$\underset{\mathrm{x}\to 0}{\lim }\frac{1 - {\cos }^3\left(4\mathrm{x}\right)}{{\mathrm{x}}^2} = ?$

• 0

• 12

• 24

• 36

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \sin \left(\cos \left(\mathrm{x}\right)\right), \mathrm{then} \mathrm{f}\text{'}\left(\mathrm{x}\right) = ?$

• $\cos \left(\cos \left(\mathrm{x}\right)\right)$

• $\sin \left( - \sin \left(\mathrm{x}\right)\right)$

• $\left(\sin \left(\mathrm{x}\right)\right)\cos \left(\cos \left(\mathrm{x}\right)\right)$

• $\left( - \sin \left(\mathrm{x}\right)\right)\cos \left(\cos \left(\mathrm{x}\right)\right)$

$$\begin{gathered} \mathrm{What} \mathrm{is} \mathrm{the} \mathrm{derivative} \mathrm{of} \mathrm{the} \mathrm{function} \\ \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{e}}^{\tan \left(\mathrm{x}\right)} + \ln \left(\sec \left(\mathrm{x}\right)\right) - {\mathrm{e}}^{\ln \left(\mathrm{x}\right)} \mathrm{at} \mathrm{x} = \frac{\mathrm{\pi }}{4} ? \end{gathered}$$

• $\frac{\mathrm{e}}{2}$

• e

• 2e

• 4e

$\mathrm{What} \mathrm{is} \underset{\mathrm{x}\to 0}{\lim }\frac{\tan \left(\mathrm{x}\right)}{\sin \left(2\mathrm{x}\right)} = ?$

• $\frac{1}{2}$

• 1

• 2

• Limit does not exist

$\mathrm{What} \mathrm{is} \underset{\mathrm{h}\to 0}{\lim }\frac{\sqrt{2\mathrm{x} + 3\mathrm{h}} - \sqrt{2\mathrm{x}}}{2\mathrm{h}} =?$

• $\frac{1}{2\sqrt{2}\mathrm{x}}$

• $\frac{3}{\sqrt{2}\mathrm{x}}$

• $\frac{3}{2\sqrt{2}\mathrm{x}}$

• $\frac{3}{2\sqrt{2}\mathrm{x}}$

What is the value of ${\log }_7\left(\sqrt{7\sqrt{7\sqrt{7}}}\right) = ?$

• $3{\log }_2\left(7\right)$

• $1 - 3{\log }_2\left(7\right)$

• $1 - 3{\log }_7\left(2\right)$

• $\frac{7}{8}$

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \sqrt{25 - {\mathrm{x}}^2}, \mathrm{then} \mathrm{what} \mathrm{is} \underset{\mathrm{x}\to 1}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(1\right)}{\mathrm{x} - 1} = ?$

• $- \frac{1}{\sqrt{24}}$

• $\frac{1}{\sqrt{24}}$

• $- \frac{1}{4\sqrt{3}}$

• $\frac{1}{4\sqrt{3}}$

$\mathrm{What} \mathrm{is} \underset{\mathrm{x}\to 0}{\lim }\frac{\sqrt{1 - \cos \left(\mathrm{\theta }\right)}}{\mathrm{\theta }} = ?$

• $\sqrt{2}$

• $2\sqrt{2}$

• $\frac{1}{\sqrt{2}}$

• $- \frac{1}{2\sqrt{2}}$

$\mathrm{If} \mathrm{y} = {\left|\sin \left(\mathrm{x}\right)\right|}^{\left|\mathrm{x}\right|}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \frac{\mathrm{dy}}{\mathrm{dx}} \mathrm{at} \mathrm{x} = - \frac{\mathrm{\pi }}{6} ?$

• $\frac{2^{ - {\displaystyle \frac{\mathrm{\pi }}{6}}}\left(6\ln \left(2\right) - \sqrt{3}\mathrm{\pi }\right)}{6}$

• $\frac{2^{{\displaystyle \frac{\mathrm{\pi }}{6}}}\left(6\ln \left(2\right) + \sqrt{3}\mathrm{\pi }\right)}{6}$

• $\frac{2^{ - {\displaystyle \frac{\mathrm{\pi }}{6}}}\left(6\ln \left(2\right) + \sqrt{3}\mathrm{\pi }\right)}{6}$

• $\frac{2^{{\displaystyle \frac{\mathrm{\pi }}{6}}}\left(6\ln \left(2\right) - \sqrt{3}\mathrm{\pi }\right)}{6}$

10.

$\mathrm{What} \mathrm{is} \frac{\mathrm{d}}{\mathrm{dx}}\sqrt{1 - \sin \left(2\mathrm{x}\right)} = ? \mathrm{where} \frac{\mathrm{\pi }}{4} < \mathrm{x} < \frac{\mathrm{\pi }}{2} ?$

• $\cos \left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right)$

• $- \left(\cos \left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right)\right)$

• $\pm \left(\cos \left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right)\right)$

• None of the above