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Continuity and Differentiability

Long Answer Type

611.

Find all points of discontinuity of f, where f is defined as following:

$f (x) = \{\begin{cases} \begin{matrix} |x| + 3, x \leq - 3 \\ - 2 x, - 3 < x < 3 \end{matrix} \\ 6 x + 2, x \geq 3 \end{cases}$

612.

$Find \frac{dy}{dx}, if y = {(cosx)}^{x} + {(sinx)}^{\frac{1}{x}}$

613.

Find the value of ‘a’ for which the function f defined as

$f (x) = \{\begin{cases} a \sin \frac{π}{2} (x + 1), x \leq 0 \\ \frac{\tan x - \sin x}{x^{3}}, x > 0 \end{cases}$

is continuous at x = 0.

614.

Differentiate $X^{x \cos x} + \frac{x^{2} + 1}{x^{2} - 1} w . r . t . x$

615.

If $x = a (θ - \sin θ), y = (1 + \cos θ), find \frac{d^{2} y}{{dx}^{2}}$

616.

If ${(\cos x)}^{y} = {(\cos y)}^{x}, find \frac{dy}{dx} .$

617.

If sin y = x sin (a + y), prove that $\frac{dy}{dx} = \frac{\sin^{2} a + y}{\sin a}$ .

618.
If y = 3 cos ( log x ) + 4 sin ( log x ), show that
$x^{2} \frac{d^{2} y}{{dx}^{2}} + x \frac{dy}{dx} + y = 0$

It is given that, y = 3 cos ( log x ) + 4 sin ( log x )

Then,

$\frac{dy}{dx} = 3 \times \frac{d}{dx} [\cos \log x] + 4 \times \frac{d}{dx} [\sin \log x] = 3 \times [- \sin \log x \times \frac{d}{dx} \log x] + 4 \times [\cos \log x \times \frac{d}{dx} \log x] = \frac{- 3 \sin \log x}{x} + \frac{4 \cos \log x}{x} = \frac{4 \cos \log x - 3 \sin \log x}{x} \frac{d^{2} y}{{dx}^{2}} = \frac{d}{dx} (\frac{4 \cos (\log x) - 3 \sin (\log x)}{x})$

$= \frac{x \{4 \cos (\log x) - 3 \sin (\log x)\}' - \{4 \cos (\log x) - 3 \sin (\log x)\} (x)'}{x^{2}} = \frac{x [- 4 \sin (\log x) \times (\log x)' - 3 \cos (\log x) \times (\log x)'] - 4 \cos (\log x) + 3 \sin (\log x)}{x^{2}} = \frac{- 4 \sin (\log x) - 3 \cos (\log x) - 4 \cos (\log x) + 3 \sin (\log x)}{x^{2}} = \frac{- \sin (\log x) - 7 \cos (\log x)}{x^{2}} ∴ x^{2} \frac{d^{2} y}{{dx}^{2}} + x \frac{dy}{dx} + y$

$= x^{2} (\frac{- \sin (\log x) - 7 \cos (\log x)}{x^{2}}) + x (\frac{4 \cos (\log x) - 3 \sin (\log x)}{x}) + 3 \cos (\log x) + 4 \sin (\log x) = - \sin (\log x) - 7 \cos (\log x) + 4 \cos (\log x) - 3 \sin (\log x) + 3 \cos (\log x) + 4 \sin (\log x) = 0$