NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.3 – Continuity and Differentiability

Class 12 Maths Chapter 5 Exercise 5.3 – Continuity and Differentiability:- Chapter 5 of Class 12 Maths, Continuity and Differentiability, explores fundamental concepts in calculus essential for understanding the behavior of functions. Continuity examines the seamless nature of functions across their domains, where a function 

𝑓(𝑥)f(x) is continuous at a point if its limit exists and equals the function's value at that point. Differentiability, on the other hand, focuses on the smoothness of functions, indicating that a function is differentiable at a point if its derivative exists at that point. 

This chapter also covers various types of discontinuities that highlight instances where functions exhibit abrupt changes. Additionally, differentiation rules like the chain rule and product rule are introduced to compute derivatives efficiently, facilitating the analysis of rates of change and slopes in practical applications. Exercise 5.3 in this chapter typically involves problems that challenge students to apply these concepts, including identifying points of continuity, checking differentiability, and solving derivative-based problems to deepen their understanding of calculus principles.

Check out: CBSE Class 12th Books

NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.3 – Continuity and Differentiability

Get NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.3 – Continuity and Differentiability below:-

Find dy/dx in the following Exercise 1 to 15.

Question 1. 2x + 3y = sin y

Solution :

The given relationship is 2x + 3y = sin y

Differentiating this relationship with respect to x, we obtain

Question 2. ax + by2 = cos y

Solution :

The given relationship is ax + by2 = cos y

Differentiating this relationship with respect to x, we obtain

Question 3. xy + y2 = tanx + y

Solution :
The given relationship is xy + y2 = tanx + y

Differentiating this relationship with respect to x, we obtain

Question 4. x2 + xy + y2 = 100

Solution :

The given relationship is x2 + xy + y2 = 100

Differentiating this relationship with respect to x, we obtain

Question 5. 2x + 3y = sin y

Solution :

The given relationship is 2x + 3y = sin y

Differentiating this relationship with respect to x, we obtain

Question 6.

Solution :
The given relationship is

Differentiating this relationship with respect to x, we obtain

Question 7. sin2 y + cos xy = π

Solution :
The given relationship is sin2 y + cos xy = π

Differentiating this relationship with respect to x, we obtain

Check out: Class 12th Sample Papers

Question 8. sin2 x + cos2 y = 1

Solution :
The given relationship is  sin2 x + cos2 y = 1
Differentiating this relationship with respect to x, we obtain

Question 9.

Solution :

We have,

Question 10.

Solution :

Question 11.

Solution :

Question 12.

Solution :

Question 13.

Solution :

Question 14.

Solution :

Question 15.

Solution :

Check Out: Class 12th Question Banks

Class 12 Maths Chapter 5 Exercise 5.3 – Continuity and Differentiability Summary

Continuity Concepts:

• Definition: A function f(x)f(x)f(x) is continuous at a point x=ax = ax=a if lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a)limx→a​f(x)=f(a).

• Types of Continuity: Discusses continuity over intervals and at specific points.

Differentiability Concepts:

• Definition: A function f(x)f(x)f(x) is differentiable at x=ax = ax=a if f′(a)f'(a)f′(a) exists.

• Relationship with Continuity: Differentiability implies continuity but not vice versa.

Types of Discontinuities:

• Identifying Jump, Infinite, and Removable Discontinuities: Examples and criteria for each type.

Differentiation Rules:

• Chain Rule, Product Rule, Quotient Rule: Techniques to differentiate composite functions, products, and quotients.

Applications of Differentiation:

• Finding Rates of Change: Using derivatives to determine slopes, velocities, and accelerations.

• Optimization Problems: Solving maximum and minimum value problems using derivative techniques.

Check out: School Books

Class 12 Maths Chapter 5 Exercise 5.3 – Continuity and Differentiability FAQs

Q1. What is continuity in mathematics?

Ans. Continuity refers to the uninterrupted nature of a function over its domain. A function f(x)f(x)f(x) is continuous at a point x=ax = ax=a if lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a)limx→a​f(x)=f(a).

Q2. How do you determine if a function is continuous at a point?

Ans. Check if the limit of the function as xxx approaches the point equals the function's value at that point f(a)f(a)f(a).

Q3. What is differentiability?

Ans. Differentiability of a function f(x)f(x)f(x) at a point x=ax = ax=a means that the derivative f′(a)f'(a)f′(a) exists. It indicates the function's smoothness at that point.

Q4. What's the relationship between continuity and differentiability?

Ans. Differentiability implies continuity, but continuity does not necessarily imply differentiability. A function must be continuous at a point for it to be differentiable at that point.

Q5. What are the types of discontinuities?

Ans. Discontinuities can be classified as jump discontinuities (where the limit from either side of the point exists but is not equal), infinite discontinuities (where the limit approaches infinity), and removable discontinuities (where the function can be made continuous by redefining it at that point).