Mathematics : Continuity and Differentiability

If f(x) = xⁿ, then the value of $\mathrm{f}\left(1\right) - \frac{\mathrm{f}\text{'}\left(1\right)}{1!} + \frac{\mathrm{f}\text{'}\text{'}\left(1\right)}{2!} - \frac{\mathrm{f}\text{'}\text{'}\text{'}\left(1\right)}{3!} + ... + \frac{{\left(- 1\right)}^{\mathrm{n}}\mathrm{f}{\text{'}}^{\mathrm{n}}\left(1\right)}{\mathrm{n}!}$ is

• 2ⁿ

• 2^{n - 1}

• 0

• 1

If 2a + 3b + 6c = 0, then at least one root of the equation ax² + bx + c = 0 lies in the inverval

• (0, 1)

• (1, 2)

• (2, 3)

• (1, 3)

If $\underset{\mathrm{x}\to 0}{\lim }\frac{\log \left(3 + \mathrm{x}\right) - \log \left(3 - \mathrm{x}\right)}{\mathrm{x}}$ = k, then the value of k will be

• 0

• $- \frac{1}{3}$

• $\frac{2}{3}$

• $- \frac{2}{3}$

The nth derivative of e^{2x + 5} is

• $\frac{2^{2\mathrm{n}}}{\mathrm{n}!} {\mathrm{e}}^{2\mathrm{x} + 5}$

• $2^{\mathrm{n}} {\mathrm{e}}^{2\mathrm{x} + 5}$

• $2^{2\mathrm{n}} {\mathrm{e}}^{2\mathrm{x} + 5}$

• $\frac{2^{\mathrm{n}}}{\mathrm{n}} {\mathrm{e}}^{2\mathrm{x} + 5}$

If $\sqrt{1 - {\mathrm{x}}^2} + \sqrt{1 - {\mathrm{y}}^2} = \mathrm{a}\left(\mathrm{x} - \mathrm{y}\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\sqrt{\frac{1 - {\mathrm{x}}^2}{1 - {\mathrm{y}}^2}}$

• $\sqrt{\frac{1 - {\mathrm{y}}^2}{1 - {\mathrm{x}}^2}}$

• $\sqrt{\frac{{\mathrm{x}}^2 - 1}{1 - {\mathrm{y}}^2}}$

• $\sqrt{\frac{{\mathrm{y}}^2 - 1}{1 - {\mathrm{x}}^2}}$

Function f(x) = $$\begin{gathered} \left\{\mathrm{x} - 1, \mathrm{x} < 2 \\ 2\mathrm{x} - 3, \mathrm{x} \ge 2\right\} \end{gathered}$$ is continuous function for

• all real values of x

• only x = 2

• only real values of x, when x $\ne$ 2

• the integer values of x

If x^y = y^x, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{\mathrm{y}\left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{y}\right) - \mathrm{y}\right)}{\mathrm{x}\left({\mathrm{ylog}}_{\mathrm{e}}\left(\mathrm{x}\right) - \mathrm{x}\right)}$

• $\frac{\mathrm{x}\left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{y}\right) + \mathrm{y}\right)}{\mathrm{y}\left({\mathrm{ylog}}_{\mathrm{e}}\left(\mathrm{x}\right) + \mathrm{x}\right)}$

• $\frac{\mathrm{x}\left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{y}\right) - \mathrm{y}\right)}{\mathrm{y}\left({\mathrm{ylog}}_{\mathrm{e}}\left(\mathrm{x}\right) - \mathrm{x}\right)}$

• $\frac{\mathrm{y}\left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{y}\right) + \mathrm{y}\right)}{\mathrm{x}\left({\mathrm{ylog}}_{\mathrm{e}}\left(\mathrm{x}\right) + \mathrm{x}\right)}$

If y = ${\mathrm{e}}^{\left(1 + {\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• e

• 1

• 0

• ${\log }_{\mathrm{e}}\left(\mathrm{x}\right){\mathrm{e}}^{{\log }_{\mathrm{e}}\left(\mathrm{ex}\right)}$

The value of m for which the function f(x) = $$\begin{gathered} \left\{{\mathrm{mx}}^2, \mathrm{x} \le 1 \\ 2\mathrm{x}, \mathrm{x} > 1\right\} \end{gathered}$$ is differentiable at x = 1, is

• 0

• 1

• 2

• does not exist

If f(x) = $$\begin{gathered} \left\{\mathrm{b} + \mathrm{ax}, \mathrm{x} < 0 \\ \mathrm{b}{\left(\mathrm{x} - 1\right)}^2, \mathrm{x} \ge 0\right\} \end{gathered}$$, is differentiable at x = 0, then (a, b) will be

• (- 3, - 1)

• (- 3, 1)

• (3, 1)

• (2, - 1)