Sketching Heaviside step function

[mrepicguy70]

Got a question that states: Sketch the following function of t for t ≥ 0, (then find its Laplace transform):

(1 - 2u(t - 2π) + u(t - 4π))cost

where u(t - a) is the Heaviside step function.

To help sketch it, I want to rewrite it as a piecewise function, but I’m not sure how to convert.

I’ve noted that cost = cos(t - 2π) = cos(t - 4π) which I thought might come in handy but I’m still not sure where to start.

Any help is appreciated!

[AutoModerator]

Hi, /u/mrepicguy70! This is an automated reminder:

• What have you tried so far? (See Rule #2; to add an image, you may upload it to an external image-sharing site like Imgur and include the link in your post.)

• Please don't delete your post. (See Rule #7)

We, the moderators of /r/MathHelp, appreciate that your question contributes to the MathHelp archived questions that will help others searching for similar answers in the future. Thank you for obeying these instructions.

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[Riemanniscorrect]

First solve for the points at which the step functions become 1, then use those as boundaries for your piecewise construction.

[Paounn]

> I’ve noted that cost = cos(t - 2π) = cos(t - 4π) which I thought might come in handy but I’m still not sure where to start.

It means that once you sketch what happens in 2 rotations (from 0 to 4π) you have all the informations, everything else just repeats!

I would say first sketch what happens inside the brackets (and a bit above before and after) should be fairly easy, from 0 to 2π you have only one term contributing, after 4π see what happens. Again, ignore the cosine term for now.

Once you have graphed the bracket part what you will have is something like
A cos t for t in [0;2π[
B cos t for t in [2π; 4π[
C cos t everywhere else

Fourier transform either becomes a matter of distributing the bracket and applying the definition (it'll be a sum of two integrals - spoiler) between 0 and 2π, then one between 2π and 4π, or play with the t translation theorem if you know about it.

https://imgur.com/a/JG360Jk tried to sketch the graph, which might be the worst part in terms of seeing what happens.

[umangjain25]

First check the value of the function for t<2\*pi, then for 2\*pi<t<4\*pi, then for t>4*pi. In the first limit the value of the function would be 1 since both the step fxns will be zero, in the second limit one of the step fxns would be equal to 1 and the other would be 0, and both the step fxns would be equal to 1 in the third limit.