If $\mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x} - 3\right|$, then f'(3) is

• - 1

• 1

• 0

• does not exist

If $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{xsin}\left(\frac{1}{\mathrm{x}}\right), \mathrm{x} \ne 0 \\ 0 , \mathrm{x} = 0\right\} \end{gathered}$$, then at x = 0 the function f(x) is

• continuous

• differentiable

• continuous but not differentiable

• None of the above

833.

If Rolle's theorem for f(x) = ${\mathrm{e}}^{\mathrm{x}}\left(\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)\right)$ is verified on $\left[\frac{\mathrm{\pi }}{4}, \frac{5\mathrm{\pi }}{4}\right]$, then the value of c is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{4}$

• $\mathrm{\pi }$

If the function f(x) defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{xsin}\left(\frac{1}{\mathrm{x}}\right), \mathrm{for} \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{for} \mathrm{x} = 0\right\} \end{gathered}$$

is continuous at x = 0, then k is equal to

• 0

• 1

• - 1

• $\frac{1}{2}$

If $\mathrm{y} = {\mathrm{e}}^{\mathrm{m}{\sin }^{-1}\left(\mathrm{x}\right)} \mathrm{and} \left(1 - {\mathrm{x}}^2\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2$ = Ay², then A is equal to

• m

• - m

• m²

• - m²

For what value of k, the function defined by

f(x) = $$\begin{gathered} \left\{\frac{\log \left(1 + 2\mathrm{x}\right)\sin \left(\mathrm{x}°\right)}{{\mathrm{x}}^2}, \mathrm{for} \mathrm{x} \ne 0 \\ \mathrm{k} , \mathrm{for} \mathrm{x} = 0\right\} \end{gathered}$$

is continuous at x = 0 ?

• 2

• $\frac{1}{2}$

• $\frac{\mathrm{\pi }}{90}$

• $\frac{90}{\mathrm{\pi }}$

If ${\log }_{10}\left(\frac{{\mathrm{x}}^2 - {\mathrm{y}}^2}{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right) = 2$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $- \frac{99\mathrm{x}}{101\mathrm{y}}$

• $\frac{99\mathrm{x}}{101\mathrm{y}}$

• $- \frac{99\mathrm{y}}{101\mathrm{x}}$

• $\frac{99\mathrm{y}}{101\mathrm{x}}$

If g(x) is the inverse function of f(x) and$\mathrm{f}\text{'}\left(\mathrm{x}\right) = \frac{1}{1 +\hspace{0.17em}{\mathrm{x}}^4}$, then g'(x) is

• 1 + [g(x)]⁴

• 1 - [g(x)]⁴

• 1 + [f(x)]⁴

• $\frac{1}{1 + {\left[\mathrm{g}\left(\mathrm{x}\right)\right]}^4}$

If the function f(x) = ${\left[\tan \left(\frac{\mathrm{\pi }}{4} + \mathrm{x}\right)\right]}^{\frac{1}{\mathrm{x}}}$ for $\mathrm{x} \ne 0$ is  = K for x = 0 continuous at x = 0, then K = ?

• e

• e^{- 1}

• e²

• e^{- 2}

If x = f(t) and y = g(t) are differentiable functions of t, then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is

• $\frac{\mathrm{f}\text{'}\left(\mathrm{t}\right) . \mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right) - \mathrm{g}\text{'}\left(\mathrm{t}\right) . \mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right)}{{\left[\mathrm{f}\text{'}\left(\mathrm{t}\right)\right]}^3}$

• $\frac{\mathrm{f}\text{'}\left(\mathrm{t}\right) . \mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right) - \mathrm{g}\text{'}\left(\mathrm{t}\right) . \mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right)}{{\left[\mathrm{f}\text{'}\left(\mathrm{t}\right)\right]}^2}$

• $\frac{\mathrm{g}\text{'}\left(\mathrm{t}\right) . \mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right) - \mathrm{f}\text{'}\left(\mathrm{t}\right) . \mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right)}{{\left[\mathrm{f}\text{'}\left(\mathrm{t}\right)\right]}^3}$

• $\frac{\mathrm{g}\text{'}\left(\mathrm{t}\right) . \mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right) + \mathrm{f}\text{'}\left(\mathrm{t}\right) . \mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right)}{{\left[\mathrm{f}\text{'}\left(\mathrm{t}\right)\right]}^3}$