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jaejoon89
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I'm trying to find a Fourier series for exp(-ax) where a is a positive constant. How is exp(-ax) piecewise smooth?
A piecewise smooth function is a mathematical function that is composed of different smooth functions defined on different intervals. This means that the function is continuous on each interval, but there may be discontinuities or sharp changes at the points where the intervals meet.
A regular function is continuous and differentiable on its entire domain, while a piecewise smooth function may have discontinuities or sharp changes at certain points. This makes it a more flexible and versatile type of function, often used in modeling real-world situations with varying behaviors.
To graph a piecewise smooth function, you first identify the different intervals where the function is defined. Then, you graph each smooth function on its respective interval and make sure to include the points where the intervals meet. This will create a graph with different segments or pieces that come together to form the complete function.
Piecewise smooth functions are commonly used in modeling real-world situations with varying behaviors, such as in economics, engineering, and physics. They are also useful in solving optimization problems, as they allow for more flexibility in the function's behavior.
To determine continuity for a piecewise smooth function, you need to check if the function is continuous at each point where the intervals meet. Differentiability, on the other hand, depends on the differentiability of each smooth function on its respective interval. If all the smooth functions are differentiable, then the piecewise smooth function is also differentiable.