Class 12 Maths - BIHAR
Continuity and Differentiability
Continuity and Differentiability bridges the gap between algebra and calculus by studying smooth, unbroken curves and their rates of change. In the Bihar Board (BSEB) Class 12 Mathematics syllabus, this chapter forms the foundation of calculus, leading directly into applications of derivatives, integrals, and differential equations. You will learn how to check if a function has any breaks or sharp turns, master advanced differentiation techniques like logarithmic and parametric differentiation, and apply crucial theorems such as Rolle's Theorem and Mean Value Theorem. Scoring high in calculus is essential for securing top marks in your board exams.
Start Learning FreeKey Concepts
Continuity of a Function
A function is continuous at a point if its left-hand limit, right-hand limit, and the value of the function at that point are all equal, meaning you can draw its graph without lifting your pen.
Differentiability
A function is differentiable at a point if its left-hand derivative and right-hand derivative exist and are equal, implying the graph has a smooth tangent without any sharp corners.
Relationship between Differentiability and Continuity
Every differentiable function is necessarily continuous, but a continuous function may not always be differentiable (for example, at sharp turns like absolute value functions).
Logarithmic Differentiation
A technique used to differentiate functions of the form f(x)^g(x) or complex products/quotients by taking logarithms on both sides to simplify powers into products.
Rolle's Theorem
If a function is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one value c in (a, b) where the derivative f'(c) = 0.
Important Formulas
Board Exam Info
In the Bihar Board (BSEB) Class 12 Mathematics exam, calculus carries a heavy weightage of around 35 to 40 marks, with Continuity and Differentiability contributing approximately 8 to 12 marks. Common question types include short-answer questions on checking continuity of piecewise functions at given points, finding derivatives using logarithmic or chain rules, and long-answer questions proving differentiability or applying Rolle's and Mean Value Theorems.
Frequently Asked Questions
Are all continuous functions 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 at x = 0 due to a sharp corner.
How do I know when to use logarithmic differentiation?
Use logarithmic differentiation when you have variables in the exponent (like x^x) or when a function is given as a complicated product or quotient of many terms, as logs turn exponents and multiplication into simpler additions.
Is it mandatory to write steps for checking limits in continuity problems?
Yes, in BSEB board exams, you must explicitly show the calculation for Left-Hand Limit (LHL), Right-Hand Limit (RHL), and the function's value at that point to get full marks.
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