NCERT Class 12th Maths Chapter 5 Continuity and Differentiability Notes

Introduction: Class 12 Maths chapter 5 notes are concerning continuity and differentiability. In continuity and differentiability, Class 12 notes include finding continuity and differentiability nothing but finding the limits and differentiating. This Class 12 Maths chapter 5 notes contains the following topics: continuity at a point, continuity in the interval, few standard limits notations, differentiability, right-hand derivative, left-hand derivative, rules of differentiation like sum, difference, product, quotient, chain rules, second-order derivatives, standard formulas, Rolle’s theorem, Mean value theorem and many more.

Notes for Class 12 Maths chapter 5 also contain standard formulas that are to be memorised for the implementation in problems. NCERT Class 12 Math chapter 5 contains a detailed explanation of topics, theorems, examples, exercises. By going through the document students can cover all the topics that are in NCERT Notes for Class 12 Math chapter 5 textbook. It also contains examples, exercises, a few interesting points and most importantly contains FAQs that are frequently asked questions by students which can clarify many other students with the same doubt.

Every concept that is in CBSE Class 12 Maths chapter 5 notes is explained here in a simple and understanding way that can reach students easily. All these concepts can be downloaded from Class 12 Maths chapter 5 notes pdf download, Class 12 notes continuity and differentiability, continuity and differentiability Class 12 notes pdf download.

Students can also refer,

• NCERT solutions for class 12 maths chapter 5 continuity and differentiability
• NCERT exemplar class 12 maths chapter 5 solutions continuity and differentiability

NCERT Class 12 Chapter 5 Notes

Continuity:

DEFINITION: suppose f be a real function, c be a point in the domain of function f in the range of real numbers. f is said to be continuous if it satisfies

x→c fx=f(c)

Theoretical: If the left-hand side limit, right-hand side limits are equal, and x=c then function f is said to be continuous else not continuous.

DEFINITION 2: A function f is said to be continuous if it is continuous at every point in domain f.

Continuity of a:

Algebra of continuous functions:

Theorem:

f and g be two real numbers that are continuous and real at point c, then

a) f + g is continuous at x = c

b) f-g is continuous at x = c

c) f. g is continuous at x = c

d) fg is continuous at x=c and g(c) ≠ 0

Proof:

From equation a :

f + g is said to be continuous at x = c

=

= f(c) + g(c)

= (f+ g) (c)

Hence proved that it is continuous.

Differentiability:

suppose f be a real function, c be a point in the domain of function f in the range of real numbers. f is said to be differentiable if it satisfies:

Derivative of f(x) is given by f’(x)

RIGHT HAND DERIVATIVE:

LEFT HAND DERIVATIVE:

Sum and difference rule: let y = f(x) ± g(x)

Product rule: Let y = f(x) g(x)

Quotient rule: Let y = f(x)/g(x); g(x) ≠ 0

Chain rule: Let y = f(u) and u = f(x)

Differentiation of Functions in Parametric Form: Relationship between two values x and y that are expressed in the forms x = f(t), y = g(t) is called to be in parametric form when,

Second-order Derivative: Deriving First order derivative will result in a second-order derivative.

Rolle’s Theorem:

Let f : [a, b] → R be a real number and is continuous on [a, b] satisfies following:

a)differentiable in the interval (a, b) so that f(a) = f(b); a, b are real numbers. Then,

b) there will be at least one number c in (a, b) that satisfies the condition f'(c) = 0.

This is called Rolle’s Theorem.

Mean Value Theorem:

Let f : [a, b] → R be a real number that is continuous in the interval [a, b] satisfies the following:

a) differentiable in the interval (a, b). Then,

b) there will be at least one number c in the interval (a, b) that satisfies the condition

This is called the mean value theorem.

With this topic we conclude NCERT class 12 chapter 5 notes.

The link for the NCERT textbook pdf is given below:

URL: ncert.nic.in/ncerts/l/lemh105.pdf

Significance of NCERT Class 12 Maths Chapter 5 Notes:

NCERT Class 12 Maths chapter 5 notes will be very much helpful for students to obtain maximum marks in their 12 board exams. In Class 12 continuity and differentiability notes, we have discussed many topics like continuity, differentiability, derivatives, standard limits formulas, sum, difference, product, quotient, chain rules with their mathematical representations, Rolle’s and Mean value theorem along with their conditions and rules of solving the problems. NCERT Class 12 Mathematics chapter 5 is also very helpful to cover major topics of the Class 12 CBSE Mathematics Syllabus.

The CBSE Class 12 Maths chapter 5 will enable us to understand the theorems, statements, rules with their conditions in detail. This pdf also contains previous year’s questions and NCERT Textbook pdf. The next part contains FAQs' most frequently asked questions along with a topic-wise explanation. By referring to the document you can get a complete idea of all the topics of Class12 chapter 5 continuity and differentiability pdf download.

NCERT Class 12 Notes Chapter Wise.

Go through the above link to learn more examples topic-wise.

Subject Wise NCERT Exampler Solutions

• NCERT Exemplar Class 12 Solutions
• NCERT Exemplar Class 12 Maths
• NCERT Exemplar Class 12 Physics
• NCERT Exemplar Class 12 Chemistry
• NCERT Exemplar Class 12 Biology

Subject Wise NCERT Solutions

• NCERT Solutions for Class 12 Mathematics
• NCERT Solutions for Class 12 Chemistry
• NCERT Solutions for Class 12 Physics
• NCERT Solutions for Class 12 Biology