Prove that the function f ( x ) = x 2 - 1 ; x ≥ 1 1 - x ; x < 1 is not differentiable at x = 1

To check the differentiability of a function at x = 1 , we need to find L H D & R H D Notice that f ( 1 ) = 1 2 - 1 = 0 R H D = lim h → 0 f ( 1 + h ) - f ( 1 ) h ⇒ R H D = lim h → 0 ( 1 + h ) 2 - 1 - 0 h ⇒ R H D = lim h → 0 h 2 + 2 h h ⇒ R H D = lim h → 0 h + 2 = 2 Now, L H D = lim h → 0 f ( 1 ) - f ( 1 - h ) h ⇒ L H D = lim h → 0 0 - ( 1 - ( 1 - h ) ) h ⇒ L H D = lim h → 0 - h h = - 1 So even though, L H D , R H D are finite, but they are not equal. So the function is not differentiable at x = 1

Let f : R → R be differentiable at c ∈ R and f c = 0 . If g x = f x , then at x = c , g is:

not differentiable if f ' c = 0

Match List - I with List - II. List-I List-II (a) Source of microwave frequency (i) Radioactive decay of nucleus (b) Source of infrared frequency (ii) Magnetron (c) Source of Gamma Rays (iii) Inner shell electrons (d) Source of X -rays (iv) Vibration of atoms and molecules (v) LASER (vi) RC circuit Choose the correct answer from the options given below:

(a)-(ii), (b)-(iv), (c)-(i), (d)-(iii)

(a)-(ii), (b)-(iv), (c)-(vi), (d)-(iii)

(a)-(vi), (b)-(iv), (c)-(i), (d)-(v)

(a)-(vi), (b)-(v), (c)-(i), (d)-(iv)

If f x = x e - 1 x + 1 x ; i f x ≠ 0 0 ; i f x = 0 then which of the following is correct?

f x is continuous and f ' 0 does not exist

f x is continuous and f ′ 0 also exists

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Which of the following is an organizational barrier to effective communication?

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Important Questions on Electromagnetic Waves

MEDIUM

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

Prove that the function $f (x) = \{\begin{cases} x^{2} - 1 & ; x \geq 1 \\ 1 - x & ; x < 1 \end{cases}$ is not differentiable at $x = 1$

HARD

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

Let

f (x) = 15 - |x - 10|; x \in R .

Then the set of all values of

x

, at which the function

g (x) = f (f (x))

is not differentiable, is:

MEDIUM

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

The greatest integer function defined by

f (x) = [x], 0 < x < 2

is not differentiable at

x =

_____

HARD

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

Write the difference between genetic biodiversity and species biodiversity that exists at all the levels of biological organisations.

EASY

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

Long distance short- wave radio broadcasting uses

EASY

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

Given below is the representation of the extent of the global diversity of invertebrates. What groups the four portions (A-D) represent respectively?
Question Image

MEDIUM

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

Let

f : R \to R

be differentiable at

c \in R

and

f (c) = 0

. If

g (x) = |f (x)|

, then at

x = c, g

is:

EASY

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

An association of individuals of different species living in the same habitat and having functional interactions is:

EASY

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

Equation for wave vector is:

MEDIUM

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

Let

f (x) = \{\begin{matrix} m a x (|x|, x^{2}), & |x| \leq 2 \\ 8 - 2 |x|, & 2 < |x| \leq 4 \end{matrix} .

Let

S

be the set of points in the interval

(- 4, 4)

at which

f

is not differentiable. Then

S

MEDIUM

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

"India has greater ecosystem diversity than Norway." Do you agree with this statement? Give reasons in support of your answer.

EASY

Physics>Modern Physics>Electromagnetic Waves>Sources of Electromagnetic Waves

Match List - I with List - II.

List-I		List-II
(a)	Source of microwave frequency	(i)	Radioactive decay of nucleus
(b)	Source of infrared frequency	(ii)	Magnetron
(c)	Source of Gamma Rays	(iii)	Inner shell electrons
(d)	Source of $X$ -rays	(iv)	Vibration of atoms and molecules
		(v)	LASER
		(vi)	RC circuit