Class 12 Maths Revision Notes for Continuity and Differentiability of Chapter 5

Class 12 Maths Revision Notes for Continuity and Differentiability of Chapter 5 – Free PDF Download

Free PDF download of Class 12 Maths revision notes & short key-notes for Continuity and Differentiability of Chapter 5 to score high marks in exams, prepared by expert mathematics teachers from latest edition of CBSE books.

 

CBSE Class 12 Mathematics Chapter 5 Continuity and Differentiability

  • Continuity of function at a point: Geometrically we say that a function  Continuity and Differentiability class 12 Notes Mathematics is continuous at  Continuity and Differentiability class 12 Notes Mathematics if the graph of the function  Continuity and Differentiability class 12 Notes Mathematics is continuous (without any break) at  Continuity and Differentiability class 12 Notes Mathematics.
  • A funciton  Continuity and Differentiability class 12 Notes Mathematics is said to be continuous at a point   if:

(i)     Continuity and Differentiability class 12 Notes Mathematics exists i.e., Continuity and Differentiability class 12 Notes Mathematics is finite, definite and real.
(ii)     exists.
(iii)   

  • A function   is continuous at    if  Continuity and Differentiability class 12 Notes Mathematics where   through positive values.
  • Continuity of a function in a closed interval: A function   is said to be continuous in the closed interval  if it is continuous for every value of   lying between a and b continuous to the right of a and to the left of   i.e.,  Continuity and Differentiability class 12 Notes Mathematics and 
  • Continuity of a function in a open interval: A function   is said to be continuous in an open interval   if it is continuous at every point in   .
  • Discontinuity (Discontinuous function): A function   is said to be discontinuous in an interval if it is discontinuous even at a single point of the interval.
  • Suppose   is a real function and   is a point in its domain. The derivative of   at   is defined by  Continuity and Differentiability class 12 Notes Mathematics provided this limit exists.
  • A real valued function is continuous at a point in its domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.
  •  is derivative of first order and is also denoted by   or  .
  • Sum, difference, product and quotient of continuous functions are continuous. i.e., if f and g are continuous functions, then  Continuity and Differentiability class 12 Notes Mathematics is continuous. (f . g) (x) = f (x) . g(x) is continuous.

Continuity and Differentiability class 12 Notes Mathematics (wherever g (x) ≠ 0) is continuous.

  • Every differentiable function is continuous, but the converse is not true.
  • Chain rule is rule to differentiate composites of functions. If f = v o u, t = u (x) and if both  exist then
  • Following are some of the standard derivatives (in appropriate domains):
  •  [Product Rule]
  • , wherever   [Quotient Rule]
  • If   then  Continuity and Differentiability class 12 Notes Mathematics [Chain Rule]
  • If  , then   [Parametric Form]
  • Continuity and Differentiability class 12 Notes Mathematics
  • Logarithmic differentiation is a powerful technique to differentiate functions of the form     Here both f (x) and u (x) need to be positive for this technique to make sense.
  • If we have to differentiate logarithmic funcitons, other than base  , then we use the result   and then differentiate R.H.S.
  • While differentiating inverse trigonometric functions, first represent it in simplest form by using suitable substitution and then differentiate simplified form.
  • If we are given implicit functions then differentiate directly w.r.t. suitable variable involved and get the derivative by readusting the terms.
  •  is derivative of second order and is denoted by   or  .
  • Rolle’s Theorem: If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f ′(c) = 0.
  • Lagrange’s Mean Value Theorem: If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that

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