Integral as Antiderivative

Q.

How do you find the derivative of $y={\tan }^2\left(x\right)$?

Q. Let $f$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1).$ If $\underset{0}{\overset{x}{\int }}\sqrt{1-\left(f^{\prime }\right(t){)}^2}dt=\underset{0}{\overset{x}{\int }}f\left(t\right)dt,0\le x\le 1$ and $f\left(0\right)=0,$ then $\underset{x\to 0}{lim}\frac{1}{x^2}\underset{0}{\overset{x}{\int }}f\left(t\right)dt$

1. Equals $1$
2. Does not exist
3. Equals $0$
4. Equals $\frac{1}{2}$

Q. If $f$ is a differentiable function satisfying $f\left(xy\right)=f\left(x\right)+f\left(y\right)+\frac{x+y-1}{xy}$ for all $x,y>0$ and $f^{\prime }\left(1\right)=2,$ then the value of $\left[f\right(e^{100}\left)\right]$ is
(where $[.]$ represents the greatest integer function)

Q.

If $z=i\left(\log \left(2-\sqrt{3}\right)\right)$, then $\cos z$ is equal to

1. $i$

2. $2i$

3. $1$

4. $2$

Q. Which of the following is an indeterminate form (where $[.]$ denotes greatest integer function)

1. $\underset{x\to \infty }{lim}\frac{\sin x}{x}$
2. $\underset{x\to 0^{+}}{lim}\frac{[x{]}^2}{x^2}$
3. $\underset{x\to 1}{lim}\frac{x-1}{x^3-1}$
4. $\underset{x\to \infty }{lim}\sqrt{x^4-1}+x$

Q. The function $f\left(x\right)=\{\begin{array}{cc}x\sin \left(\ln x^2\right),& x\ne 0\\ 0,& x=0\end{array}$ is:

1. non differentiable at $x=0$
2. differentiable at $x=0$
3. discontinuous at $x=0$
4. none of the above

Q. If $f\left(x\right)=\underset{0}{\overset{x}{\int }}{e}^t\sin (x-t)dt,$ then $f^{\prime \prime }x-f\left(x\right)$ is equal to

1. $\sin x+\cos x$
2. $2\cos x$
3. $\sin x-\cos x$
4. $2\sin x$

Q. For the function $f\left(x\right)=\frac{1}{\ln \left|x\right|}$ which of the following statements is/are true

1. $f\left(x\right)$ has removable discontinuity at $x=0$
2. $f\left(x\right)$ has non-removable discontinuity at $x=1$
3. $f\left(x\right)$ has removable discontinuity at $x=-1$
4. $f\left(x\right)$ has non-removable discontinuity at $x=0$

Q.

If $\underset{x\to 0}{lim}{\left\{2-\cos x\sqrt{\cos 2x}\right\}}^{\frac{x+2}{x^2}}=e^{\alpha }$, then $\alpha =?$

Q. If $f\left(x\right)=\mathrm{sgn}(2\sin x+a)$ is continuous for all $x\in R$, then the possible values of $a$ are

1. $a\in R$
2. $a\in (-\infty ,-2)\cup (2,\infty )$
3. $a\in (-2,2)$
4. $a\in (0,\infty )$

Q.

Find the local maxima and local minima, if any of the following functions. Also, find the local maximum and the local minimum values, as the case may be as follows.
$f\left(x\right)=x^2$

$g\left(x\right)=x^3-3x$

$h\left(x\right)=sinx+cosx,0<x<\frac{\pi }{2}$

$f\left(x\right)=sinx-cosx,0<x<2\pi$

$f\left(x\right)=x^3-6x^2+9x+15$

$g\left(x\right)=\frac{x}{2}+\frac{2}{x},x>0$

$g\left(x\right)=\frac{1}{x^2+2}$

$f\left(x\right)=x\sqrt{1-x},x>0$

Q. $\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{20}xdx=a^{-b}\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{20}x{\cos }^{20}xdx,$ where $a$ is a prime number. Then the value of $a-b$ is

1. $21$
2. $22$
3. $23$
4. $20$

Q. If $f\left(x\right)$ be a function such that $\left(f\right(x){)}^{2007}=\underset{0}{\overset{x}{\int }}\frac{\left(f\right(t){)}^{2006}}{2+t^2}dt,$ then which of the following is/are true?

1. There are only two such functions are possible.
2. There are infinite number of such functions are possible.
3. $f\left(x\right)=\frac{1}{2007}{\tan }^{-1}\left(\frac{x}{\sqrt{2}}\right)$
4. $f\left(x\right)=\frac{1}{2007\sqrt{2}}{\tan }^{-1}\left(\frac{x}{\sqrt{2}}\right)$

Q. Suppose a differentiable function $f\left(x\right)$ satisfies the identity $f(x+y)=f\left(x\right)+f\left(y\right)+x{y}^2+x^{2}y$ for all real $x$ and $y.$ If $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}=1$, then $f^{\prime }\left(3\right)$ is equal to

Q. Assertion (A): The principal amplitude of complex number $x+ix$ is $\frac{\pi }{4}$.
Reason (R): The principal amplitude of a complex number $x+iy$ is $\frac{\pi }{4}$ if $y=x$.

1. Both A and R are false
2. Both A and R are true and R is the correct explanation of A
3. A is true R is false
4. A is false, R is true

Q. If $3\int {\sec }^{4}xdx=f\left(x\right)+2\tan x,$ then the value of $f\left(\frac{\pi }{4}\right)$ is equal to
(Assume integration constant to be zero)

Q. In the intergal $\underset{0}{\overset{10}{\int }}\frac{\left[\sin 2\pi x\right]}{e^{x-\left[x\right]}}dx=\alpha e^{-1}+\beta e^{-\frac{1}{2}}+\gamma ,$ where $\alpha ,\beta ,\gamma$ are integers and $\left[x\right]$ denotes the greatest integer less than or equal to $x,$ then the value of $\alpha +\beta +\gamma$ is equal to:

1. $20$
2. $0$
3. $25$
4. $10$

Q. For function $f\left(x\right)=$${\tan }^{-1}\left(\frac{1}{x}\right),x\ne 0$ which of the following statement is TRUE

1. $f\left(x\right)$ has missing point discontinuity
2. $f\left(x\right)$ has finite type discontinuity
3. $f\left(x\right)$ has infinite type discontinuity
4. $f\left(x\right)$ has isolated point discontinuity

Q. If $f\left(x\right)$ be a function such that $\left(f\right(x){)}^{2007}=\underset{0}{\overset{x}{\int }}\frac{\left(f\right(t){)}^{2006}}{2+t^2}dt,$ then which of the following is/are true?

1. There are only two such functions are possible.
2. There are infinite number of such functions are possible.
3. $f\left(x\right)=\frac{1}{2007}{\tan }^{-1}\left(\frac{x}{\sqrt{2}}\right)$
4. $f\left(x\right)=\frac{1}{2007\sqrt{2}}{\tan }^{-1}\left(\frac{x}{\sqrt{2}}\right)$

Q. Let $[.]$ denote the greatest integer function and $f\left(x\right)=\{\begin{array}{cc}\left[x\right]\left(\frac{e^{1/x}-1}{e^{1/x}+1}\right),& x<0\\ b,& x=0\\ \left[x\right]\left(\frac{e^{1/x}-1}{e^{1/x}+1}\right)+a,& x>0\end{array}$. If $f\left(x\right)$ is continuous at $x=0$, then the value of $a+b$ is

1. $2$
2. $\frac{1}{2}$
3. $0$
4. $1$

Q. The $f\left(x\right)=\sqrt{x},g\left(x\right)=e^x-1\forall x\in (0,\infty )$ and $\int fog\left(x\right)dx=Afog\left(x\right)+B{\tan }^{-1}\left(fog\right(x\left)\right)+C$, then $A+B$ is

1. $2$
2. $4$
3. $0$
4. $8$

Q. If $f\left(x\right)=\sqrt{1-\sin 2x},$ then $f^{\prime }\left(x\right)$ is equal to

1. $(\cos x-\sin x)$, for $x\in (\pi /4,\pi /2)$
2. $-(\cos x+\sin x)$, for $x\in (0,\pi /4)$
3. $-(\cos x+\sin x)$, for $x\in (\pi /4,\pi /2)$
4. $\cos x+\sin x,$ for $x\in (0,\pi /4)$

Q. If $x+|y|=2y,$ then $y$ as a function of $x$ is
$(\text{where}y=f(x\left)\right)$

1. defined for all real $x$
2. continuous at $x=0$
3. differentiable at $x=0$
4. $f^{\prime }\left(0^{-}\right)=\frac{1}{3}$

Q. The function $f\left(x\right)$ is defined by $f\left(x\right)=\{\begin{array}{cc}{(x^2+e^{\frac{1}{2-x}})}^{-1},& x\ne 2\\ k,& x=2\end{array}$.
If it is continuous from right at the point $x=2$, then $k$ is equal to

1. $0$
2. $\frac{1}{4}$
3. $-\frac{1}{2}$
4. None of these

Q. The function $f\left(x\right)$ is defined by $f\left(x\right)=\{\begin{array}{cc}{(x^2+e^{\frac{1}{2-x}})}^{-1},& x\ne 2\\ k,& x=2\end{array}$.
If it is continuous from right at the point $x=2$, then $k$ is equal to

1. $0$
2. $\frac{1}{4}$
3. $-\frac{1}{2}$
4. None of these

Q. The value of $I=\underset{-10}{\overset{3}{\int }}\mathrm{sgn}({\cot }^{-1}x+e^{-x}+1-\sin x)dx$ is

1. $15$
2. $19$
3. $17$
4. $13$

Q. If $f:R\to R$ is a function defined by $f\left(x\right)=[x-1]\cos \left(\frac{2x-1}{2}\right)\pi$, where [.] denotes the greatest integer function, then $f$ is:

1. discontinuous only at $x=1$
2. discontinuous at all integral values of $x$ except at $x=1$
3. continuous only at $x=1$
4. continuous for every real $x$

Q. Find the derivative with respect to ${}^{\prime }{x}^{\prime }$ of the function ${\sin }^{-1}\left(2x\right)$ at $x=\frac{\pi }{4}$

Q.

How do you find the derivative of $cot\left(x\right)$?

Q. Let $f\left(x\right)$ be a differentiable function satisfying $f\left(\frac{x+y}{2}\right)=\frac{f\left(x\right)+f\left(y\right)}{2}\forall x,y\in R$, where $f\left(0\right)=0$ and $I=\underset{0}{\overset{2\pi }{\int }}{\left(f\right(x)-\sin x)}^{2}dx.$
Which of the following is/are correct?

1. When $I$ is minimum, then $f(–4{\pi }^2)=3$
2. $f\left(x\right)$ is a odd function
3. $I$ is constant.
4. When $I$ is minimum, then $f(–4{\pi }^2)=-3$