Maclaurin and Power Series

2023 CJC P2 Q2 It is given that $y={{text{e}}^{-x}}sin x+x-1$. (a) Show that $frac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}=k{{text{e}}^{-x}}cos x$, where $k$ is a constant to be determined. [2] (b) By further differentiation of this

2023 ASRJC P1 Q8 It is given that $y=frac{1}{3+sin 2x}$. (a) Show that $frac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}-4{{y}^{2}}sin 2x+4yfrac{text{d}y}{text{d}x}cos 2x=0$. Hence find the Maclaurin series for $y$, up to and including the term in

2018 AJC P1 Q5  It is given that $text{f}left( x right)=ln left( sin x+cos x right)$ for $-frac{pi }{4}<xle frac{pi }{4}$. (i) Show that $text{f},text{ }!!’!!text{ }!!’!!text{ }left( x right)+{{left[

2022 VJC Promo Q7 (b) An arc $PQ$ of a circle subtends an angle $left( pi -theta right)$ radians at the centre $O$. The length of chord $PQ$ is $L$

2023 RI Promo Q5 In this question you may use expansions from the List of Formulae (MF26). (a) Find the first four non-zero terms of the Maclaurin series of ${{text{e}}^{-x}}left(

2023 CJC Promo Q5 It is given that $y={{sin }^{-1}}x$, $-1le xle 1$. (a) Show that $frac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}=x{{left( frac{text{d}y}{text{d}x} right)}^{3}}$. [3] (b) Using the result in part (a), find the Maclaurin

RI Maclaurin Series Tutorial Q4 (i) Given that $y=tan left( {{text{e}}^{2x}}-1 right)$, show that $frac{text{d}y}{text{d}x}=k{{text{e}}^{2x}}left( 1+{{y}^{2}} right)$, where $k$ is to be found. Hence find the values of $frac{text{d}y}{text{d}x}$, $frac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}$

2023 SAJC P2 Q2 In triangle $PQR$, $angle PRQ=frac{2pi }{3}$, $angle RPQ=frac{pi }{6}+theta $, and $PR=a$ units, where $a$ is a positive real constant. (i) Show that $PQ=frac{sqrt{3}a}{cos theta -sqrt{3}sin