If $x=a(\theta+\sin\theta),\ y=a(1-\cos\theta)$, find $\dfrac{d^{2}y}{dx^{2}}$ at $\theta=\dfrac{\pi}{2}$.

CBSE Class 12 · 2005 · DifferentiationLong Answer

x = a (θ + sin θ), y = a (1 - cos θ)

, find

\frac{d ^{2} y}{d x ^{2}}

θ = \frac{π}{2}

Answer: $\frac{1}{a}$

Differentiate w.r.t.

θ

\frac{d x}{d θ} = a (1 + cos θ), \frac{d y}{d θ} = a sin θ

First derivative, using

\frac{sin θ}{1 + cos θ} = tan \frac{θ}{2}

\frac{d y}{d x} = \frac{a sin θ}{a ( 1 + cos θ )} = tan \frac{θ}{2}

Second derivative (chain through

θ

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d θ} (tan \frac{θ}{2}) \cdot \frac{1}{d x / d θ} = \frac{1}{2} sec^{2} \frac{θ}{2} \cdot \frac{1}{a ( 1 + cos θ )}

Since

1 + cos θ = 2 cos^{2} \frac{θ}{2}

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{4 a cos ^{4} \frac{θ}{2}}

θ = \frac{π}{2}

cos \frac{π}{4} = \frac{1}{2}

cos^{4} \frac{π}{4} = \frac{1}{4}

\frac{d ^{2} y}{d x ^{2}}_{θ = π /2} = \frac{1}{a}

CBSE Class 12 Board · 2005 · Subjective