Differentiability

Q.

If $x+\frac{1}{x}=2$, then the principal value of ${\sin }^{-1}x$ is

1. $\frac{\mathrm{\pi }}{4}$

2. $\frac{\mathrm{\pi }}{2}$

3. $\mathrm{\pi }$

4. $\frac{3\mathrm{\pi }}{2}$

Q.

The value of $b$ for which function $f\left(x\right)=\sin x-bx+c$ is decreasing in the interval $\left(-\infty ,\infty \right)$ is given by

1. $b<1$

2. $b\ge 1$

3. $b>1$

4. $b\le 1$

Q.

For some constants a and b, find the derivative of:

(i) (x-a)(x-b)

(ii) $(a{x}^2+b{)}^2$

(iii) $\frac{x-a}{x-b}$

Q.

Shifting one term from one side of an equation to another side with a change of sign is known as

1. commutativity

2. transposition

3. distributivity

4. associativity

Q.

The derivative of x³ – 20 x at x = 10 is:

1. 250

2. 280

3. 180

4. 320

Q.

The function f : R → R defined by y = f(x) = 5 for each x ϵ R is

1. Modulus function

2. Constant function

3. Signum function

4. Identity function

Q.

$limi{t}_{x\to \infty }\frac{\sin \left(x\right)}{x}=$

1. $\infty$

2. $1$

3. $0$

4. does not exist

Q.

If f(x) = x² and g(x) = x are two functions from R to R then
(fg)(2) is

1. 2

2. 8

3. 1

4. 4

Q. Why cant we have negative bases in logarithm? We can for an equation (-3)^3=-27 therefore log(-27) with base -3 is 3. That way log of some negative numbers can be defined

Q.

If f(x) = x² - x + 1, g(x) = 7x - 3, be two real functions then (f - g)(8) is

1. 53

2. 57

3. 64

4. 4

Q. APPLICATION OF DERIVATIVES : Integer Type Let f(x) be a non-constant thrice differentiable function defined on R such that f(x) = f(6-x) and p(0)=0=p(2)=p(5).If n is the minimum number of roots of (g(x)) + p(x) h(x) =0 in the interval [0, 6] then the value of n/2 is? p, g, h are the first, second and third derivatives of f(x) w.r.t. x.

Q. 4. How to find the value of Sin (7.07)?

Q.

Solution of$x\sin \left(\frac{y}{x}\right)dy=\left(y\sin \left(\frac{y}{x}\right)-x\right)dx$ is

1. $\cos \left(\frac{y}{x}\right)=c$

2. $\log \left(\frac{y}{x}\right)+\log x=c$

3. $\log \left(\frac{y}{x}\right)-\log x=c$

4. None of these

Q. The domain of definition of $f\left(x\right)=\sqrt{4x-x^2}$ is

(a) R − [0, 4]
(b) R − (0, 4)
(c) (0, 4)
(d) [0, 4]

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

1. continuous at x = 0.
2. not continuous at x = 0.
3. differentiable at x = 0.
4. not differentiable at x = 0.

Q. The domain of definition of the function $f\left(x\right)=\sqrt{x-1}+\sqrt{3-x}$ is

(a) [1, ∞)
(b) (−∞, 3)
(c) (1, 3)
(d) [1, 3]

Q. Find the derivative of the following function (it is to be understood that a, b, c, d are fixed non-zero constants):
f(x)= $(ax+b)(cx+d{)}^2$

Q. Write the domain and range of the function $f\left(x\right)=\frac{x-2}{2-x}$.

Q. The eccentricity of the hyperbola x² − 4y² = 1 is

(a) $\frac{\sqrt{3}}{2}$

(b) $\frac{\sqrt{5}}{2}$

(c) $\frac{2}{\sqrt{3}}$

(d) $\frac{2}{\sqrt{5}}$

Q.

The value of a, for which the coefficients of the middle terms of (1+ax)⁴ and (1-ax)⁶ are equal, will be:

1. 

2. 

3. 

4. 

Q.

Find the domain of the function f(x) = x |x|

1. R – {0}

2. R – {-1}

3. R⁺

4. R

Q.

The derivative of an even function if exists is

1. Even

2. Neither even nor odd

3. Can't be determined

4. odd

Q.

The derivative of the function at x = – 2 is:

1. –10

2. 10

3. 5

4. – 5

Q.

The derivative of the function sec² x is:

1. 

2. 

3. 

4. 

Q.

The set of all points, where the function $f\left(x\right)=\frac{x}{1+|x|}$ is differentiable, is

1. $\left(-\infty ,\infty \right)$
2. $[0,\infty )$
3. $(-\infty ,0)\cup (0,\infty )$
4. $\left(0,\infty \right)$

Q. $\underset{x\to \mathrm{\pi }/4}{\lim }\frac{4\sqrt{2}-{\left(\cos x+\sin x\right)}^5}{1-\sin 2x}$ is equal to

(a) $5\sqrt{2}$

(b) $3\sqrt{2}$

(c) $\sqrt{2}$

(d) none of these

Q. The maximum value of ${\sin }^2\left(\frac{2\pi }{3}+x\right)+{\sin }^2\left(\frac{2\pi }{3}-x\right)$ is
(a) 1/2
(b) 3/2
(c) 1/4
(d) 3/4

Q. what is cos^-1 x derivative ?

Q.

Evaluate the expression when $x=4,y=-6$and $z=-3$.
$\frac{3x-2y}{z}$

Q. Find the derivative of the following functions (it is to be understood that a, b, c, d are fixed non-zero constants):
f(x)= $\frac{ax+b}{cx+d}$