University of Florida/Egm4313/s12.team8.dupre/R6.1

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R6.1

Problem Statement[edit]

Given :

(1) Find the (smallest) period of and .

(2) Show that these functions also have period .

(3) Show that the constant is also a periodic function with period .

Solution (1)[edit]

We know that the period of a normal or is . When there are values or variables being multiplied by this variable, the period becomes divided by the values or variables. We know that:

(6.1.1)

Where is the period.

Using this relation, along with our variable , we can solve for the period of and as follows:



(6.1.2)

Since the period will be smallest at , plugging into equation (6.1.2) shows that the smallest period of and is:

     (6.1.3)

Solution (2)[edit]

We are also given that:

(6.1.4)

Using this relation, we can solve for the period again as follows:



(6.1.5)

We know that the period is smallest at , and plugging this value into (6.1.5) proves that:

     (6.1.6) 

Solution (3)[edit]

We know that, starting at 0:

(6.1.7)

Where the period is represented by 0 to 2L. We are also given that:

(6.1.8)

Rearranging (6.1.8) allows us to solve for L:

(6.1.9)

Multiplying (6.1.9) by 2 allows us to find the period of , as follows:

     (6.1.10)

This shows that is indeed a periodic function with a period of . This also shows that at any given value or period throughout the periodic function, holds its constant value.