# Maclaurin and Power Series

###### 5 Essential Questions

Here is where we provide free online Revision materials for your H2 Math. We have compiled 5 essential questions from each topic for you and broken down the core concepts with video explanations. Please download the worksheet and try the questions yourself! Have fun learning with us, consider joining our tuition classes or online courses.

• Q1
• Q2
• Q3
• Q4
• Q5
##### 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]

• I
• II
• I
• II

##### 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]

• (i)
• (ii)
• (iii)
• (i)
• (ii)
• (iii)

##### 2013 ACJC P1 Q12
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]

##### 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]

• I
• II

• I
• II

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

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