A-Level H2 Math | 5 Essential Questions
Maclaurin and Power Series
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 (1698–1746), 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.
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2009 HCI Promo P1 Q3 Modified
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]
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2020 MI P1 Q6
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]
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2013 ACJC P1 Q12
$\left( 1-{{x}^{2}} \right){{\left( \frac{\text{d}y}{\text{d}x} \right)}^{2}}=4y$
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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2017 PJC Promo Q9 (a)
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]
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How to show ${{\text{e}}^{\text{i}\theta }}=\cos \theta +\text{i}\sin \theta $ using Standard Series
Using the standard series in MF26, show ${{\text{e}}^{\text{i}\theta }}=\cos \theta +\text{i}\sin \theta $.
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