Is exp(-ax) a Piecewise Smooth Function?

Thread starter jaejoon89
Start date Apr 14, 2009
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Functions Smooth

In summary, a piecewise smooth function is a mathematical function composed of different smooth functions on different intervals, with potential discontinuities at the points where the intervals meet. It differs from a regular function in its versatility and is commonly used in modeling real-world situations. To graph it, you identify intervals and graph each function on its respective interval. It is commonly used in economics, engineering, and physics, and determining continuity and differentiability depends on the smoothness of the individual functions on their respective intervals.

Apr 14, 2009

jaejoon89

I'm trying to find a Fourier series for exp(-ax) where a is a positive constant. How is exp(-ax) piecewise smooth?

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Apr 14, 2009

CompuChip

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It is piecewise smooth in one big piece

If you take piecewise smooth to mean: smooth in all but finitely many points, it satisfies the definition because the number of points in which it is not smooth is zero ([itex]< \infty[/itex]).

1. What is a piecewise smooth function?

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.

2. How is a piecewise smooth function different from a regular function?

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.

3. How do you graph a piecewise smooth function?

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.

4. What are some common applications of piecewise smooth functions?

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.

5. How do you determine continuity and differentiability for a piecewise smooth function?

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.