Maclaurin Series 1

Using the standard Maclaurin series for  $\cos x$ , find the series expansion  $\cos2x.$      (TKQ10B)

• $1-x^2+2x^4-...$

• $1-2x^2+\frac{2}{3}x^4-...$

• $1+2x^2-4x^4+...$

• $1+\frac{1}{2}x^2-\frac{1}{4}x_{ }^4+...$

 Find the Maclaurin series for   $y=\frac{\sin3x}{x}$ .  (TKQ2C) 

•  $y=1-\frac{9}{2}x^2+...$  

•  $y=1+\frac{3}{2}x^2+...$  

•  $y=3-\frac{9}{2}x^2+...$  

•  $y=3+\frac{3}{2}x^2+...$  

The Maclaurin series for  $e^x\sin x=ax+bx^2+cx^3+...$   
Find the values of a, b and c.   (TKQ5)
 

•  $a=-1,\ b=2,\ c=3$  

•  $a=1,\ b=1,\ c=\frac{1}{3}$  

•  $a=3,\ b=-1,\ c=\frac{1}{2}$  

•  $a=2,\ b=\frac{1}{2},\ c=-3$  

Given  $\ \cos2x=1-2\sin^2x$ and Maclaurin series for   $\cos\ x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+...$ . 
(i) Find the Maclaurin series for  $\sin^2x$  and hence,      
(ii) evaluate  $\lim_{x\rightarrow0}\ \left(\frac{\sin^2x-x^2}{2x^4}\right).$    
Choose the correct answer for (i) and (ii).              (TKQ17)

•  $\left(i\right)\sin^2x=x^2+\frac{1}{3}x^4-\frac{2}{45}x_{ }^6+...$  

•  $\left(i\right)\sin^2x=x^2-\frac{1}{3}x^4+\frac{2}{45}x^6-...$  

•  $\left(ii\right)\frac{1}{6}$  

•  $\left(ii\right)-\frac{1}{6}$  

Using the standard Maclaurin series for  $\ln\left(1+x\right)$ , find the series expansion for  $\ln\left(1-2x\right)$ . 

•  $x-\frac{2}{3}x^2+\frac{3}{4}x^3-...$  

•  $x-\frac{1}{2}x^2+\frac{4}{3}x^3-...$  

•  $-2x+2x^2-\frac{8}{3}x^3+...$  

•  $-2x^2+4x^3-\frac{5}{4}x^4+...$  

Using the Maclaurin series for  $\ln\left(1+x\right)$ , evaluate  $\lim_{x\rightarrow0}\ \left[\frac{x-\ln\left(1+x\right)}{5x^2}\right].$        (TKC)

•  $\frac{1}{5}$  

•  $\frac{1}{7}$  

•  $\frac{1}{10}$  

•  $\frac{1}{15}$  

Given  $y=\sqrt{2e^x-1}$  . Choose all the TRUE mathematical statements.     (TKQ7)

•  $y\frac{\text{d}y}{\text{d}x}=e^x$  

•  $y\frac{d^2y}{dx^2}+\left(\frac{\text{d}y}{\text{d}x}\right)^2=e^x$  

•  $y\frac{d^3y}{dx^3}+3\left(\frac{\text{d}y}{\text{d}x}\right)\left(\frac{d^2y}{dx^2}\right)=e^x$  

•  $x=0,\ y=1,\frac{\text{dy}}{\text{d}x}=1,\frac{d^2y}{dx^2}=0,\frac{d^3y}{dx^3}=1$  

•  $Maclaurin\ series,\ y=1+x+\frac{1}{6}x^3+...$  

Given $y=\ \ln\left[\frac{1+e^{-x}}{4}\right]$ .  Which of the followings are TRUE to find the the Maclaurin series for y.        (TKQ8)

•  $\frac{\text{d}y}{\text{d}x}=\frac{1}{4}e^{-y}\ -1$  

•  $\frac{\text{d}^2y}{\text{d}x^2}=\frac{1}{4}e^{-y}\ \frac{\text{d}y}{\text{d}x}$  

•  $\frac{\text{d}^3y}{\text{d}x^3}=\frac{1}{4}e^{-y}\left(\frac{dy}{dx}\right)^2\ -\frac{1}{4}e^{-y}\left(\frac{d^2y}{dx^2}\right)$  

•  $x=0,\ y=\ln2,\frac{\text{d}y}{\text{d}x}=\frac{1}{2},\frac{d^2y}{dx^2}=-\frac{1}{4},\frac{d^3y}{dx^3}=0$  

• Maclaurin series, $y=-\ln2-\frac{1}{2}x+\frac{1}{8}x^2+...$  

Given  $y=\tan^{-1}x$ . Which of the followings are TRUE to find the Maclaurin series for y.  (TKQ14) 

•  $\frac{\text{d}y}{\text{d}x}=\frac{2}{1+x^2}$  

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

•  $\frac{d^3y}{\text{d}x^3}=-2\left(\frac{\text{d}y}{\text{d}x}\right)^2-4x\left(\frac{dy}{dx}\right)\left(\frac{d^2y}{\text{d}x^2}\right)$  

•  $\tan^{-1}x=x-\frac{1}{3}x^3+...$  

Given $f\left(x\right)=3xe^{x^2}$ . 
Which of the followings are TRUE to find the Maclaurin series for f(x).    (TKQ22)

•  $f'\left(x\right)=\ 2x\ f\left(x\right)+3e^{x^2}$  

•  $f''\left(x\right)=2x\ f'\left(x\right)\ +\ 4\ f\left(x\right)$  

•  $f'''\left(x\right)=2x\ f''\left(x\right)+6\ f'\left(x\right)$  

•  $x=0,f=0,f'=3,f''=0,f'''=18$  

•  $f\left(x\right)=3x+3x^2+3x^3+...$