यदि ` x=a (theta +sin theta ), y=a (1-cos theta ),` तो `theta =(pi)/(4)` पर ` (d^(2)y)/(dx^(2))` का मान ज्ञात कीजिए|

Explanation

दिया है `" "x=a (theta +sin theta ) `
` rArr" "(dx)/(d theta )=a(1+cos theta ) `
तथा ` " "y=a (1-cos theta )rArr (dy)/(d theta)=asin theta `
` therefore " "(dy)/(dx) =(dy//d theta )/(dx//d theta ) =(asin theta )/(a(1+cos theta ) )`
` " "=(2sin theta //2cos theta //2)/(2cos ^(2)theta //2)=tan"" (theta)/(2)`
`rArr (d^(2)y)/(dx^(2))=(sec ^(2) ""(theta )/(2))*(1)/(2)(d theta )/(dx ) =(1)/(2) sec ^(2) ""(theta )/(2) (1)/(a(1+cos theta ))`
अब ` ((d^(2)y)/(dx^(2)))_ (theta=(pi)/(4) ) =(1)/(2) sec ^(2) ""(pi)/(4) *(1)/(a(1+cos ""(pi)/(2)))`
` " "=(1)/(2) (sqrt(2))^(2)*(1)/(a(1+0) )= (1)/(a) `