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

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

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

MEDIUM

Earn 100

Which measure of dispersion is defined as the average of the squared differences from the Mean?

50% studentsanswered this correctly

Important Questions on Continuity and Differentiability

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

The graph between angle of deviation

(δ)

and angle of incidence

(i)

for a triangular prism is represented by :

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

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

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

The angle of incidence for a ray of light at a refracting surface of a prism is

45^{o}

. The angle of prism is

60^{o}

. If the ray suffers minimum deviation through the prism, the angle of minimum deviation and refractive index of the material of the prism respectively, are:

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

For a glass prism, the angle of minimum deviation will be smallest for the light of

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

f (x) = \{\begin{matrix} x e^{- (\frac{1}{|x|} + \frac{1}{x})}; & i f & x \neq 0 \\ 0; & i f & x = 0 \end{matrix}

then which of the following is correct?

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

A ray of light on passing through an equilateral glass prism, suffers a minimum deviation equal to the angle of the prism. The value of refractive index of the material of the prism is _____.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

Prove that function $f (x) = |x|$ is not differentiable at $x = 0$ .

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

Let

[x]

be the greatest integer less than or equals to

x

. Then, at which of the following point(s) the function

f (x) = x \cos (π (x + [x]))

is discontinuous?

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

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

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

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:

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

A prism has a refracting angle of

60^{o} .

When placed in the position of minimum deviation, it produces a deviation of

30^{o} .

The angle of incidence is

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

A small angled prism of

4 °

deviates a ray through

2.48 °

. Find the refractive Index of the prism.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

If the function

f

defined as

f (x) = \frac{1}{x} - \frac{k - 1}{e^{2 x} - 1}, x \neq 0

is continuous at

x = 0

, then ordered pair

(k, f (0))

is equal to

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

An equilateral glass prism has a refractive index

1.6

in the air. Calculate the angle of minimum deviation of the prism, when kept in a medium of refractive index

\frac{4 \sqrt{2}}{5}

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

The value of

k

which the function

f (x) = \{\begin{matrix} {(\frac{4}{5})}^{\frac{\tan 4 x}{\tan 5 x}}, & 0 < x < \frac{π}{2} \\ k + \frac{2}{5}, & x = \frac{π}{2} \end{matrix}

is continuous at

x = \frac{π}{2},

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

If the function

f

defined on

(\frac{π}{6}, \frac{π}{3})

f (x) = \{\begin{matrix} \frac{\sqrt{2} c o s x - 1}{c o t x - 1}, x \neq \frac{π}{4} \\ k, x = \frac{π}{4} \end{matrix}

is continuous, then

k

is equal to

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

Let

f : [- 1,3] \to R

be defined as

f (x) = \{\begin{matrix} |x| + [x], \\ x + |x|, \\ x + [x], \end{matrix} \begin{matrix} - 1 \leq x < 1 \\ 1 \leq x < 2 \\ 2 \leq x \leq 3, \end{matrix}

Where

[t]

denotes the greatest integer less than or equal to

t

. Then,

f

is discontinuous at:

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

The greatest integer function defined by

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

is not differentiable at

x =

_____

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

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

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

A thin prism having refracting angle

10^{o}

is made of glass of refractive index 1.42. This prism is combined with another thin prism made of glass of refractive index 1.7. This combination produces dispersion without deviation. The refracting angle of second prism should be

Which measure of dispersion is defined as the average of the squared differences from the Mean?

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !