Question

100 points !!! Please help

If x=a ( -sin theta) and
y=a (1-cos theta),
Find d^2y/dx^2 at theta =pi

Answer 1

`DERIVATIVE \: \: OF \: \: \\ \: \: \: \: \:PARAMETRIC \: \: FUNCTIONS\\ \\ \\ x = a( \beta - \sin( \beta ) ) \\ y = a(1 - \cos( \beta ) \\ \\ (dy)/(d \beta ) = a \sin( \beta ) \\ \\ (dx)/(d \beta ) = a(1 - \cos( \beta ) \\ \\ (dy)/(dx) = (a \sin( \beta ) )/(a(1 - \cos( \beta ) ) = \cot( ( \beta )/(2) ) \\ \\ \\ \frac{d^(2)y }{d {x}^(2) } = (d)/(dx) ( \ \ \cot ( ( \beta )/(2) ) ) = ( - (1)/(2) cosec ^(2) ( \beta )/(2) ) * (d \beta )/(dx) \\ \\ \\ = (- (1)/(2) cosec^(2) ( \beta )/(2) ) * (1)/(a(1 - \cos( \beta ) ) \\ \\ \\ At \: \beta = \pi \\ \\ = ( - ( 1)/(2) {cosec}^(2)(\pi)/(2) ) * (1)/(a(1 - \cos(\pi)) ) \\ \\ = - (1)/(2) . {1}^(2) * (1)/(a(1 - ( - 1))) \\ \\ \\ = - (1)/(4a) \: \: \: \: \: \: \: \: \: Ans.`