###### Project 2.4 Part I

A periodic function is one whose values repeat at evenly spaced intervals, or periods, of the input variable. Periodic functions are used to model phenomena that exhibit cyclical behavior, such as growth patterns in plants and animals, radio waves, and planetary motion. In this project, we consider some applications of periodic functions.

###### Example 2.72

Which of the functions in Figure 2.73 are periodic? If the function is periodic, give its period.

This graph is periodic with period \(360\text{.}\)

This graph is not periodic.

This graph is periodic with period \(8\text{.}\)

## Section

### Subsection Exercise Set 1

###### 1

Which of the functions are periodic? If the function is periodic, give its period.

###### 2

Which of the following graphs are periodic? If the graph is periodic, give its period.

###### 3

Match each of the following situations with the appropriate graph.

When the heart contracts, blood pressure in the arteries rises rapidly to a peak (systolic blood pressure) and then falls off quickly to a minimum (diastolic blood pressure). Blood pressure is a periodic function of time.

After an injection is given to a patient, the amount of the drug present in his bloodstream decreases exponentially. The patient receives injections at regular intervals to restore the drug level to the prescribed level. The amount of the drug present is a periodic function of time.

The monorail shuttle train between the north and south terminals at Gatwick Airport departs from the south terminal every 12 minutes. The distance from the train to the south terminal is a periodic function of time.

Delbert gets a haircut every two weeks. The length of his hair is a periodic function of time.

###### 4

A patient receives regular doses of medication to maintain a certain level of the drug in his body. After each dose, the patient's body eliminates a certain percent of the medication before the next dose is administered. The graph shows the amount of the drug, in milliliters, in the patient's body as a function of time in hours.

How much of the medication is administered with each dose?

How often is the medication administered?

What percent of the drug is eliminated from the body between doses?

###### 5

You are sitting on your front porch late one evening, and you see a light coming down the road tracing out the path shown below, with distances in inches. You realize that you are seeing a bicycle light, fixed to the front wheel of the bike.

Approximately what is the period of the graph?

How far above the ground is the light?

What is the diameter of the bicycle wheel?

###### 6

The graph shows arterial blood pressure, measured in millimeters of mercury (mmHg), as a function of time.

What are the maximum (systolic) and minimum (diastolic) pressures? The pulse pressure is the difference of systolic and diastolic pressures. What is the pulse pressure?

The mean arterial pressure is the diastolic pressure plus one-third of the pulse pressure. Calculate the mean arterial pressure and draw a horizontal line on the graph at that pressure.

The blood pressure graph repeats its cycle with each heartbeat. What is the heart rate, in beats per minute, of the person whose blood pressure is shown in the graph?

For Problems 7–10, sketch a periodic function that models the situation.

###### 7

At a ski slope, the lift chairs take \(5\) minutes to travel from the bottom, at an elevation of \(3000\) feet, to the top, at elevation \(4000\) feet. The cable supporting the ski lift chairs is a loop turning on pulleys at a constant speed. At the top and bottom, the chairs are at a constant elevation for a few seconds to allow skiers to get on and off.

Sketch a graph of \(h(t)\text{,}\) the height of one chair at time \(t\text{.}\) Show at least two complete up-and-down trips.

What is the period of \(h(t)\text{?}\)

###### 8

The heater in Paul's house doesn’t have a thermostat; it runs on a timer. It uses \(300\) watts when it is running. Paul sets the heater to run from 6 a.m. to noon, and again from 4 p.m. to 10 p.m.

Sketch a graph of \(P(t)\text{,}\) the power drawn by the heater as a function of time. Show at least two days of heater use.

What is the period of \(P(t)\text{?}\)

###### 9

Francine adds water to her fish pond once a week to keep the depth at \(30\) centimeters. During the week, the water evaporates at a constant rate of \(0.5\) centimeter per day.

Sketch a graph of \(D(t)\text{,}\) the depth of the water, as a function of time. Show at least two weeks.

What is the period of \(D(t)\text{?}\)

###### 10

Erin's fox terrier, Casey, is very energetic and bounces excitedly at dinner time. Casey can jump \(30\) inches high,and each jump takes him \(0.8\) second.

Sketch a graph of Casey's height, \(h(t)\text{,}\) as a function of time. Show at least two jumps.

What is the period of \(h(t)\text{?}\)