Class 12 Maths Revision Notes for Continuity and Differentiability of Chapter 5 â€“ 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 functionÂ Â is continuous atÂ Â if the graph of the functionÂ Â is continuous (without any break) atÂ .
- A funcitonÂ Â is said to be continuous at a pointÂ Â if:
(i)Â Â Â Â Â Â exists i.e.,Â Â is finite, definite and real.
(ii)Â Â Â Â Â exists.
(iii)Â Â Â
- A functionÂ Â is continuous atÂ Â ifÂ Â whereÂ Â through positive values.
- Continuity of a function in a closed interval: A functionÂ Â is said to be continuous in the closed interval Â if it is continuous forÂ every value ofÂ Â lying between a and b continuous to the right of a and to the left ofÂ Â i.e.,Â Â andÂ
- Continuity of a function in a open interval: A functionÂ Â is said to be continuous in an open intervalÂ Â if it is continuous at every point inÂ Â .
- Discontinuity (Discontinuous function): A functionÂ Â is said to be discontinuous in an interval if it is discontinuous even at a single point of the interval.
- SupposeÂ Â is a real function andÂ Â is a point in its domain. The derivative ofÂ Â atÂ Â is defined byÂ Â 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.
- Â is derivative of first order and is also denoted byÂ Â orÂ .
- Sum, difference, product and quotient of continuous functions are continuous. i.e., if f and g are continuous functions, thenÂ Â is continuous. (f . g) (x) = f (x) . g(x) is continuous.
Â (wherever g (x) â‰ 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 bothÂ exist then
- Following are some of the standard derivatives (in appropriate domains):
- Â [Product Rule]
- , whereverÂ Â [Quotient Rule]
- IfÂ Â thenÂ Â [Chain Rule]
- IfÂ , thenÂ Â [Parametric Form]
- Logarithmic differentiation is a powerful technique to differentiate functions of the formÂ Â Â Â 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 baseÂ , then we use the resultÂ Â 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.
- Â is derivative of second order and is denoted byÂ Â orÂ .
- Rolleâ€™s Theorem:Â If f: [a, b] â†’ 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 â€²(c) = 0.
- Lagrangeâ€™s Mean Value Theorem: If f: [a, b] â†’ R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that