Class 12 Maths - ODISHA
Continuity and Differentiability
The chapter 'Continuity and Differentiability' builds the foundation of calculus in Class 12 Mathematics under the Odisha Board (BSE). Students learn to determine whether a function's graph breaks at a point (continuity) and whether it has a smooth, non-vertical tangent (differentiability). This chapter covers limits, algebraic and trigonometric continuity, derivatives of composite functions via the Chain Rule, logarithmic and implicit differentiation, parametric forms, and Second Order Derivatives. Mastering these concepts is crucial for board exams as they form the core of application of derivatives and calculus scoring sections.
Start Learning FreeKey Concepts
Continuity at a Point
A function f(x) is continuous at x = a if the left-hand limit, right-hand limit, and the value of the function at that point are all equal.
Differentiability at a Point
A function is differentiable at x = a if the left-hand derivative and right-hand derivative exist and are finite at that point.
Chain Rule
A rule used to find the derivative of composite functions by multiplying the derivative of the outer function by the derivative of the inner function.
Implicit Differentiation
A technique used to find the derivative of y with respect to x when the function is given implicitly in an equation containing both x and y.
Logarithmic Differentiation
Using logarithms to simplify complex functions involving products, quotients, or variable powers before differentiating.
Important Formulas
Board Exam Info
In the Odisha Board (BSE) Class 12 Mathematics examination, this chapter typically carries around 8 to 12 marks. Common question types include checking continuity of piecewise functions at specific points, finding derivatives of implicit or parametric functions, and proving second-order derivative relations.
Frequently Asked Questions
Are all continuous functions also differentiable?
No. While every differentiable function is continuous, the converse is not true. For example, f(x) = |x] is continuous at x = 0, but not differentiable there.
When should I use logarithmic differentiation?
You should use it when a function is given in the form of a variable raised to another variable, like y = x^x, or when there is a complicated product and quotient of many terms.
How do I prove continuity at a point in board exams?
You must evaluate three things: Left Hand Limit (LHL), Right Hand Limit (RHL), and the direct function value f(a). If all three are equal, the function is continuous.
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