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FAQs on Differentiability (With Solved Examples) - Topic-wise Tests & Solved Examples for Mathematics
| 1. What is differentiability in mathematics? |  |
Differentiability is a concept in calculus that describes the smoothness of a function at a specific point. A function is said to be differentiable at a point if it has a derivative at that point. The derivative represents the rate at which the function is changing at that point.
| 2. How is differentiability determined? | |
Differentiability of a function at a specific point is determined by checking if the derivative of the function exists at that point. If the derivative exists, the function is differentiable at that point. The derivative can be found using various methods such as the limit definition of derivative or using differentiation rules.
| 3. What is the significance of differentiability? | |
Differentiability is significant in calculus because it allows us to analyze the behavior of a function at a specific point. Differentiable functions have many useful properties, such as being continuous and having local extrema. It also enables us to approximate the function using linear approximations, which is essential in optimization and numerical methods.
| 4. Can a function be continuous but not differentiable? | |
Yes, it is possible for a function to be continuous but not differentiable at a specific point or over an interval. This occurs when there is a sharp corner, a vertical tangent, or a cusp in the graph of the function. These irregularities prevent the existence of a derivative at those points, even though the function remains continuous.
| 5. What are some real-life applications of differentiability? | |
Differentiability has numerous applications in various fields. It is used in physics to analyze the motion of objects, in economics to study optimization problems, in engineering to design efficient systems, and in computer science for image and signal processing. For example, the concept of differentiability is essential in modeling the flow of fluids, analyzing stock market trends, designing smooth curves in computer graphics, and optimizing algorithms.
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