First slide

Differentiability

Question

If $x = k (t - \sin t), y = k (1 - \cos t) (k \neq 0)$ then $\frac{d^{2} y}{d x^{2}}$ at $t = \frac{π}{2}$

Moderate

A

$- \frac{1}{k}$
B

$\frac{1}{2 k}$
C

$\frac{1}{k^{2}}$
D

$2 k$

Solution

$\frac{d x}{d t} = k (1 - \cos t), \frac{d y}{d t} = k \sin t$
so $\frac{d y}{d x} = \frac{k \sin t}{k (1 - \cos t)} = \frac{2 \sin \frac{t}{2} \cos \frac{t}{2}}{2 \sin^{2} \frac{t}{2}} = \cot \frac{t}{2}$
$\begin{array}{l} \frac{d^{2} y}{d x^{2}} = \frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d t} (\frac{d y}{d x}) \frac{d t}{d x} \\ = \frac{1}{2} (- {cosec}^{2} (\frac{t}{2})) \frac{1}{k (2 \sin^{2} (\frac{t}{2}))} \\ = - \frac{1}{4 k} {cosec}^{4} \frac{t}{2} \end{array}$
${\frac{d^{2} y}{d x^{2}}|}_{t = π / 2} = - \frac{1}{4 k} (\sqrt{2})^{4} = - \frac{1}{k}$

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