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Continuity & Differentiability Class 12 Notes Maths Chapter 5

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FAQs on Continuity & Differentiability Class 12 Notes Maths Chapter 5

1. What is the definition of continuity in calculus?
Ans. Continuity in calculus refers to the property of a function where it is defined and has no abrupt breaks or jumps. Formally, a function f(x) is said to be continuous at a point c if three conditions are satisfied: (1) f(c) is defined, (2) the limit of f(x) as x approaches c exists, and (3) the limit of f(x) as x approaches c is equal to f(c).
2. How do you determine if a function is differentiable?
Ans. To determine if a function is differentiable, we need to check whether its derivative exists at every point in its domain. If the derivative exists for all x in the domain, the function is said to be differentiable. It means that the function has a well-defined instantaneous rate of change at each point.
3. What is the relationship between continuity and differentiability?
Ans. Continuity is a necessary condition for differentiability. If a function is differentiable at a point, it must be continuous at that point. However, a function can be continuous at a point without being differentiable at that point. In other words, differentiability implies continuity, but continuity does not imply differentiability.
4. Can a function be continuous but not differentiable?
Ans. Yes, a function can be continuous but not differentiable. This occurs when there is a sharp corner, a vertical tangent, or a cusp at a certain point on the function's graph. In such cases, the function fails to have a well-defined instantaneous rate of change at that point, leading to a lack of differentiability.
5. How can we determine if a function is continuous and differentiable on an interval?
Ans. To determine if a function is continuous and differentiable on an interval, we need to check the conditions of continuity and differentiability at every point within the interval. For continuity, we ensure that the function is defined and has no abrupt breaks or jumps within the interval. For differentiability, we need to check if the derivative exists at every point within the interval and is continuous within the interval. If both conditions are satisfied, the function is continuous and differentiable on the interval.
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