NCERT Solutions Continuity & Differentiability

Exercise 5.1

Q1: Prove that the function 

Ans: 

Therefore, f is continuous at x = 5.

Q2: Examine the continuity of the function 

.
Ans:

Thus, f is continuous at x = 3.

Q3: Examine the following functions for continuity.
(a) 

(b)
(c) 
(d) 
Ans: 

(c) The given function is
For any real number c ≠ -5, we obtain


Therefore, f is continuous at all real numbers greater than 5.
Hence, f is continuous at every real number and therefore, it is a continuous function.

Q4: Prove that the function 

  is continuous at x = n, where n is a positive integer.
Ans: The given function is f (x) = xⁿ.
It is evident that f is defined at all positive integers n, and its value at n is nⁿ.

Therefore, f is continuous at n, where n is a positive integer.

Q5: Is the function f defined by

 continuous at x = 0? At x = 1? At x = 2?
Ans: The given function f is
At x = 0,
It is evident that f is defined at 0 and its value at 0 is 0.

The right hand limit of f at x = 1 is,

Therefore, f is continuous at x = 2.

Q6: Find all points of discontinuity of f, where f is defined by

Ans: The given function f is
It is evident that the given function f is defined at all the points of the real line.
Let c be a point on the real line. Then, three cases arise.
(i) c < 2
(ii) c > 2
(iii) c = 2
Case (i) c < 2

Therefore, f is continuous at all points x such that x < 2.
Case (ii) c > 2

It is observed that the left and right hand limits of f at x = 2 do not coincide.
Therefore, f is not continuous at x = 2.
Hence, x = 2 is the only point of discontinuity of f.

Q7: Find all points of discontinuity of f, where f is defined by 

Ans: The given function f is
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x such that x < -3.
Case II:

It is observed that the left and right hand limits of f at x = 3 do not coincide.
Therefore, f is not continuous at x = 3.
Case V:

Therefore, f is continuous at all points x such that x > 3.
Hence, x = 3 is the only point of discontinuity of f.

Q8: Find all points of discontinuity of f, where f is defined by

Ans: The given function f is

It is observed that the left and right hand limits of f at x = 0 do not coincide.
Therefore, f is not continuous at x = 0.
Case III:

Therefore, f is continuous at all points x such that x > 0.
Hence, x = 0 is the only point of discontinuity of f.

Q9: Find all points of discontinuity of f, where f is defined by

Ans: The given function f is

Therefore, the given function is a continuous function.
Hence, the given function has no point of discontinuity.

Q10: Find all points of discontinuity of f, where f is defined by

Ans: 
The given function f is
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x such that x > 1.
Hence, the given function f has no point of discontinuity.

Q11: Find all points of discontinuity of f, where f is defined by 

Ans: The given function f is
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x such that x > 2.
Thus, the given function f is continuous at every point on the real line.
Hence, f has no point of discontinuity.

Question 12: Find all points of discontinuity of f, where f is defined by 

Ans: The given function f is
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x such that x > 1.
Thus, from the above observation, it can be concluded that x = 1 is the only point of discontinuity of f.

Q13: Is the function defined by

a continuous function?
Ans: The given function is
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x such that x > 1.
Thus, from the above observation, it can be concluded that x = 1 is the only point of
discontinuity of f.

Q14: Discuss the continuity of the function f, where f is defined by

Ans: The given function is
The given function is defined at all points of the interval [0, 10].
Let c be a point in the interval [0, 10].
Case I:

Therefore, f is continuous at all points of the interval (1, 3).
Case IV:

Therefore, f is continuous at all points of the interval (3, 10].
Hence, f is not continuous at x = 1 and x = 3.

Q15: Discuss the continuity of the function f, where f is defined by

Ans: The given function is
The given function is defined at all points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points of the interval (0, 1).
Case IV:

Therefore, f is continuous at all points x such that x > 1.
Hence, f is not continuous only at x = 1.

Q16: Discuss the continuity of the function f, where f is defined by

Ans: The given function f is
The given function is defined at all points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at x = -1.
Case III:

Therefore, f is continuous at all points x such that x > 1.
Thus, from the above observations, it can be concluded that f is continuous at all points of the real line.

Q17: Find the relationship between a and b so that the function f defined by is continuous at x = 3.

Ans: The given function f is
If f is continuous at x = 3, then

Q18: For what value of is the function defined by

 continuous at x = 0? What about continuity at x = 1?
Ans: The given function f is
If f is continuous at x = 0, then

Therefore, there is no value of λ for which f is continuous at x = 0. At x = 1,
f (1) = 4x + 1 = 4 × 1 + 1 = 5.

Therefore, for any values of λ, f is continuous at x = 1.

Q19: Show that the function defined by g(x)= x-[x] is discontinuous at all integral point. Here [x] denotes the greatest integer less than or equal to x.
Ans: The given function is g(x)= x-[x].
It is evident that g is defined at all integral points.
Let n be an integer.
Then,

It is observed that the left and right hand limits of f at x = n do not coincide.
Therefore, f is not continuous at x = n.
Hence, g is discontinuous at all integral points.

Q20: Is the function defined by 

  continuous at x = p?
Ans: The given function is
It is evident that f is defined at x = p.

Therefore, the given function f is continuous at x = π.

Q21: Discuss the continuity of the following functions.
(a) f (x) = sin x + cos x
(b) f (x) = sin x - cos x
(c) f (x) = sin x × cos x
Ans: It is known that if g and h are two continuous functions, then
g+h, g-h, and g·h are also continuous.
It has to be proved first that g (x) = sin x and h (x) = cos x are continuous functions.
Let g (x) = sin x.
It is evident that g (x) = sin x is defined for every real number.
Let c be a real number. Put x = c + h.
If x → c, then h → 0.

Therefore, g is a continuous function.
Let h (x) = cos x.
It is evident that h (x) = cos x is defined for every real number.
Let c be a real number. Put x = c + h.
If x → c, then h → 0. h (c) = cos c.

Therefore, h is a continuous function.
Therefore, it can be concluded that
(a) f (x) = g (x) + h (x) = sin x + cos x is a continuous function.
(b) f (x) = g (x) - h (x) = sin x - cos x is a continuous function.
(c) f (x) = g (x) × h (x) = sin x × cos x is a continuous function.

Q22: Discuss the continuity of the cosine, cosecant, secant and cotangent functions,
Ans: It is known that if g and h are two continuous functions, then

It has to be proved first that g (x) = sin x and h (x) = cos x are continuous functions. Let g (x) = sin x.
It is evident that g (x) = sin x is defined for every real number.
Let c be a real number. Put x = c + h. Therefore, g is a continuous function.
Let h (x) = cos x.
It is evident that h (x) = cos x is defined for every real number.
Let c be a real number. Put x = c + h.
If x → c, then h → 0.
Therefore, h (x) = cos x is a continuous function.


Therefore, cotangent is continuous except at x = nπ, n ∈ Z.

Q23: Find the points of discontinuity of f, where 

Ans:

Therefore, f is continuous at x = 0.
From the above observations, it can be concluded that f is continuous at all points of the real line.
Thus, f has no point of discontinuity.

Q24:Determine if f defined by

is a continuous function?
Ans: The given function f is
It is evident that f is defined at all points of the real line.
Let c be a real number.
Case I:


Therefore, f is continuous at x = 0.
From the above observations, it can be concluded that f is continuous at every point of the real line.
Thus, f is a continuous function.

Q25: Examine the continuity of f, where f is defined by

Ans: The given function f is
It is evident that f is defined at all points of the real line.
Let c be a real number.
Case I:

Therefore, f is continuous at x = 0.
From the above observations, it can be concluded that f is continuous at every point of the real line.
Thus, f is a continuous function.

Q26: Find the values of k so that the function f is continuous at the indicated point.

Ans:


Therefore, the required value of k is 6.

Question 27: Find the values of k so that the function f is continuous at the indicated point.

Ans: The given function is  
The given function f is continuous at x = 2, if f is defined at x = 2 and if the value of f at x = 2 equals the limit of f at x = 2.
It is evident that f is defined at x = 2 and f(2) = k(2)² = 4k.

Q28: Find the values of k so that the function f is continuous at the indicated point.

Ans: The given function is
The given function f is continuous at x = p, if f is defined at x = p and if the value of f at x = p equals the limit of f at x = p.

Q29: Find the values of k so that the function f is continuous at the indicated point.

Ans: The given function f is
The given function f is continuous at x = 5, if f is defined at x = 5 and if the value of f at x = 5 equals the limit of f at x = 5.

Q30: Find the values of a and b such that the function defined by

, is a continuous function.
Ans: The given function f is
It is evident that the given function f is defined at all points of the real line.
If f is a continuous function, then f is continuous at all real numbers.
In particular, f is continuous at x = 2 and x = 10.
Since f is continuous at x = 2, we obtain

Therefore, the values of a and b for which f is a continuous function are 2 and 1 respectively.

Q31: Show that the function defined by f(x) = cos (x²) is a continuous function.
Ans: The given function is f (x) = cos (x²).
This function f is defined for every real number and f can be written as the composition of two functions as,
f = g ∘ h, where g (x) = cos x and h (x) = x².

Therefore, g (x) = cos x is continuous function.
h (x) = x².
Clearly, h is defined for every real number.
Let k be a real number, then h (k) = k².

Therefore, h is a continuous function.
It is known that for real-valued functions g and h, such that (g ∘ h) is defined at c, if g is continuous at h(c) and if h is continuous at c, then (g ∘ h) is continuous at c.
Therefore, is a continuous function.

Q32: Show that the function defined by 

 is a continuous function.
Ans: The given function is  
This function f is defined for every real number and f can be written as the composition of two functions as,

Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:

Therefore, g is continuous at all points x such that x > 0.

Q33: Examine that sin|x| is a continuous function.
Ans:

This function f is defined for every real number and f can be written as the composition of two functions as,

Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:

Therefore, g is continuous at x = 0.
From the above three observations, it can be concluded that g is continuous at all points. h (x) = sin x.
It is evident that h (x) = sin x is defined for every real number.
Let c be a real number. Put x = c + k.
If x → c, then k → 0. h (c) = sin c.

Therefore, h is a continuous function.  
It is known that for real-valued functions g and h, such that (g ∘ h) is defined at c, if g is continuous at h(c) and if h is continuous at c, then (g ∘ h) is continuous at c.  
 is a continuous function.

Q34: Find all the points of discontinuity of f defined by 

.
Ans: The given function is
The two functions, g and h, are defined as

Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:


From the above three observations, it can be concluded that g is continuous at all


Case II:  

Clearly, h is defined for every real number.
Let c be a real number.
Case I:

Therefore, h is continuous at all points x such that x < -1.
Case III:

Therefore, h is continuous at x = -1.
From the above three observations, it can be concluded that h is continuous at all points of the real line. g and h are continuous functions. Therefore, f = g - h is also a continuous function.
Therefore, f has no point of discontinuity.

Exercise 5.2

Continuity & Differentiability

 Question 1: Differentiate the functions with respect to x.

Answer

Question 2: Differentiate the functions with respect to x. cos(sinx)

Answer

Thus, f is a composite function of two functions. 

Put t = u (x) = sin x. 

By chain rule,
Alternate method  

Question 3: Differentiate the functions with respect to x.

sin(ax + b)

Answer

Alternate method

Question 4: Differentiate the functions with respect to x.

Answer

Hence, by chain rule, we obtain

Question 5: Differentiate the functions with respect to x.

Answer

The given function is

Put y = p (x) = cx + d.

Question 6: Differentiate the functions with respect to x. 

Answer

Question 7:  Differentiate the functions with respect to x.

Answer

Question 8: Differentiate the functions with respect to x. 

Answer

Clearly, f is a composite function of two functions, u and v, such that

Alternate method

Question 9:
 Prove that the function f given by

 is not differentiable at x = 1.

Answer
The given function is

It is known that a function f is differentiable at a point x = c in its domain if both

 are finite and equal.

To check the differentiability of the given function at x = 1,  

consider the left hand limit of f at x = 1  

Since the left and right hand limits of f at x = 1 are not equal, f is not differentiable at x = 1.

Question 10:
Prove that the greatest integer function  

 defined by is not differentiable at x = 1 and x = 2.

Answer
The given function f is  

It is known that a function f is differentiable at a point x = c in its domain if both

 are finite and equal. 

To check the differentiability of the given function at x = 1, consider the left hand limit  of f at x = 1  

Since the left and right hand limits of f at x = 1 are not equal, f is not differentiable at x = 1.

To check the differentiability of the given function at x = 2, consider the left hand limit of f at x = 2

Since the left and right hand limits of f at x = 2 are not equal, f is not differentiable at x = 2.

Exercise 5.3

Continuity & Differentiability

 Question 1:

Answer

Question 2:

         

Answer

Question 3:

Answer

Question 4:

Answer
The given relationship is
Differentiating this relationship with respect to x, we obtain

Question 5: 

Find 

Answer

The given relationship is x² +xy +y² = 100.

Differentiating this relationship with respect to x, we obtain 

[Derivative of constant function is 0]. 

Question 6:

Answer

Question 7:

Answer

Using chain rule, we obtain

Question 8:

Find 

Answer
The given relationship is

Differentiating this relationship with respect to x, we obtain

Question 9:

Find  

Answer

Therefore, by quotient rule, we obtain

Question 10:

 

Answer

 

Question 11:

 

Answer
The given relationship is,

On comparing L.H.S. and R.H.S. of the above relationship, we obtain  tan(y/2) = x.

Differentiating this relationship with respect to x, we obtain 

Question 12:

 

Answer
The given relationship is

 

From (1), (2), and (3), we obtain

Differentiating this relationship with respect to x, we obtain

Question 13:

Answer

Question 14:

Find 

Answer

Differentiating this relationship with respect to x, we obtain

 

Question 15:

Find dy/dx  

Answer

Differentiating this relationship with respect to x, we obtain 

Exercise 5.4

Q1: Differentiate the following w.r.t. x: 

Ans: Let y =
By using the quotient rule, we obtain

Q2: Differentiate the following w.r.t. x: e^sin-1x
Ans: Let y = e^sin-1x.
By using the chain rule, we obtain

Q3: Differentiate the following w.r.t. x: 

Ans: Let y =
By using the chain rule, we obtain

Q4: Differentiate the following w.r.t. x: sin(tan^-1e^-x)
Ans: Let y = sin(tan^-1e^-x).
By using the chain rule, we obtain

Q5: Differentiate the following w.r.t. x: log(cose^x)
Ans: Let y = log(cose^x).
By using the chain rule, we obtain

Q6: Differentiate the following w.r.t. x:

Ans:

Q7: Differentiate the following w.r.t. x: 
Ans: Let

Q8: Differentiate the following w.r.t. x:

Ans: Let y =
By using the chain rule, we obtain

Q9: Differentiate the following w.r.t. x:

Ans: Let y =
By using the quotient rule, we obtain

Q10: Differentiate the following w.r.t. x: 

, x>0
Ans: Let y =
By using the chain rule, we obtain
 

Exercise 5.5

Continuity & Differentiability 

Question 1: Differentiate the function with respect to x.

Answer

Question 2: Differentiate the function with respect to x.

Answer

Let y =

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain 

Question 3: Differentiate the function with respect to x. 

Answer

Let y =

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Question 4: Differentiate the function with respect to x. x^x - 2sinx

Answer

Let y =

u = x^x.
Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

 

Question 5: Differentiate the function with respect to x.

Answer

Let y =

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

 

Question 6: Differentiate the function with respect to x.

Answer

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

Therefore, from (1), (2), and (3), we obtain

Question 7: Differentiate the function with respect to x.

Answer

Let y =

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

Question 8: Differentiate the function with respect to x.

Answer

Let y =

Differentiating both sides with respect to x, we obtain

Therefore, from (1), (2), and (3), we obtain

Question 9: Differentiate the function with respect to x.

Answer
Let y =

Differentiating both sides with respect to x, we obtain

Question 10: Differentiate the function with respect to x.

Answer

Let y =

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

Question 11: Differentiate the function with respect to x.

Answer

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

 

Question 12:
 Find 

 of function .

Answer

Differentiating both sides with respect to x, we obtain

Question 13:

Find 

Answer

Differentiating both sides with respect to x, we obtain

Question 14:
 Find

 of function

 .

Answer

Differentiating both sides, we obtain

Question 15:
 Find 

  of function .

Answer

Differentiating both sides with respect to x, we obtain

Question 16:
 Find the derivative of the function given by  

  and
 hence find .

Answer
The given relationship is

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Question 17:
 Differentiate   in three ways mentioned below
 (i) By using product rule.
 (ii) By expanding the product to obtain a single polynomial.
 (iii By logarithmic differentiation.
 Do they all give the same answer?

Answer

Let y =

(i)

(ii)

 

( iii)

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

From the above three observations, it can be concluded that all the results of

  are the same.

Question 18: If u, v and w are functions of x, then show that

in two ways-first by repeated application of product rule, second by logarithmic
 differentiation.

Answer
Let

By applying product rule, we obtain

Exercise 5.6

Continuity & Differentiability

 Question 1:
 If x and y are connected parametrically by the equation, without eliminating the

Answer

Question 2: If x and y are connected parametrically by the equation, without eliminating the
 parameter, find 

.
 .x = a cos θ, y = b cos θ

Answer
The given equations are x = a cos θ and y = b cos θ.

Question 3: If x and y are connected parametrically by the equation, without eliminating the
 parameter, find 

.
 .x = sin t, y = cos 2t

Answer
The given equations are x = sin t and y = cos 2t.

 

Question 4:
 If x and y are connected parametrically by the equation, without eliminating the

.

Answer

Question 5: If x and y are connected parametrically by the equation, without eliminating the 

 Answer
The given equations are

Question 6: If x and y are connected parametrically by the equation, without eliminating the

Answer

The given equations are

Question 7: If x and y are connected parametrically by the equation, without eliminating the

parameter,  find  

.

Answer
The given equations are

 

Question 8: If x and y are connected parametrically by the equation, without eliminating the

parameter, find 

Answer
The given equations are

 Question 9: If x and y are connected parametrically by the equation, without eliminating the
 parameter, find .

Answer

 

Question 10: If x and y are connected parametrically by the equation, without eliminating the

parameter, find  

.

Answer

Question 11:
 If 

Answer

Hence, proved.

Exercise 5.7

Continuity & Differentiability

Question 1: Find the second order derivatives of the function.

Answer
Let y =

Then,

Question 2: Find the second order derivatives of the function. x²⁰

Answer
Let y = x²⁰.
Then,

Question 3: Find the second order derivatives of the function. x.cos x

Answer
Let y = x.cos x.
Then,

Question 4: Find the second order derivatives of the function. log x

Answer
Let y = log x.
Then,

Question 5: Find the second order derivatives of the function. x³ log x

Answer
Let y = x³ log x.
Then,

Question 6: Find the second order derivatives of the function. 

Answer

Let y =

 

Question 7: Find the second order derivatives of the function.

Answer
Let y =

Then,

Question 8: Find the second order derivatives of the function.

Answer
Let y =

Then,

 

Question 9: Find the second order derivatives of the function.

Answer
Let y =

Then,

Question 10: Find the second order derivatives of the function.

Answer
Let y =

Then,

 

Question 11: If 

Answer
It is given that,

Then,

Hence, proved.

Question 12:
 If  

  in terms of y alone.

Answer
It is given that,

Then,

Question 13:
 If 

Answer
It is given that,  

Then,

Hence, proved.
Question 14:
 If , show that  

Answer
It is given that,

Then,

Hence, proved.

Question 15:
 If 

 , show that  

Answer
It is given that,

Then,

Hence, proved.

Question 16:
 If  

, show that 

Answer
The given relationship is

Taking logarithm on both the sides, we obtain

Differentiating this relationship with respect to x, we obtain

Hence, proved.

Question 17:
 If 

, show that 

Answer
The given relationship is

Then,

 
Hence, proved.