Study MaterialsCBSE NotesContinuity and Differentiability Class 12 Notes Maths Chapter 5

Continuity and Differentiability Class 12 Notes Maths Chapter 5

CBSE Class 12 Maths Notes Chapter 5 Continuity and Differentiability

Continuity at a Point: A function f(x) is said to be continuous at a point x = a, if
Left hand limit of f(x) at(x = a) = Right hand limit of f(x) at (x = a) = Value of f(x) at (x = a)
i.e. if at x = a, LHL = RHL = f(a)
where, LHL = \(\lim _{ x\rightarrow { a }^{ – } }{ f(x) }\) and RHL = \(\lim _{ x\rightarrow { a }^{ + } }{ f(x) }\)
Note: To evaluate LHL of a function f(x) at (x = o), put x = a – h and to find RHL, put x = a + h.

Continuity in an Interval: A function y = f(x) is said to be continuous in an interval (a, b), where a < b if and only if f(x) is continuous at every point in that interval.

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    • Every identity function is continuous.
    • Every constant function is continuous.
    • Every polynomial function is continuous.
    • Every rational function is continuous.
    • All trigonometric functions are continuous in their domain.

    Standard Results of Limits
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 1
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 2

    Algebra of Continuous Functions
    Suppose f and g are two real functions, continuous at real number c. Then,

    • f + g is continuous at x = c.
    • f – g is continuous at x = c.
    • f.g is continuous at x = c.
    • cf is continuous, where c is any constant.
    • (\(\frac { f }{ g }\)) is continuous at x = c, [provide g(c) ≠ 0]

    Suppose f and g are two real valued functions such that (fog) is defined at c. If g is continuous at c and f is continuous at g (c), then (fog) is continuous at c.

    If f is continuous, then |f| is also continuous.

    Differentiability: A function f(x) is said to be differentiable at a point x = a, if
    Left hand derivative at (x = a) = Right hand derivative at (x = a)
    i.e. LHD at (x = a) = RHD (at x = a), where Right hand derivative, where
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 3

    Note: Every differentiable function is continuous but every continuous function is not differentiable.

    Differentiation: The process of finding a derivative of a function is called differentiation.

    Rules of Differentiation
    Sum and Difference Rule: Let y = f(x) ± g(x).Then, by using sum and difference rule, it’s derivative is written as
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 4

    Product Rule: Let y = f(x) g(x). Then, by using product rule, it’s derivative is written as
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 5

    Quotient Rule: Let y = \(\frac { f(x) }{ g(x) }\); g(x) ≠ 0, then by using quotient rule, it’s derivative is written as
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 6

    Chain Rule: Let y = f(u) and u = f(x), then by using chain rule, we may write
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 7

    Logarithmic Differentiation: Let y = [f(x)]g(x) ..(i)
    So by taking log (to base e) we can write Eq. (i) as log y = g(x) log f(x). Then, by using chain rule
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 8

    Differentiation of Functions in Parametric Form: A relation expressed between two variables x and y in the form x = f(t), y = g(t) is said to be parametric form with t as a parameter, when
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 9
    (whenever \(\frac { dx }{ dt } \neq 0\))
    Note: dy/dx is expressed in terms of parameter only without directly involving the main variables x and y.

    Second order Derivative: It is the derivative of the first order derivative.
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 10

    Some Standard Derivatives
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 11

    Rolle’s Theorem: Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b) such that f(a) = f(b), where a and b are some real numbers. Then, there exists at least one number c in (a, b) such that f'(c) = 0.

    Mean Value Theorem: Let f : [a, b] → R be continuous function on [a, b]and differentiable on (a, b). Then, there exists at least one number c in (a, b) such that
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 12
    Note: Mean value theorem is an expansion of Rolle’s theorem.

    Some Useful Substitutions for Finding Derivatives Expression
    Continuity and Differentiability Class 12 Notes Maths Chapter 5 13

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