# Maclaurin and Power Series

#### Timothy Gan

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### 2023 MI PU3 P2 Q2

2023 MI PU3 P2 Q2 (a) Given that $text{f}left( x right)={{sec }^{2}}x$, find $text{f},text{ }!!’!!text{ }left( x right)$ and $text{f},text{ }!!’!!text{ }!!’!!text{ }left( x right)$. Hence, find the Maclaurin series

### 2019 TJC P1 Q5

2019 TJC P1 Q5 In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system such as a

### 2023 YIJC P2 Q5

2023 YIJC P2 Q5 In the triangle $ABC$, $AC=1$, angle $BAC=x$ radians and angle $ACB=frac{1}{6}pi$ radians (see diagram). (a) Show that $AB=frac{1}{cos x+sqrt{3}sin x}$. [3] (b) Given that $x$

### 2013 HCI P2 Q2

2013 HCI P2 Q2 (i) Given that $ln left( ky right)={{tan }^{-1}}left( kx right)$, where $k$ is a non-zero constant, show that $left( 1+{{k}^{2}}{{x}^{2}} right)frac{text{d}y}{text{d}x}=ky$. By further differentiation of this

### 2017 HCI P1 Q7

2017 HCI P1 Q7 (i) It is given that $ln y=2sin x$. Show that $frac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}=-yln y+frac{1}{y}{{left( frac{text{d}y}{text{d}x} right)}^{2}}$. [2] (ii) Find the first four terms of the Maclaurin series for

### 2019 NJC P1 Q4

2019 NJC P1 Q4 (i) Show that $frac{p{{x}^{2}}+left( 4p-q right)x+left( 4p+q right)}{left( 1-x right){{left( 2+x right)}^{2}}}=frac{p}{1-x}+frac{q}{{{left( 2+x right)}^{2}}}$. [1] (ii) Find the values of $p$, $q$ and $r$ such that

### 2017 AJC P1 Q5

2017 AJC P1 Q5 A curve $C$ has equation $y=text{f}left( x right)$. The equation of the tangent to the curve $C$ at the point where $x=0$ is given by $2x-ay=3$

2009 TPJC P1 Q11 (i) Given that $y={{text{e}}^{{{tan }^{-1}}x}}$, find $frac{text{d}y}{text{d}x}$ and show that $left( 1+{{x}^{2}} right)frac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}+left( 2x-1 right)frac{text{d}y}{text{d}x}=0$. Obtain the Maclaurin series for ${{text{e}}^{{{tan }^{-1}}x}}$ up to and including