Differentiability in an Interval

Q.

(Idempotent laws) For any set A, prove that:
$\left(i\right)A\cup A=A$

$\left(ii\right)A\cap A=A$

Q.

Find the domain and range of the real function $f\left(x\right)=\frac{1}{1-x^2}$

Q. If $f\left(x\right)=|1-x|$, then the points where $si{n}^{-1}\left(f|x|\right)$ is non-differentiable are

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

Q. The number of points at which $f\left(x\right)=\{\begin{array}{cc}min(x,x^2),& \text{if}-\infty <x<1\\ min(2x-1,x^2),& \text{if}x\ge 1\end{array}$ is not differentiable is

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

Q.

(Commultative laws) For any two sets a and B, prove that:
$I.A\cup B=B\cup A$ [Commutative law for union of sets]

$II.A\cap B=B\cap A$ [Commutative law for intersection of sets]

Q. A discontinuous function can be made continuous by redefining the domain to exclude the points of discontinuities.

1. False

2. True

Q.

The second derivative of a single valued function parametrically represented by $x=\varphi \left(t\right)andy=\psi \left(t\right)$, ( where $\varphi \left(t\right)and\psi \left(t\right)$ are different functions and ${\varphi }^{\prime }\left(t\right)\ne 0$) is given by

1. 

2. 

3. 

4. 

Q.

Check whether the following probabilities P(A) and P(B) are consistently defined
(i) $P\left(A\right)=0.5,P\left(B\right)=0.7,P(A\cap B)=0.6$
(ii) $P\left(A\right)=0.5,P\left(B\right)=0.4,P(A\cup B)=0.8$

Q.

(De Morgan's laws) For any two sets A and B, prove that:
$I.(A\cup B{)}^{\prime }=(A^{\prime }\cap B^{\prime })$

$II.(A\cap B{)}^{\prime }=(A^{\prime }\cup B^{\prime })$

Q. If $f\left(x\right)=|1-x|,$ then the number of points (excluding the boundary points if exist) where $g\left(x\right)={\sin }^{-1}\left(f\right(|x|\left)\right)$ is non-differentiable, is

1. $5$
2. $2$
3. $4$
4. $3$

Q.

$\text{Let f (x)be a twice differentiable function and f"(0)=5, then}\underset{x\to 0}{lim}\frac{3f\left(x\right)-4f\left(3x\right)+f\left(9x\right)}{x^2}\text{is equal to:}$

1. 120

2. 40

3. 5

4. 30

Q. The function $y=||x-1|-1|$ is differentiable for all real numbers except the points

1. {0, 1, 2}
2. {0, 1}
3. {1, 2}
4. {-1, 1}

Q. Number of point(s) where the function $f\left(x\right)=max(\sqrt{2x-x^2},2-x)$ is non-differentiable, is/are

1. \N
2. 1
3. 2
4. 3

Q. Let $f$ be a differentiable function satisfying $f(x+2y)=2yf\left(x\right)+xf\left(y\right)-3xy+1\forall x,yϵR$ such that $f^{\prime }\left(0\right)=1$ then $f\left(2\right)$ is

1. 4
2. 1
3. 5
4. 3

Q. If f(x) is a differentiable function such that F : R $\to$ R and $f\left(\frac{1}{n}\right)=0\forall n\le 1,nϵI$ then
[IIT Screening 2005]

1. $f\left(x\right)=0\forall xϵ(0,1)$
2. $|f\left(x\right)|\le 1\forall xϵ(0,1)$
3. f(0) = 0 = f'(0)
4. f(0) = 0 but f (0) may or may not be 0

Q. If f is a real- valued differentiable function satisfying $|f\left(x\right)-f\left(y\right)|\le (x-y{)}^2,x,yϵRandf(0)=0,thenf(1)$ equal

1. 2
2. 1
3. -1
4. 0

Q.

Find the domain of the function:

$f\left(x\right)=\frac{x^2+2x+1}{x^2-8x+12}$

Q. Among the given functions, the one which is non-differentiable is .

1. $f\left(x\right)=cosx$
2. $f\left(x\right)=x^2-5$
3. $f\left(x\right)=|x|$
4. $f\left(x\right)=x$

Q.

In which of the following cases is the function f(x) discontinuous at a?

1. 

2. 

3. 

4. 

Q.

If arg$⟮\frac{z-i}{z+i}⟯$ =$\frac{\pi }{4}$, then z represents a point on

1. A pair of straight line

2. None of these

3. A circle

4. A straight line

Q.

Find the value of $\underset{x\to 0}{lim}\frac{|x|}{x}$

1. 1

2. 0

3. Does not exist

4. -1

Q.

Let f be a subset of $Z\times Z$ defined by f = {(ab, a + b): a, b $ϵZ$}. Is f a function from Z to Z? Justify your answer.

Q. Write the domain and range of function f(x) given by $f\left(x\right)=\sqrt{\left[x\right]-x}$.

Q.

If $A\subseteq B$ then for any set C, prove that $(C-B)\subseteq (C-A)$

Q. The number of points at which the function f(x) = |x - 0.5| + |x - 1| + tan x does not have a derivative in the interval (0, 2), is
[MNR 1995]

1. 2
2. 3
3. 1
4. 4

Q.

The set of all those points, where the function

$f\left(x\right)=\frac{x}{1+|x|}$

is differentiable, is

1. $(-\infty ,\infty )$

2. [0, $\infty$]

3. $(-\infty ,0)\cup (0,\infty )$

4. $(0,\infty )$

Q.

$\frac{1}{2-|x|}\ge 1,xϵR-\{-2,2\}$

Q. The function $f\left(x\right)={\cos }^{-1}(4x^3-3x)$ is

1. always differentiable
2. not differentiable at 2 points only
3. not continuous at 2 points only
4. not differentiable at 4 points only

Q. Let $f\left(x\right)=\{\begin{array}{cc}-1,& -2\le x<0\\ & \\ x^2-1,& 0\le x\le 2\end{array}$

and $g\left(x\right)=|f\left(x\right)|+f\left(|x|\right)$. Then, in the interval $(-2,2)$, $g$ is :

1. not continuous
2. not differentiable at one point
3. not differentiable at two points
4. differentiable at all points

Q. For some constant $a$ and $b$, find the derivative of the following functions:
$(x-a)(x-b)$.