12 class Maths Notes Chapter 5 Continuity and Differentiability free PDF| Quick revision Continuity and Differentiability Notes class 12 maths
CBSE Revision Notes for CBSE Class 12 Mathematics Continuity and Differentiability Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation.
Class 12 Maths Chapter-5 Continuity and Differentiability Quick Revision Notes Free Pdf
CBSE Revision Notes for CBSE Class 12 Mathematics Continuity and Differentiability Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation.
Class 12 Maths Chapter-5 Continuity and Differentiability Quick Revision Notes Free Pdf
🔷 Chapter - 5 🔷
👉 Continuity and Differentiability 👈
✳️ DIFFERENTIABILITY IN A SET
1. A function f (x) defined on an open interval (a, b) is said to be differentiable or derivable in open interval (a, b), if it is differentiable at each point of (a, b).
2. A function f (x) defined on closed interval [a, b] is said to be differentiable or derivable. "If f is derivable in the open interval (a, b) and also the end points a and b, then f is said to be derivable in the closed interval [a, b]".
A function f is said to be a differentiable function if it is differentiable at every point of its domain.
✳️ Note :-
1. If f (x) and g (x) are derivable at x = a then the functions f(x)+g (x), f(x)-g(x), f(x). g (x) will also be derivable at x=a and if g (a) ≠ 0 then the function f (x)/g(x) will also be derivable at x=a.
2. If f (x) is differentiable at x = a and g (x) is not differentiable at x = a, then the product function F (x) = f (x). g (x) can still be differentiable at x=a.
E.g. f (x) =x and g (x) = |x|.
3. If f (x) and g (x) both are not differentiable at x = a then the product function; F (x) = f(x). g (x) can still be differentiable at x = a.
E.g., f(x) = |x| and g (x) = |x|.
4. If f (x) and g (x) both are not differentiable at x=a then the sum function F (x) =f (x)+g (x) may be a differentiable function.
E.g., f (x) = |x| and g (x)=-|xl.
5. If f(x) is derivable at x= a
• F' (x) is continuous at x=a.
✳️ RELATION B/W CONTINUITY & DIFFERENNINBILINY
In the previous section we have discussed that if a function is differentiable at a point, then it should be continuous at that point and a discontinuous function cannot be differentiable. This fact is proved in the following theorem.
✳️ Theorem : If a function is differentiable at a point, it is necessarily continuous at that point. But the converse is not necessarily true,
Or f(x) is differentiable at x=c
f(x) is continuous at x = c.
✳️ Note.
🔹 Converse : The converse of the above theorem is not necessarily true i.e., a function may be continuous at a point but may not be differentiable at that point.
E.g., The function f (x) = |x| is continuous at x = 0 but it is not differentiable at x = 0, as shown in the figure.
The figure shows that sharp edge at x = 0 hence, function is not differentiable but continuous at x = 0.
✳️ Note:-
(a) Let f´⁺(a) = p & f´⁻(a)=q where p & q are finite then
(b) If a function f is not differentiable but is continuous at x = a it geometrically implies a sharp corner at X=a.
Theorem 2 : Let fand g be real functions such that fog is defined if g is continuous at x = a and f is continuous at g (a), show that fog is continuous at x = a.
✳️ DIFFERENTIATION
✳️ DEFINITION
(a) Let us consider a function y=f(x) defined in a certain interval. It has a definite value for each value of the independent variable x in this interval.
Now, the ratio of the increment of the function to the increment in the independent variable,
✳️ DERIVATIVE OF STANDARD FUNCTION
✳️ THEOREMS ON DERIVATIVES
✳️ METHODS OF DIFFERENTIATION
🔹 4.1 Derivative by using Trigonometrical Substitution
🔹 4.2 Logarithmic Differentiation
✳️ DERIVATIVE OF ORDER TWO & THREE
Let a function y =f (x) be defined on an open interval (a, b). It's derivative, if it exists on (a, b), is a certain function f'(x) [or (dy/dx) or y'] & is called the first derivative of y w.r.t. x. If it happens that the first derivative has a derivative on (a, b) then this derivative is called the second derivative of y w.r.t. x & is denoted by f"(x) or (d²y/dx²) or y".
Similarly, the 3rd order derivative of y w.r.t. x, if it exists, is
✳️ CONTINUITY
✳️ DEFINITION
🔹 A function f (x) is said to be continuous at x = a; where a ∈ domain of f(x), if
i.e., LHL=RHL = value of a function at x= a
✳️ Reasons of discontinuity
🔹 If f (x) is not continuous at x = a, we say that f (x) is discontinuous at x = a.
There are following possibilities of discontinuity
✳️ PROPERTIES OF CONTINUOUS FUNCTIONS
✳️ THE INTERMEDIATE VALUE THEOREM
🔹 Suppose f(x) is continuous on an interval I, and a and b are any two points of I. Then if y₀ is a number between f(a) and f(b), their exits a number c between a and b such that f(c)=y₀.
✴️ Note :-
🔹 That a function f which is continuous in [a, b] possesses the following properties:
( i ) If f (a) and f(b) possess opposite signs, then there exists at least one solution of the equation f(x) =0 in the open interval (a, b).
(ii) If K is any real number between f(a) and f(b), then there exists at least one solution of the equation f (x) = K in the open interval (a, b).
✳️ CONTINUITY IN AN INTERVAL
(a) A function f is said to be continuous in (a, b) if f is continuous at each and every point ∈ (a, b).
(b) A function f is said to be continuous in a closed interval [a, b] if:
(1) f is continuous in the open interval (a, b) and
(2) f is right continuous at 'a' i.e. limit ₓ→ₐ⁺ f(x)=f(a) = a finite quantity.
(3) f is left continuous at 'b'; i.e. limit ₓ→b⁻f(x) = f(b) a finite quantity.
✳️ A LIST OF CONTINUOUS FUNCTIONS
✳️ TYPES OF DISCONTINUITIES
Type-1: (Removable type of discontinuities)
In case, limit ₓ→c f(x) exists but is not equal to f (c) then the function is said to have a removable discontnuity or discontinuity of the first kind. In this case, we can redefine the function such that limit ₓ→c f(x) = f (c) and make it continuous at x=c. Removable type of discontinuity can be further classified as:
(a) Missing Point Discontinuity :
Where limit ₓ→ₐ f(x) exists finitely but f(a) is not defined.
✳️ DEFINITION
🔹 A function f (x) is said to be continuous at x = a; where a ∈ domain of f(x), if
i.e., LHL=RHL = value of a function at x= a
✳️ Reasons of discontinuity
🔹 If f (x) is not continuous at x = a, we say that f (x) is discontinuous at x = a.
There are following possibilities of discontinuity
✳️ PROPERTIES OF CONTINUOUS FUNCTIONS
✳️ THE INTERMEDIATE VALUE THEOREM
🔹 Suppose f(x) is continuous on an interval I, and a and b are any two points of I. Then if y₀ is a number between f(a) and f(b), their exits a number c between a and b such that f(c)=y₀.
✴️ Note :-
🔹 That a function f which is continuous in [a, b] possesses the following properties:
( i ) If f (a) and f(b) possess opposite signs, then there exists at least one solution of the equation f(x) =0 in the open interval (a, b).
(ii) If K is any real number between f(a) and f(b), then there exists at least one solution of the equation f (x) = K in the open interval (a, b).
✳️ CONTINUITY IN AN INTERVAL
(a) A function f is said to be continuous in (a, b) if f is continuous at each and every point ∈ (a, b).
(b) A function f is said to be continuous in a closed interval [a, b] if:
(1) f is continuous in the open interval (a, b) and
(2) f is right continuous at 'a' i.e. limit ₓ→ₐ⁺ f(x)=f(a) = a finite quantity.
(3) f is left continuous at 'b'; i.e. limit ₓ→b⁻f(x) = f(b) a finite quantity.
✳️ A LIST OF CONTINUOUS FUNCTIONS
✳️ TYPES OF DISCONTINUITIES
Type-1: (Removable type of discontinuities)
In case, limit ₓ→c f(x) exists but is not equal to f (c) then the function is said to have a removable discontnuity or discontinuity of the first kind. In this case, we can redefine the function such that limit ₓ→c f(x) = f (c) and make it continuous at x=c. Removable type of discontinuity can be further classified as:
(a) Missing Point Discontinuity :
Where limit ₓ→ₐ f(x) exists finitely but f(a) is not defined.
✴️ Isolated Point Discontinuity:
Type-2: (Non-Removable type of discontinuities)
In case, limit ₓ→ₐ f(x) does not exist, then it is not possible to make the function continuous by redefining it. Such discontinuities are known as non-removable discontinuity or discontinuity of the 2nd kind. Non-removable type of discontinuity can be further classified as :
✳️ (a) Finite Discontinuity :
✳️ (b) Infinite Discontiunity:
✳️ (c) Oscillatory Discontinuity :
From the adjacent graph note that
- f is continuous at x =-1
- f has isolated discontinuity at x = 1
- f has missing point discontinuity at x = 2
- f has non-removable (finite type) discontinity at the origin.
- f is continuous at x =-1
- f has isolated discontinuity at x = 1
- f has missing point discontinuity at x = 2
- f has non-removable (finite type) discontinity at the origin.
✴️ Note :-
🔹 (a) In case of dis-continuity of the second kind the non- negative difference between the value of the RHL at x=a and LHL at x=a is called the jump of discontinuity. A function having a finite number of jumps in a given interval I is called a piece wise continuous or sectionally continuous function in this interval.
(b) All Polynomials, Trigonometrical functions, exponential and Logarithmic functions are continuous in their domains.
(c) If f (x) is continuous and g (x) is discontinuous at x = a then the product function ∅ (x)=f(x). g (x) is not necessarily be discontinuous at x = a. e.g.
🔹 (a) In case of dis-continuity of the second kind the non- negative difference between the value of the RHL at x=a and LHL at x=a is called the jump of discontinuity. A function having a finite number of jumps in a given interval I is called a piece wise continuous or sectionally continuous function in this interval.
(b) All Polynomials, Trigonometrical functions, exponential and Logarithmic functions are continuous in their domains.
(c) If f (x) is continuous and g (x) is discontinuous at x = a then the product function ∅ (x)=f(x). g (x) is not necessarily be discontinuous at x = a. e.g.
(d) If f (x) and g (x) both are discontinuous at x = a then the product function ∅ (x) =f (x). g (x) is not necessarily be discontinuous at x =a. e.g.
✳️ DIFFERENTIABILITY
✳️ DEFINITION
🔹 Let f (x) be a real valued function defined on an open interval (a, b) where c ∈ (a, b). Then f(x) is said to be differentiable or derivable at x =c,
✳️ DEFINITION
🔹 Let f (x) be a real valued function defined on an open interval (a, b) where c ∈ (a, b). Then f(x) is said to be differentiable or derivable at x =c,
✳️ DIFFERENTIABILITY IN A SET
1. A function f (x) defined on an open interval (a, b) is said to be differentiable or derivable in open interval (a, b), if it is differentiable at each point of (a, b).
2. A function f (x) defined on closed interval [a, b] is said to be differentiable or derivable. "If f is derivable in the open interval (a, b) and also the end points a and b, then f is said to be derivable in the closed interval [a, b]".
A function f is said to be a differentiable function if it is differentiable at every point of its domain.
✳️ Note :-
1. If f (x) and g (x) are derivable at x = a then the functions f(x)+g (x), f(x)-g(x), f(x). g (x) will also be derivable at x=a and if g (a) ≠ 0 then the function f (x)/g(x) will also be derivable at x=a.
2. If f (x) is differentiable at x = a and g (x) is not differentiable at x = a, then the product function F (x) = f (x). g (x) can still be differentiable at x=a.
E.g. f (x) =x and g (x) = |x|.
3. If f (x) and g (x) both are not differentiable at x = a then the product function; F (x) = f(x). g (x) can still be differentiable at x = a.
E.g., f(x) = |x| and g (x) = |x|.
4. If f (x) and g (x) both are not differentiable at x=a then the sum function F (x) =f (x)+g (x) may be a differentiable function.
E.g., f (x) = |x| and g (x)=-|xl.
5. If f(x) is derivable at x= a
• F' (x) is continuous at x=a.
✳️ RELATION B/W CONTINUITY & DIFFERENNINBILINY
In the previous section we have discussed that if a function is differentiable at a point, then it should be continuous at that point and a discontinuous function cannot be differentiable. This fact is proved in the following theorem.
✳️ Theorem : If a function is differentiable at a point, it is necessarily continuous at that point. But the converse is not necessarily true,
Or f(x) is differentiable at x=c
f(x) is continuous at x = c.
✳️ Note.
🔹 Converse : The converse of the above theorem is not necessarily true i.e., a function may be continuous at a point but may not be differentiable at that point.
E.g., The function f (x) = |x| is continuous at x = 0 but it is not differentiable at x = 0, as shown in the figure.
The figure shows that sharp edge at x = 0 hence, function is not differentiable but continuous at x = 0.
✳️ Note:-
(a) Let f´⁺(a) = p & f´⁻(a)=q where p & q are finite then
(b) If a function f is not differentiable but is continuous at x = a it geometrically implies a sharp corner at X=a.
Theorem 2 : Let fand g be real functions such that fog is defined if g is continuous at x = a and f is continuous at g (a), show that fog is continuous at x = a.
✳️ DIFFERENTIATION
✳️ DEFINITION
(a) Let us consider a function y=f(x) defined in a certain interval. It has a definite value for each value of the independent variable x in this interval.
Now, the ratio of the increment of the function to the increment in the independent variable,
✳️ DERIVATIVE OF STANDARD FUNCTION
✳️ THEOREMS ON DERIVATIVES
✳️ METHODS OF DIFFERENTIATION
🔹 4.1 Derivative by using Trigonometrical Substitution
🔹 4.2 Logarithmic Differentiation
✳️ DERIVATIVE OF ORDER TWO & THREE
Let a function y =f (x) be defined on an open interval (a, b). It's derivative, if it exists on (a, b), is a certain function f'(x) [or (dy/dx) or y'] & is called the first derivative of y w.r.t. x. If it happens that the first derivative has a derivative on (a, b) then this derivative is called the second derivative of y w.r.t. x & is denoted by f"(x) or (d²y/dx²) or y".
Similarly, the 3rd order derivative of y w.r.t. x, if it exists, is
