CBSE Class 12 Maths Revision Notes Chapter 5

Class 12 Mathematics Chapter 5 Notes

Mathematics can be easy or difficult, depending on how well a student understands the concepts. It is essential to have a strong foundation so that it becomes easier to grasp knowledge about high-level topics. This subject cannot be mastered with memorisation. To excel in Mathematics, students need to practice regularly. The Class 12 Mathematics chapter 5 notes- Continuity and Differentiability give a clear understanding of different concepts, such as the algebra of continuous functions. It also includes differentiability of different functions such as inverse trigonometric, implicit, composite and more. In this chapter, students will learn about second-order derivatives and the Mean Value theorem. 

Extramarks, an online learning platform, provides one of the best academic materials to help students understand concepts and ace their exams. The Class 12 Mathematics notes chapter 5 help to understand the chapter. Differentiability and continuity are crucial concepts that help students learn other mathematical branches. The notes act as a helping guide and aid in quick revision. 

Visit Extramarks for all the latest news and updates about CBSE exams and get access to various study materials like the Class 12 Mathematics chapter 5 notes in addition to various other study materials. Students can also solve many practice tests and mock tests available on the website for free. 

List Of The Topics Covered In Class 12 Mathematics Chapter 5 Notes

The list of topics covered in Extramarks class 12 Mathematics notes chapter 5 Continuity and Differentiability include the following.

• Introduction
• Continuity and Algebra of Continuous Function
• Differentiability
• Derivatives of Composite, Implicit and Inverse Trigonometric Function
• Logarithmic and Exponential Function
• Logarithmic Differentiation
• Parametric Form Differentiation
• Second-Order Derivative
• Mean Value Theorem and Rolle’s Theorem

Key Topics Covered In Class 12 Mathematics Chapter 5 Notes

The key topics covered under Class 12 Mathematics Chapter 5 notes include

Definition of Continuity at a point: 

The continuity of a function (f) at any point x = c, where c belongs to the domain of the function, is given as, Left-hand limit of f(x = c) = Right-hand limit of f(x = c) = Value of f(x=c) , i.e., 

If x = c, then LHL = RHL = f(c).  

Therefore, xa–f(x) = xa+f(x) = f (x = c) = f(c) 

REMEMBER: 

1. To find the solution for LHL of f(x) at x=0, substitute x = a – h and for RHL, substitute x = a + h.
2. If xa–f(x) and xa+f(x) exists but LHL RHL, then the function is not continuous. 
3. If xa–f(x) and xa+f(x) exists but LHL = RHL f(c), then the function is discontinuous. 
4. At least one limit of the f does not exist; then, the function is discontinuous. 

Definition of Continuity in an Interval (INTERMEDIATE VALUE THEOREM):

The continuity of a function (f) in an interval [a, b] where a<b then, if the function is continuous at all points, including the end-points in the given interval. 

Continuity at point ‘a’ is given as, 

 xaf(x) = f (a)

Continuity at point ‘b’ is given as, 

 xbf(x) = f (b)

Algebra of Continuous Functions

If f(x) and g(x) be two real functions. If they are continuous at any point c R, then

1. f(x) + g(x) is continuous at c.
2. f(x) – g(x) is also continuous at c.
3. f(x) . g(x) is continuous at c.
4. f(x) / g(x) is continuous at c, g(c) 0.
5. The product c.f(x) is continuous, where c is a constant. 
6. Consider a composite function fog(x), if g (x) is continuous at x and f(x) is continuous at g(c), then fog(x) is continuous at point c. 
7. If f(x) is continuous, then f(x) is also continuous. 

Definition and formula of differentiability: 

If f(x) is a real function. It is said to be differentiable at a point c in its domain if the left-hand derivative is equal to the right-hand derivative of the function at point c, i.e.

LHD f(x=c) = RHD f(x=c).

The derivative of a function is given ash0f(c+h) – f(c)h 

Therefore, RHD f’(c) = h0f(c+h) – f(c)h  and LHD f’(c) = h0f(c-h) – f(c)-h 

Types of Discontinuity: 

REMOVABLE DISCONTINUITY:

If xaf(x) exists but is not equal to f(c), for c R, then function f(x) has removable discontinuity. In such a case, we can redefine the function and make it continuous. 

1. Missing point discontinuity: 

In this type of discontinuity, xaf(x) exists but f(c) does not exist. 

2. Isolated point discontinuity: 

In this type of discontinuity, xaf(x) exists and f(c) exist, but  xaf(x) f(c).

NON-REMOVABLE DISCONTINUITY:

In this case, xaf(x) does not exist, and so the function cannot be redefined to make it continuous. 

1. Finite Discontinuity: 

This type of discontinuity exists when the LHL and RHL limits do not exist, but their values are both finite, yet LHL RHL. 

2. Infinite Discontinuity: 

This type of discontinuity exists when either LHL or RHL or both limits tend toward infinity.

3. Oscillatory Discontinuity:

This type of discontinuity exists when limits oscillate between two finite values.

Relation between continuity and differentiability: 

• Every f(x) is differentiable; then it is also continuous. But, if f(x) is continuous, then it is not differentiable. 
• A function that is not continuous (discontinuous) will not be differentiable. 
• If f is function it is differentiable a a point x = c, then f is also continuous at x = c. 

REMEMBER:

Differentiable Continuous 

Not Differentiable Not Continuous

Not Continuous Not Differentiable

Derivatives of inverse functions: 

If f and g are two real-valued functions such that y = f(x) and x= g(y) then

g(f(x)) = x ddx g(f(x)) = g’(f(x)). f’(x) = 1

Using this result, we can say that dydx . dxdy= 1 or dydx = 1dxdy or dxdy = 1dydx  where dxdy0

Rules of Differentiation: 

1. Sum and difference rule:

The derivative of y= f(x) g(x) is given as

dydx= ddyf(x) ddy g(x)

2. Product Rule:

The derivative of y= f(x). g(x) can be remembered with the help of the u-v rule. It is defined as

ddxu.v = (ddxu) v + (ddxv) u i.e.,

dydx= (ddyf(x)) g(x) + (ddyg(x)) f(x) 

3. Quotient Rule:

The derivative of y= f(x) / g(x) can be remembered with the help of the the u-v rule. It is defined as

ddx(uv) = u (ddxv)  + v (ddx u)v2 i.e.,

ddx(f(x)g(x)) = (ddxg(x)) f(x) + (ddx f(x)) g(x)g(x)2

4. Chain Rule:

i) for two functions:

Let y = f(u), u= f(x) and if dduy and ddxu exists then, 

We can write  ddxy = dduy . ddxu

ii) for three functions: 

Let y = f(u), u= f(w), w= f(x) and if dduy, ddwu and ddxw exists then, 

We can write  ddxy = dduy . ddxu. ddxw

1. Differentiation in Parametric Form:

If x and y are expressed in the form x = f(t), y = g(t), where t is a parameter. It is given as

ddxy = ddtyddtx, provided ddtx 0. 

2. Logarithmic Differentiation: 

Consider y = f(x)g(x). This function can be treated in the following way:

Taking loge on both sides, we get loge y = g(x) loge f(x). The derivation can further be calculated by using the chain rule, given as 

ddxy= f(x)g(x) g(x)f(x)f'(x) + g'(x) log f(x)

Second-Order Derivative: 

It is denoted by y2 or y’’. 

The derivative of the first-order derivative of a function is called the second-order derivative. 

d2dx2y= ddx(ddxy)

Derivative of infinite series: 

If y = f(x) +f(x) +f(x) +…. ∞

Then y can be treated as y= f(x) + y 

By differentiating, we get, (2y-1) ddxy = f’(x) 

Rolle’s Theorem: 

If function f:[a, b] R is continuous of [a, b] and differentiable on (a, b) and f(a) = f(b), then at least one number c in the interval (a, b) such that f’(c) = 0, a,b,c R. 

Lagrange’s Mean Value Theorem:

It is an expansion of Rolle’s Theorem. It states that if function f:[a, b] R is continuous of [a, b] and differentiable on (a, b) and f(a) = f(b), then at least one number c in the interval (a, b) such that 

f’(c) = f(b) – f(a)b – a, where a,b,c R.

Useful substitutions for finding derivatives: 

Derivatives of Trigonometric functions:  

Derivatives of Inverse Trigonometric functions:

Derivatives of standard functions: 

List of Continuous Functions: 

In addition to Class 12 Mathematics chapter 5 notes, students may also access various other study materials on Extramarks pertaining to Continuity and Differentiability.

Class 12 Mathematics chapter 5 notes Exercises & Answer Solutions.

Chapter 5 Mathematics class 12 notes begin with a recall of all concepts students learnt in Class 11. To revise the previse topics, students can refer to CBSE Solutions for class 11. With the help of the notes, students can learn in detail about Continuity and Differentiation. Topics such as second-order derivatives, Rolle’s theorem, and MVT theorem are explained in detail in the Class 12 Mathematics chapter 5 notes. Students will also gain knowledge about Derivatives of standard functions and Continuous Functions.

To help students gain knowledge of the chapter, Extramarks provides detailed and well-structured Class 12 Mathematics chapter 5 notes. All important definitions, formulas, properties, and theorems are included in the notes. With the help of the CBSE solutions, it will become easier for students to tackle the most difficult questions in no time. The step-by-step solutions in the Class 12 Mathematics chapter 5 notes help students to understand each step and how to solve the problems ahead. It empowers students to get closer to pursuing their dream careers. 

Visit the Extramarks link given below to access the Exercises & Answer Solutions of this chapter. 

Extramarks is an online learning platform that focuses on providing a fun learning experience through technology. The academic materials provided give a 360-degree coverage of every concept and ensure deeper comprehension. The CBSE revision notes, such as Class 12 Mathematics chapter 5 notes, are important reference material that will help students to reach their full potential and do well in their examinations. 

Every study material provided by Extramarks is as per the CBSE syllabus and with reference to the NCERT Books. In addition to Class 12 Mathematics chapter 5 notes, students may also access the following study materials on Extramarks.

NCERT Exemplar Class 12 Mathematics

The DESM (Department of Education in Science and Mathematics), along with the NCERT board, has curated the NCERT Exemplar, which includes an extensive set of practice questions. Students are advised to use the Class 12 Mathematics chapter 5 notes to gain deeper knowledge, master the basic techniques and lay a strong foundation in the subject. The notes include unlimited practice questions to create an upward learning graph. Study materials such as NCERT Exemplar, Class 12 Mathematics chapter 5 notes and other NCERT books enable engaging learning modules and prepare them for the future. Students can access themselves based on their performance through adaptive and progressive tests. Studying using the Class 12 Mathematics chapter 5 notes enables detailed and quick learning. The notes also teach several short-cut methods, tips and tricks to tackle difficult questions which require higher-order thinking skills (HOTS).  

Students can perform outstandingly in the Class 12 CBSE board exams. The NCERT Exemplar enhances learning skills and develops problem-solving abilities and analytical thinking. Get access to the best academic materials from Extramarks. 

Key Features of Class 12 Mathematics chapter 5 notes

The key features of Extramarks Class 12 Mathematics Chapter 5 notes include

• Class 12 Mathematics chapter 5 notes are focused on the fundamental basic concepts. 
• The notes strictly follow the CBSE Syllabus and the latest guidelines given by the board.
• It is curated by the in-house team of experts at Extramarks.
• Extensive research is carried out to provide apt knowledge and authentic information to every student in an easy language and detailed manner. 
• It enables a thorough understanding and aims to clear all the doubts. 
• The students get an idea of the type of questions that will be asked in the CLass 12 board exams. 
• The Class 12 Mathematics chapter 5 notes offer solutions to a variety of problems such as the exercise questions, solved examples, etc. 
• The notes help students to have an easy hand at solving twisted questions, thus reducing stress and boosting self-confidence. 
• It also reduces the chances of making simple mistakes and enhances time management skills. 

For assistance and guidance, students can visit the Extramarks web portal. On our platform, find all the latest updates about the major boards, such as ICSE, CBSE, etc. and the competitive examinations. The unlimited practice materials such as the Class 12 Mathematics chapter 5 notes help students to get exam ready. Optimise your preparation with Extramarks!