Question

which of the following best describes an application of periodic functions?

a music genre characterized by songs that start off with slow tempos which gradually increase as the songs continue.

a music genre characterized by songs that start off with fast tempos which gradually decrease as the songs continue.

an electrocardiogram (ekg) showing the electrical signals corresponding to a heartbeat with output values that repeat themselves over consecutive equal-sized intervals of time.

an electrocardiogram (ekg) showing the electrical signals corresponding to a heartbeat with output values that gradually increase over consecutive equal-sized intervals of time.

Explanation:

Define periodic functions

A periodic function is a function that repeats its values in regular, repeating intervals or cycles. Mathematically, a function \(f\) is periodic if there exists a non-zero constant \(P\) (the period) such that \(f(t + P) = f(t)\) for all values of \(t\) in the domain.

Analyze the first option

The first option describes a music genre where songs start with slow tempos and gradually increase. This represents a monotonic or strictly increasing trend over time, rather than a repeating cycle. Thus, it does not describe a periodic function.

Analyze the second option

The second option describes a music genre where songs start with fast tempos and gradually decrease. Similar to the first option, this represents a strictly decreasing trend over time. It does not exhibit repeating cycles, so it is not periodic.

Analyze the third option

The third option describes an electrocardiogram (EKG) showing electrical signals of a heartbeat with output values that repeat over consecutive equal-sized intervals of time. This matches the definition of a periodic function perfectly, where the repeating electrical pattern represents the function values and the equal-sized time intervals represent the period \(P\).

Analyze the fourth option

The fourth option describes an EKG where output values gradually increase over consecutive equal-sized intervals of time. Because the values are continuously increasing rather than repeating the same pattern, this represents a growing trend rather than a periodic cycle.

Answer:

• A music genre characterized by songs that start off with slow tempos which gradually increase as the songs continue.
• A music genre characterized by songs that start off with fast tempos which gradually decrease as the songs continue.
• An electrocardiogram (EKG) showing the electrical signals corresponding to a heartbeat with output values that repeat over consecutive equal-sized intervals of time. (Correct answer)
• An electrocardiogram (EKG) showing the electrical signals corresponding to a heartbeat with output values that gradually increase over consecutive equal-sized intervals of time.