Question

1Using Lagrange’s Mean value theorem Prove that log10(x + 1) =

x log10e

(1+θx)

Q.2 Show that excosx = 1 + x + (x2

/2) - (x3

/3) - (11x4

/24)

Answer 1

1.By mean value theorem:

$f'(c)=\frac{f(b)-f(a)}{b-a},\ c\isin [a,b]$

Let $f(x)=log(x+1)$

Then:

$\frac{1}{x_0+1}=\frac{log(x+1)}{x},\ x_0\isin [0,x],\ x>0$

$log(x+1)=\frac{x}{x_0+1}$

$log(x+1)=xloge^{-(x_0+1)}$

$log10(x+1)=log10+xloge^{-(x_0+1)}$

$log10(x+1)=log10+xloge^{-(\theta x+1)},\ \theta\isin [0,1]$

2.Maclauren series:

$f(x)=\displaystyle\sum_{n=0}^{\infin}\frac{f^{(n)}(0)}{n!}x^n$

Then:

$y(0)=1$

$y'=e^xcosx-e^xsinx=e^x(cosx-sinx)$

$y'(0)=1$

$y''=e^x(cosx-sinx)-e^x(cosx+sinx)=-2e^xsinx$

$y'''=-2e^x(cosx+sinx)$

$y'''(0)=-2$

$y^{(4)}=-2(e^x(cosx+sinx)+e^x(cosx-sinx))=-4e^xcosx$

$y^{(4)}(0)=-4$

$e^xcosx=1+x-\frac{x^3}{3}-\frac{x^4}{6}+...$