djeitnstine said:I was just reflecting upon my math courses and wondered why can we transform any piecewise continuous functions by using transforms such as laplace transforms or converting to Fourier series by simply adding the required integrals on the respective bounds?
djeitnstine said:Yes but why can the integrals simply be added in both cases?
Example:
[tex]f(t)= \left\{^{5, t<1}_{sin(t), t>1}[/tex] So the laplace transform of this would be
[tex]L\left\{ f(t) \right\} = \int_0^1 5e^{-st}dt + \int_1^{\infty} sin(t)e^{-st}dt[/tex]
Why can they just be added?
A piecewise continuous function is a mathematical function that is defined by different equations on different intervals of its domain. This means that the function has different rules for different parts of its input values, but it is still continuous overall.
To transform a piecewise continuous function, you can apply different transformations such as translations, reflections, or stretches to each piece of the function. These transformations will affect the shape and position of the function, but each piece will still be continuous.
Piecewise continuous functions are useful for modeling real-world situations where different rules apply in different scenarios. They allow for more flexibility and accuracy in representing complex relationships between variables.
To graph a piecewise continuous function, you can first graph each individual piece of the function on its respective interval. Then, you can connect the pieces together to create a continuous graph. It is also helpful to label the points of intersection between the different pieces.
Piecewise continuous functions are commonly used in fields such as physics, engineering, and economics to model situations where different rules apply in different scenarios. They can also be used in computer programming and data analysis to represent complex relationships between variables.