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Continuity and Differentiability: Important formulae Video Lecture | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
FAQs on Continuity and Differentiability: Important formulae Video Lecture - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is continuity in calculus? |  |
Ans. Continuity in calculus refers to the property of a function where there are no abrupt changes or breaks in its graph. It means that the function can be drawn without lifting the pen from the paper. Mathematically, a function is continuous at a point if the limit of the function as it approaches that point exists and is equal to the value of the function at that point.
| 2. What is differentiability in calculus? | |
Ans. Differentiability in calculus is the property of a function where it has a derivative at every point within its domain. A function is said to be differentiable at a point if its derivative exists at that point. Geometrically, a function is differentiable at a point if the tangent line to its graph at that point exists and is unique.
| 3. What is the continuity and differentiability formula for a constant function? | |
Ans. For a constant function, both continuity and differentiability are satisfied. The continuity formula for a constant function f(x) = c, where c is a constant, is:
f(x) = c for all x in the domain of f(x)
The differentiability formula for a constant function is:
f'(x) = 0 for all x in the domain of f(x)
| 4. How do you determine if a function is continuous using limits? | |
Ans. To determine if a function is continuous using limits, you need to check three conditions:
1. The function must be defined at the point you are testing for continuity.
2. The limit of the function as it approaches that point from the left must exist and be equal to the value of the function at that point.
3. The limit of the function as it approaches that point from the right must exist and be equal to the value of the function at that point.
If all three conditions are met, then the function is continuous at that point. If any of the conditions fail, the function is not continuous at that point.
| 5. 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, continuity alone does not guarantee differentiability. A function can be continuous at a point but not differentiable if it has a sharp corner or a vertical tangent line at that point. Differentiability requires not only continuity but also the existence of a unique tangent line at that point, which is determined by the derivative of the function.
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