♦ 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
No comments:
Post a Comment