if x=a (theta +sin theta) and y=a (1-cos theta), find d^2y/dx^2 at theta =pi/2
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x= theta+sintheta....y=a(1-costheta)
dx/dtheta=a(1+costheta)
dy/dtheta=asintheta
dy/dx=asintheta/a(1+costheta)
dy/dx=2sin(theta/2)cos(theta/2)/2cos^2(theta/2)
=tan(theta/2)
d^2(y)/dx^2=sec^2(theta/2)×1/2dtheta/dx
=sec^2theta/2×1/2(a(1+costheta) at theta=pi/2 now =1/4a(1+0)
=a/4
dx/dtheta=a(1+costheta)
dy/dtheta=asintheta
dy/dx=asintheta/a(1+costheta)
dy/dx=2sin(theta/2)cos(theta/2)/2cos^2(theta/2)
=tan(theta/2)
d^2(y)/dx^2=sec^2(theta/2)×1/2dtheta/dx
=sec^2theta/2×1/2(a(1+costheta) at theta=pi/2 now =1/4a(1+0)
=a/4
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