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###### A-Level H2 Math | 5 Essential Questions

# Maclaurin and Power Series

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**2009 HCI Promo P1 Q3 Modified**

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**2020 MI P1 Q6**

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**2013 ACJC P1 Q12**

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**2017 PJC Promo Q9 (a)**

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##### How to show ${{\text{e}}^{\text{i}\theta }}=\cos \theta +\text{i}\sin \theta $ using Standard Series

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Maclaurin series refers to the expansion of a series of functions where the estimated value of the function is determined as a sum of the derivatives of that function when evaluated at zero. It is named after Colin Maclaurin (1698â€“1746), a Scottish mathematician who made extensive use of this special case of Taylor series in the 18th century.

Maclaurin series are very frequently used to approximate functions.

- Q1
- Q2
- Q3
- Q4
- Q5

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 is valid.

[3]

Find, in the simplest form, the coefficient of ${{x}^{2017}}$ in this expansion.

[2]

- I
- II

- I
- II

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It is given that $y=\sqrt{{{\text{e}}^{x}}\cos x}$.

(i)

Show that $2y\frac{\text{d}y}{\text{d}x}={{y}^{2}}-{{\text{e}}^{x}}\sin x$.

[2]

(i) Show that $2y\frac{\text{d}y}{\text{d}x}={{y}^{2}}-{{\text{e}}^{x}}\sin x$.

[2]

(ii)

By further differentiation of the result in part (i), find the Maclaurin series for $y$, up to and including the term in ${{x}^{2}}$.

[4]

(ii) By further differentiation of the result in part (i), find the Maclaurin series for $y$, up to and including the term in ${{x}^{2}}$.

[4]

(iii)

Using the standard series from the List of Formulae (MF 26). Expand $\sqrt{{{\text{e}}^{x}}\cos x}$ as far as the term in ${{x}^{2}}$ and verify that the same result is obtained in part (ii).

[3]

(iii) Using the standard series from the List of Formulae (MF 26). Expand $\sqrt{{{\text{e}}^{x}}\cos x}$ as far as the term in ${{x}^{2}}$ and verify that the same result is obtained in part (ii).

[3]

- (i)
- (ii)
- (iii)

- (i)
- (ii)
- (iii)

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Given that $y={{({{\sin }^{-1}}x)}^{2}}$, show that

$\left( 1-{{x}^{2}} \right){{\left( \frac{\text{d}y}{\text{d}x} \right)}^{2}}=4y$

and $\left( 1-{{x}^{2}} \right)\frac{{{\text{d}}^{\text{2}}}y}{\text{d}{{x}^{2}}}-x\frac{\text{d}y}{\text{d}x}=2$.

[3]

By further differentiation of these results, find the Maclaurin series of $y$ up to including the term in ${{x}^{4}}$.

[3]

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In the triangle $ABC$ as shown below, $BC=3$, angle $BAC=\frac{\pi }{3}+\theta $ radians and angle $ACB=\frac{\pi }{2}$ radians.

Show that $AC=\frac{3\left( 1-\sqrt{3}\tan \theta \right)}{\sqrt{3}+\tan \theta }$.[3]

Given that $\theta $ is a sufficiently small angle, deduce that $AC\approx \sqrt{3}+a\theta $, where $a$ is a constant to be determined.[3]

- I
- II

- I
- II

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Using the standard series in MF26, show ${{\text{e}}^{\text{i}\theta }}=\cos \theta +\text{i}\sin \theta $.

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