Question

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

Answer 1

★ DERIVATIVE OF PARAMETRIC FUNCTIONS ★

$x = a( \beta - \sin( \beta ) ) \\ y = a(1 - \cos( \beta ) \\ \\ \frac{dy}{d \beta } = a \sin( \beta ) \\ \frac{dx}{d \beta } = a(1 - \cos( \beta ) \\ \\ \frac{dy}{dx} = \frac{a \sin( \beta ) }{a(1 - \cos( \beta ) } = \cot( \frac{ \beta }{2} ) \\ \\ \frac{d^{2}y }{d {x}^{2} } = \frac{d}{dx} ( \ \ \cot ( \frac{ \beta }{2} ) ) = ( - \frac{1}{2} cosec ^{2} \frac{ \beta }{2} ) \times \frac{d \beta }{dx} \\ \\ = (- \frac{1}{2} cosec^{2} \frac{ \beta }{2} ) \times \frac{1}{a(1 - \cos( \beta ) } \\ \\ at \: \beta = \pi \\ = ( - \frac{ 1}{2} {cosec}^{2}\frac{\pi}{2} ) \times \frac{1}{a(1 - \cos(\pi)) } \\ \\ = - \frac{1}{2} . {1}^{2} \times \frac{1}{a(1 - ( - 1))} \\ \\ = - \frac{1}{4a}$

Answer 2

Answer:

Step-by-step explanation: