Periodic function and substitution question

This will allow you to replace f(x)dx with f(x+T)dx. Then, use a substitution u = x+T in the first part to simplify the integral. Finally, combine the two parts to get the desired result. In summary, use substitution and the fact that f is periodic to prove that the integral of f from a to a+T is equal to the integral of f from 0 to T.
  • #1
1
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Let f: R-> R be a continuous function. Let T>0 be such that
f(x+T)= f(x) for all x.
We say that f is a periodic function with a period T>0.
Use an appropriate substitution to prove that for all real numbers a
[tex]\int^{a+T}_{a}f(x)dx[/tex] = [tex]\int^{T}_{0}f(x)dx[/tex].

I have no idea how to do this question.
thanks for helping me!
 
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  • #2
cjl28 said:
Let f: R-> R be a continuous function. Let T>0 be such that
f(x+T)= f(x) for all x.
We say that f is a periodic function with a period T>0.
Use an appropriate substitution to prove that for all real numbers a
[tex]\int^{a+T}_{a}f(x)dx[/tex] = [tex]\int^{T}_{0}f(x)dx[/tex].

I have no idea how to do this question.
thanks for helping me!

Try breaking the integral on the left into two parts:

[tex]\int_{a}^{a+T} = \int_{a}^{T} + \int_{T}^{T+a}[/tex]

and in the second part, use the fact that f is periodic.
 

1. What is a periodic function?

A periodic function is a mathematical function that repeats its values in regular intervals. This means that the function has a specific pattern that continues infinitely in both positive and negative directions.

2. How do you determine the period of a periodic function?

The period of a periodic function is the distance between two consecutive points on the graph where the function repeats itself. To determine the period, you can find the distance between two consecutive peaks or troughs on the graph.

3. What is the difference between a periodic function and a non-periodic function?

A periodic function repeats itself in regular intervals, while a non-periodic function does not have a specific pattern and does not repeat itself. This means that a non-periodic function does not have a period.

4. How do you use substitution to solve a periodic function?

To use substitution, you can replace a variable in a periodic function with another expression or value. This can help simplify the function and make it easier to solve or evaluate.

5. Can any function be written as a periodic function?

No, not all functions can be written as periodic functions. Only functions that have a specific pattern and repeat themselves can be written as periodic functions.

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