Continuous Time Function

In summary, the conversation discusses modifying an equation to become a continuous-time function, y(t), when the function f is changing with time t. The equation given is y=\alpha e^{\beta W}, where \alpha = a e^{b f} and \beta = c f + d, and W = \int^{T}_{0}f dt. The solution involves replacing each variable with its explicit dependence on t, but this may not add any new information without knowing the explicit form of f.
  • #1
sooyewguan
1
0
Lets say I have an equation,

[tex]y=\alpha e^{\beta W}[/tex]

where,
[tex]\alpha = a e^{b f}[/tex] and [tex]\beta = c f + d[/tex]

[tex]W = \int^{T}_{0}f dt[/tex]

My problem now is, what happen if [tex]f[/tex] is changing with time [tex]t[/tex], [tex]f(t)[/tex]

How do I modify my main equation, [tex]y[/tex], so that it become an continuous-time function, [tex]y(t)[/tex].

Thank you.
 
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  • #2
I'm not sure what you mean: what you give
[tex]y= \alpha e^\beta W(t)[/itex]
is a "continuous-time function"- or at least a continuous function of t.

If you want to you can replace each of [itex]\alpha[/itex], [itex]\beta[/itex], and W with their explicit dependence on t:
[tex]y(t)= ae^{bf(t)} e^{(cf+d)\int_0^t f(u)du}[/tex]
(I've changed the dummy variable in the integral to u so as not to confuse it with the variable t.)

But I don't think that really adds anything as long as you don't know the explicit form of f.
 

What is a continuous time function?

A continuous time function is a mathematical representation of a relationship between two variables, where the independent variable (usually time) can take on any value within a given interval. This means that the function has a defined value at every point in the interval, without any gaps or jumps.

How is a continuous time function different from a discrete time function?

A discrete time function is defined only at specific, discrete points in time, while a continuous time function is defined at every point in a given interval. This means that a discrete time function can have gaps or jumps in its values, whereas a continuous time function does not.

What is the importance of continuous time functions in science?

Continuous time functions are important in science because they allow us to model and understand real-world phenomena that change continuously over time. They are used in fields such as physics, engineering, and biology to describe and predict the behavior of systems and processes.

What are some common examples of continuous time functions?

Some common examples of continuous time functions include the position of a moving object over time, the temperature of a room, and the growth of a population. These functions can be represented by mathematical equations and graphs, which allow us to analyze and understand their behavior.

What are the limitations of continuous time functions?

One limitation of continuous time functions is that they assume a smooth and continuous relationship between the variables, which may not always be the case in real-world situations. Additionally, some phenomena may be better modeled using discrete time functions, such as computer simulations or digital signals.

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