i) Show that $text{f}”left( x right)=frac{k}{1+sin x}$, where $k$ is a constant to be found.

2021 MI P2 Q1 Given that $theta $ is a sufficiently small angle, show that $frac{1}{sin 2theta +cos theta }approx 1+atheta +b{{theta }^{2}}$, where $a$ and $b$ are constants to

2021 DHS P1 Q2 A triangle $ABC$ is such that $AC=sqrt{2}$, $BC=4$ and angle $ACB=frac{1}{4}pi +theta $. Given that $theta $ is sufficiently small for ${{theta }^{3}}$ and higher powers

2013 YJC P1 Q8 (i) Expand $frac{sqrt{4+x}}{1-x}$ in ascending powers of $x$, up to and including the term in ${{x}^{2}}$. State the set of values of x for which the

2021 HCI P1 Q9 Given that $y={{left( tan x+sec x right)}^{2}}$, show that $cos xfrac{text{d}y}{text{d}x}=2y$. [2] (i) By repeat differentiation, find the Maclaurin series for y up to and including

2021 YIJC P1 Q8 It is given that $y=sin left( ln left( 1+text{e}x right) right)$. (i) Show that ${{left( 1+text{e}x right)}^{2}}frac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}+text{e}left( 1+text{e}x right)frac{text{d}y}{text{d}x}=-{{text{e}}^{2}}y$. [2] (ii) By further differentiation of the

2021 NJC P1 Q5 Two straight corridors, $P$ and $Q$, each of width $2$ m, meet at right angles. A banner is hung across the ceiling of the corridors using

2015 MI P1 Q3 (i) Expand $frac{sqrt[4]{1+3{{x}^{2}}}}{2+x}$ in ascending powers of $x$, up to and including the term in ${{x}^{2}}$. [3] (ii) Find the set of values of $x$ for