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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

$x^{2} \frac{d^{2} y}{{dx}^{2}} + x \frac{dy}{dx} + y = 0$

Multiple Choice Questions

619.
$If z = \frac{y}{x} [\sin (\frac{x}{y}) + \cos (1 + \frac{y}{x})], then x \frac{\partial z}{\partial x} is equal to$

$y \frac{\partial z}{\partial y}$

$- y \frac{\partial x}{\partial y}$

$2 y \frac{\partial z}{\partial y}$

$2 y \frac{\partial z}{\partial x}$

$- y \frac{\partial x}{\partial y}$

$We have, z = (\frac{y}{x}) [\sin (\frac{x}{y}) + \cos (1 + (\frac{y}{x}))] \frac{\partial z}{\partial x} = (\frac{y}{x}) [\cos (\frac{x}{y}) (\frac{1}{y}) - \sin (1 + (\frac{y}{x})) (\frac{- y}{x^{2}})] + (\frac{- y}{x^{2}}) [\sin (\frac{x}{y}) + \cos (1 + (\frac{y}{x}))]$

$x \frac{\partial z}{\partial x} = y [(\frac{1}{y}) \cos (\frac{x}{y}) + (\frac{y}{x^{2}}) \sin (1 + (\frac{y}{x}))] - (\frac{y}{x}) [\sin (\frac{x}{y}) + \cos (1 + (\frac{y}{x}))] \Rightarrow x \frac{\partial z}{\partial x} = \cos (\frac{x}{y}) + (\frac{y^{2}}{x^{2}}) \sin (1 + (\frac{y}{x})) - z . . . (i) Similarly, y \frac{\partial z}{\partial y} = \cos (\frac{x}{y}) - (\frac{y^{2}}{x^{2}}) \sin (1 + (\frac{y}{x})) + z . . . (ii) On adding Eqs . (i) and (ii) x \frac{\partial z}{\partial x} = - y \frac{\partial z}{\partial y}$