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A piecewise defined function is a mathematical function that is defined by different equations on different intervals of its domain. This means that the function may have different rules or equations for different parts of its domain.
Differentiability is a property of functions that allows us to find the instantaneous rate of change at any point on the function. This is important because it helps us understand the behavior of the function and its relationship to other functions.
The first step is to determine the intervals where the function is not differentiable. Then, we need to find the equations for the function on each of those intervals. Next, we can use the limit definition of the derivative to find the derivative of each equation. Finally, we can combine these derivatives to create a single piecewise defined function that is differentiable.
Piecewise defined functions are commonly used in economics and finance to model different scenarios and make predictions. They can also be used in engineering to describe systems with changing parameters or in computer graphics to create smooth curves and surfaces.
Yes, there are some limitations. In some cases, it may not be possible to make a piecewise defined function differentiable without changing the original function. Additionally, the process of finding the derivative of a piecewise defined function can be complex and time-consuming, especially for functions with many intervals or complex equations.