Today I am going to give you the Free PDF solutions of Continuity and Differentiability class 12, that too with pdf which you can easily make your notes. This is the most important chapter of NCERT maths solutions because questions come from this chapter in most of the CBSE and ICSE Board Exams.
There are a total of Eight exercises in Chapter 5 Continuity and Differentiability class 12 which you will get in the form of bullet points below. We will learn about each exercise in detail further. This exercise is very important, so you must prepare its solutions well.
- Exercises 5.1
- Exercises 5.2
- Exercises 5.3
- Exercises 5.4
- Exercises 5.5
- Exercises 5.6
- Exercises 5.7
- Exercises 5.8
- Exercises Miscellaneous
Now let us know through a table in which exercise of Continuity and Differentiability class 12, which topics are there so that we can understand each exercise well and solve all the questions in a better way.
| Exercise | Topic Covered |
|---|---|
| Ex 5.1 | Introduction to Continuity and Differentiability |
| Ex 5.2 | Understanding Continuity and its Types |
| Ex 5.3 | Understanding Differentiability and its Types |
| Ex 5.4 | Exponential and Logarithmic Functions and their Derivatives |
| Ex 5.5 | Logarithmic Differentiation and its Applications |
| Ex 5.6 | Derivatives of Functions in Parametric Forms |
| Ex 5.7 | Second Order Derivatives and their Applications |
| Ex 5.8 | Mean Value Theorem and its Applications |
| Others | Miscellaneous Questions and Answers |
You must be seeing the Continuity and Differentiability class 12 table above, in which it has been shown very well which topics have been covered in which exercise, we will learn more about it now. I have made available PDFs of all the chapters of NCERT Maths Solutions for Class 12th on this blog, which you can go and see.
Class 12th NCERT Maths Solutions Chapter 5 PDF: Continuity and Differentiability Class 12
The NCERT Maths Solutions Chapter 5 PDF for Class 12 Continuity and Differentiability is an important resource for students who are studying calculus. This chapter deals with the concepts of continuity and differentiability, which are essential for understanding the behaviour of functions in different scenarios.
The NCERT Maths Solutions Chapter 5 Free PDF for Continuity and Differentiability Class 12 provides step-by-step solutions to all the problems in the chapter. These solutions are designed to help students understand the concepts better and solve problems with ease.
The chapter begins with an introduction to the concept of continuity and its three types: point continuity, uniform continuity, and local continuity. The PDF then goes on to explain the concept of differentiability, its definition, and how it is related to continuity. The chapter also includes detailed explanations of the differentiability of composite functions and implicit functions.
The NCERT Maths Solutions Chapter 5 PDF for Continuity and Differentiability Class 12 also covers the mean value theorem and its applications, as well as the concept of Rolle’s theorem and its applications.
The PDF also includes numerous examples and practice problems, along with their solutions, to help students apply the concepts they have learned and develop their problem-solving skills.
The NCERT Maths Solutions Chapter 5 PDF for Continuity and Differentiability Class 12 is an essential resource for students who want to understand calculus better. It provides comprehensive explanations, examples, and practice problems, which help students to grasp the concepts and develop their problem-solving skills.
We have prepared a comprehensive guide on Continuity and Differentiability that is designed to help you ace your Class 12 Maths exam. In this guide, we will cover all the essential topics related to Continuity and Differentiability Class 12 that are included in the NCERT syllabus.
Introduction to Continuity and Differentiability Class 12 Exercise 5.1
Continuity and Differentiability are two of the most important concepts in calculus. They are closely related and play a significant role in many areas of mathematics, science, and engineering. In this guide, we will provide an in-depth understanding of these concepts, along with examples and exercises to help you master them.
Understanding Continuity Class 12 Exercise 5.2
The concept of Continuity deals with the behaviour of a function when the input changes. In simple terms, it tells us whether a function can be drawn without lifting the pen. A function is said to be continuous if it is possible to draw it without lifting the pen. Mathematically, we can define continuity as follows:
A function f(x) is said to be continuous at a point x = A if and only if the following conditions are satisfied:
- f(a) is defined
- The limit of f(x) as x approaches a exists
- The limit of f(x) as x approaches a is equal to f(a)
In this guide, we will discuss the different types of continuity, along with examples and exercises to help you understand the concept better.
Types of Continuity
There are three types of continuity: point continuity, uniform continuity, and local continuity. Let’s look at each of them in detail.
Point Continuity
A function is said to be point continuous if it is continuous at a single point. In other words, if the function is continuous at x = a, then it is said to be point continuous at x = a.
Uniform Continuity
A function is said to be uniformly continuous if it is possible to choose a single value of δ for all values of ε. In other words, the value of δ does not depend on the choice of x.
Local Continuity
A function is said to be locally continuous if it is continuous in a small neighbourhood around a point. In other words, if the function is continuous in a small interval around x = a, then it is said to be locally continuous at x = a.
Exercise 5.2 Differentiability
Differentiability is another important concept in calculus. It deals with the rate of change of a function at a point. A function is said to be differentiable at a point if the slope of the tangent at that point exists. Mathematically, we can define differentiability as follows:
A function f(x) is said to be differentiable at a point x = A if and only if the following conditions are satisfied:
- f(a) is defined
- The limit of [f(x) – f(a)] / [x – a] as x approaches a exists
- The limit of [f(x) – f(a)] / [x – a] as x approaches a is finite
In this guide, we will discuss the different types of differentiability, along with examples and exercises to help you understand the concept better.
Types of Differentiability
There are two types of differentiability: point differentiability and uniform differentiability. Let’s look at each of them in detail.
Point Differentiability
A function is said to be point differentiable if it is differentiable at a single point. In other words, if the function has a tangent at x = a, then it is said to be point differentiable at x = a.
Uniform Differentiability
A function is said to be uniformly differentiable if the derivative is continuous. In other words, the value of the derivative does not change abruptly.
Dear students, I hope you all have liked “Continuity and Differentiability Class 12” very much. I have also given all of you Continuity and Differentiability Class 12th PDF Solutions above, with the help of which you can read Class 12 Maths Chapter 5 in a better way. Let us now know the answers to some of the questions going on in your mind.
FAQ Related to Continuity and Differentiability Class 12
Frequently asked questions (FAQs) related to Continuity and Differentiability Class 12 are a valuable resource for students who may have doubts or questions about the concepts covered in the chapter. Some of the commonly asked questions include the definition of continuity and differentiability, how to find the derivative of a function, and the various types of functions that are continuous and differentiable.
Q: What is continuity?
A: Continuity is a concept that describes the behaviour of a function when the input changes. A function is said to be continuous if it can be drawn without lifting the pen. In other words, if there are no breaks or holes in the graph of a function, then the function is said to be continuous.
Q: What are the types of continuity?
A: There are three types of continuity: point continuity, uniform continuity, and local continuity.
Q: What is differentiability?
A: Differentiability is a concept that describes the smoothness of a function. A function is said to be differentiable at a point if it has a well-defined tangent line at that point.
Q: What is the relationship between continuity and differentiability?
A: If a function is differentiable at a point, then it is also continuous at that point. However, a function can be continuous at a point but not differentiable at that point.
Q: What is the mean value theorem?
A: The mean value theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists at least one point c in (a,b) such that f'(c) = [f(b) – f(a)]/(b – a).
Q: What is Rolle’s theorem?
A: Rolle’s theorem states that if a function f(x) is continuous on a closed interval [a,b], differentiable on the open interval (a,b), and f(a) = f(b), then there exists at least one point c in (a,b) such that f'(c) = 0.
Q: How can I practice continuity and differentiability problems?
A: You can practice continuity and differentiability problems by solving examples and practice problems from textbooks or online resources, and by taking mock tests and quizzes to assess your understanding of the concepts. The NCERT Maths Solutions Chapter 5 PDF for Class 12 Continuity and Differentiability is a useful resource for practice problems and solutions.
