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RS Aggarwal Solutions Class 12 Chapter-9 Continuity and Differentiability

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Last updated date: 29th Mar 2024
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MVSAT 2024

Class 12 RS Aggarwal Chapter-9 Continuity and Differentiability Solutions - Free PDF Download

Continuity and Differentiability Class 12 RS Aggarwal is an important topic for the students preparing for the Class 12th Board examination. The textbook provides stepwise solutions for the students so that they can easily solve various problems. It will be a great resource for practise and revision purposes that will level them up in the growing competitive scenario. The concepts that are part of this chapter also seem to be helpful for future studies. The simple and easy interface of the exercise with questions and solutions make it uncomplicated to grasp the topic. Experts from the concerned discipline are involved in framing the chapter along with a detailed explanation. So, by following the RS Aggarwal Class 12 Maths Chapter 9 Solutions PDF by Vedantu students can excel in the examination with higher grades.

Competitive Exams after 12th Science

Class 12 RS Aggarwal Chapter-9 Continuity and Differentiability Solutions - Free PDF Download

Concept of Continuity and Differentiation

Continuity and Differentiation explained in RS Aggarwal Solutions Class 12 Maths Ch 12 is one of the vital concepts for students preparing for board exams. It deals with concepts like the derivative of functions, continuity of certain points, continuity of given intervals, and others. 


The continuity of a function determines the attributes of a function and its functional values.  In a given domain or interval, a function is accepted as continuous if the curve has no breaking points or missing points. That is the curve has to be continuous at every point in its domain. 


A function f(x) is said to be continuous at a point x= y if it meets the following three conditions.

  1.  f(y) is continuous if the value of f(y) is finite

  2.  f(x) limx→af is continuous at the point if the right-hand limit is the same as that of the left-hand limit. Therefore, R.H.S = L.H.S. 

  3.  lim x→af f(x)= f(y)

In a given interval, f(x) can only be continuous if it is equal to x1,x2. All the conditions need to be satisfied with each and every point. 


In differentiation, f(x) is said to be differentiable at the point x = y if the derivative f ‘(y) exists at each point in its interval or domain.

The differentiability formula is given by

f’ (y) = {f(y+h)−f(y)}/ h


State the Differentiability Formula for the Derivatives of the Basic Trigonometric Function

In a particular point, if a function is continuous, then the function can be differentiable at any point x=y, in its domain. However, the vice-versa may not be true every time. 


In the given table, the derivatives of the basic trigonometric functions are explained below from the aspect of differentiability formula: 


d/dx (sin x) →

cos x

d/dx (cos x) →

-sin x

d/dx (tan x)→

sec²x

d/dx (cot x)→

cosec²x

d/dx (sec x)→

sec x tan x

d/dx (cosec x)→

Cosec x cot x


Highlights of the RS Aggarwal Solutions for Class 12 Continuity and Differentiability Chapter

  • The meaning of continuous functions has been explained in this chapter with reference to graphs. 

  • The distinction between continuous and discontinuous functions can be expressed with the help of a graph.

  • By practising the solutions given in this chapter, one can understand the proofs of different theorems and the behaviour of continuous functions when these are subjected to algebraic calculations. 

  • Students can study several corollaries extracted from theorems and prove them. 


Solved Exercise Questions of RS Aggarwal Class 12 Chapter 9 

Question: 

f(x) = x²

LHL at x = 2

Limx→2 f(x)= Limx→0 f(2-h)

                    = Limx→0 f(2 - h)²

                    = Limx→0 f (h² - 4h + 4)

                    = (0-4) x (0+4)

                    = 4

RHL at x = 2

Limx→2 f(x)= Limx→0 f(2-h)

                    = Limx→0 f (2 - h)²

                    = Limx→0 f (h² - 4h + 4)

                    = (0-4) x (0+4)

                    = 4

Therefore, fx is continuous at 2.


Students studying this maths topic should know about the relationship between differentiability and continuity. This will be an interesting thing to learn in their course of the syllabus.

  • If x is a differentiable function, then x has to be continuous. 

  • A function can be continuous without it being differentiable.


Exercise-Wise Discussion of RS Aggarwal Solutions for Class 12 Chapter 9 Continuity and Differentiability

  • In the exercise solutions of 9A, you will learn about the conditions required to prove whether a function is continuous or discontinuous by checking the mandatory condition. 

  • You have to check the continuity of more complex functions in the exercise solution of 9B.

  • You need to check both the continuity and differentiability of the given functions in the exercise solution of 9C.

 

Key Features of RS Aggarwal Solutions Class 12 Maths Chapter 9 

Some of the key features of RS Aggarwal Solutions of Class 12 Maths Chapter 9 are listed below.

  • The solutions of Chapter 9 of Class 12 Maths are developed by our subject-matter experts who have years of experience in this field to deliver you complete and precise solutions for all the questions. 

  • The answers are written in a very simple language and step-by-step manner, which will help build strong calculus fundamentals. 

  • Practising these questions will help improve your marks and boost your analytical skills.

  • The solutions are available in free PDF format to download, and you can access this free PDF by clicking on “Download PDF”. 


Exercise-Wise Discussion of RS Aggarwal Solutions for Class 12 Chapter 9 Continuity and Differentiability

  • In the exercise of 9A, you will learn about the conditions required to prove whether a function is continuous or discontinuous by checking the mandatory conditions. 

  • You have to check the continuity of more complex functions in exercise 9B.

  • You need to check both the continuity and differentiability of the given functions in exercise 9C.


Key Features of RS Aggarwal Solutions Class 12 Maths Chapter 9 

Some of the key features of RS Aggarwal Solutions of Class 12 Maths Chapter 9 are listed below.

  • The solutions of Chapter 9 of Class 12 Maths are developed by our subject-matter experts who have years of experience in this field to deliver you complete and precise solutions for all the questions. 

  • The answers are written in a very simple language and step-by-step manner, which will help build strong calculus fundamentals. 

  • Practising these questions will help improve your marks and boost your analytical skills.

  • The solutions are available in free PDF format to download, and you can access this free PDF by clicking on “Download PDF”. 

FAQs on RS Aggarwal Solutions Class 12 Chapter-9 Continuity and Differentiability

1. What are Important Concepts to Learn from RS Aggarwal Class 12 Solutions Continuity and Differentiability?

In chapter Continuity and Differentiability Class 12 RS Aggarwal, there are questions formatted on proving an equation as continuous with given different values of x. 


Some of the crucial aspects of the chapter are listed below:

  • The sum, difference, quotient, and product of continuous functions are continuous. 

  • The differentiable function at a given point can be continuous but the opposite may not be true. 

  • Rolle’s Theorem: If f: [ x,y]→ R is continuous on [x,y] and differentiable on [x,y] such that f (x) = f(y). There should also be a point z somewhere in ( x,y) and f (z)= 0.

  • Mean Value Theorem: If f : [x, y] → R is continuous on [x, y] and differentiable on [x, y]. There should also be a point z somewhere in ( x,y) and f (z)= (f(y) – f(x))/(y-x).

2. How RS Aggarwal Solutions Class 12 Maths Ch 12 Exercise Help in Exam Preparation?

The RS Aggarwal Class 12 Maths Chapter 9 Solutions will be great for CBSE students preparing for their 12th Boards. 

  • It will provide a brief analysis of the mandatory conditions that are necessary to prove a function whether it is continuous or discontinuous. 

  • In the exercise of Chapter 9 Solutions, you can check continuity of more complex functions. 

  • As you make progress in Chapter 9 of RS Aggrawal look over both the continuity and differentiability of a given function. 

  • Moreover, the exercise questions are designed similar to paper patterns of competitive exams. This will be beneficial for advance preparation and will boost the analytic skills of the students. 

  • Students preparing for boards, need to solve these questions if they wish to score better in the examination. 

3. Where can I download revision notes for CBSE mathematics class 12 chapter 9 Differential Equations?

Students can download the Class 12 revision notes for Mathematics Chapter 9 from the Vedantu website where they can avail themselves of all of it for free. Vedantu offers free of cost study materials for students to download and use at their convenience on any device. These revision notes are put together by expert teachers who cover all the concepts of the chapter as well as any important formulas along with illustrative examples to help students understand them better. These revision notes make last-minute studying and revision quicker as students don’t have to go through the entire chapter and can just look at the important points from these notes.

4. Why should I refer to Class 12 mathematics RS Aggarwal Solutions for Chapter 9 for my board exam or JEE preparation?

Students of class 12 must refer to the solutions of RS Aggarwal for chapter 9 from Vedantu while preparing for their boards or any other entrance exam. These solutions help students understand the higher-order thinking skills sums provided in the reference book such as RS Aggarwal which tend to be more difficult than the questions in NCERT. While preparing for entrance exams, it is important that students expose themselves to more practice outside of the NCERT textbooks, and solving the RS Aggarwal questions and exercises can help them gain a deeper understanding of the topic as well. All the steps of the answers created by Vedantu experts, also available in the app, are laid out in clear steps so that students do not have any remaining doubts. These questions have a high chance of appearing in the class 12 board exam or the JEE examination and therefore solving these solutions can help students prepare for their exams.

5. How many exercises are there in Chapter 9 of Class 12 Mathematics RS Aggarwal?

There are a total of 3 exercises in Class 12 Chapter 9 of  Aggarwal which focuses on the concept of Continuity and Differentiability. Exercise 9A has questions that ask students to prove the conditions that define if the function is continuous or discontinuous while Exercise 9B exposes students to more difficult complex functions. Exercise 9C has questions that require the students to check for both continuity and differentiability of the functions. The questions cover concepts such as proofs of various theorems and the behaviours of continuous and discontinuous functions from graphs.