Exercises: Trends and periodicity

Observe the graph of the following function describing the height of a Ferris wheel cabin relative to its central position over time. Determine its period $T$ and, knowing that the pattern repeats indefinitely, calculate the value of the function at time $x = 22$.

Knowing that a function $f(x)$ is periodic with a period $T = 6$ and that, in the first cycle, it reaches a value of $f(2) = 5$, what will the value of $f(20)$ be?

The following graph shows how the temperature of a freshly poured cup of coffee changes over time. What value does the temperature tend towards as time increases indefinitely? What is the mathematical name for the horizontal line that the curve approaches?

Given the following table of values for a function $g(x)$ obtained using a spreadsheet, what do you think its trend is as the independent variable $x$ tends to $+\infty$?

We take a can of soft drink out of the fridge at $4^\circ\text{C}$ and leave it on the dining room table, where the room temperature remains constant at $23^\circ\text{C}$. What temperature will the soft drink tend towards if we leave it there unopened for several days?

Classify the following real-life phenomena according to whether they correspond to periodic or non-periodic functions:

• A person's height over their lifetime.

• The movement of the minute hand on a wall clock.

• The available balance in a bank account throughout the year.

• The cycle of the Moon's phases.

A periodic function has a period $T = 5$. If we know its values in the first cycle: $f(1) = 3$, $f(2) = 7$, $f(3) = 0$, $f(4) = -2$ and $f(5) = 1$, calculate with reasoning the value of $f(14)$.

The time $t$ (in hours) it takes to empty the irrigation tank of a farm depends on the number of identical water pumps $n$ connected simultaneously, according to the function $t(n) = \frac{120}{n}$. What value does the emptying time tend towards if the number of pumps used tends to infinity?

The famous "London Eye" Ferris wheel has a diameter of approximately $120\text{ m}$. Passengers board the cabins at the lowest point, which is practically at ground level ($0\text{ m}$). The wheel turns at a constant speed and takes exactly $30\text{ minutes}$ to complete one full revolution. What is the period of the function describing the height of a cabin over time? At what height will a passenger be $45\text{ minutes}$ after boarding if they have not got off?

The concentration of an antibiotic in a patient's bloodstream (measured in $\text{mg/L}$) $t$ hours after administration is given by the mathematical expression $C(t) = \frac{15t}{t^2 + 5}$. Calculate the concentration after $2\text{ hours}$, $10\text{ hours}$ and $100\text{ hours}$. From these results, what is the long-term trend of the concentration?

The following chart shows a simplified electrocardiogram (ECG) of a patient at rest. Determine the period of the function in seconds and, from this data, calculate the patient's heart rate expressed in beats per minute (bpm).

The tide table for a fishing port indicates that the first high tide of the day occurs at 04:00 in the morning and the following low tide occurs at 10:15 in the morning. Assuming that sea level behaves as a periodic function in which rising and falling take the same time, what is its period in hours and minutes? At what exact time will the next high tide occur?

A small sports shoe factory has daily fixed costs of $16\,000\text{ \$}$ (rent, electricity, insurance) and manufacturing each pair of shoes has a materials cost of $100\text{ \$}$. The average cost per shoe when manufacturing $x$ pairs a day is given by the function $C_m(x) = \frac{16\,000 + 100x}{x}$. Calculate the average cost if $10$, $100$ and $5000$ shoes are manufactured in one day. What value does the average cost tend toward if the production volume tends to infinity?

The population of a protected species in a nature reserve has been modelled using the function $P(t) = \frac{6000}{1 + 4 \times 0.75^t}$, where $t$ is the time elapsed in years from the start of the study. What was the initial population in the year $t=0$? What value does the population tend towards in the long term as $t \to +\infty$? Does this limit make ecological sense?

A lighthouse signalling beacon emits a flash of bright light lasting exactly $2\text{ seconds}$, followed by a period of total darkness of $7\text{ seconds}$, repeating this cycle continuously. If the lighthouse is switched on for the first time at night at 22:00:00, what is the period of this light function? Will the light be on or off at 22:01:15?

Look at the following graph showing the water level (in meters) in an automated industrial tank over time (in minutes). Determine the period of the function and calculate what the exact water level will be at minute $t = 86$.

The following graph shows the height of a bungee jumper over time, subject to the oscillation of the elastic rope. Analyse the graph and answer: what is the approximate time "period" between the first two bounces? Can it be considered a strictly periodic function? What value does the jumper's height tend towards as time tends to infinity, and what does this value physically represent?

The monthly profits of a start-up company (in thousands of pesos) since its creation are given by the rational function $B(x) = \frac{ax + 1\,000}{x + 2}$, where $x$ is the number of months elapsed. If we know that in the initial month ($x=0$) the profit was $500\text{ mil \$}$ and that, in the long term, market studies indicate that profits tend to stabilize at $2\,400\text{ mil \$}$ per month, determine the exact value of the parameter $a$ and calculate the expected profit for month $8$.

In an automated industrial plant, machine A requires a maintenance stoppage every $45\text{ minutes}$ and machine B requires a stoppage every $60\text{ minutes}$. If both machines undergo maintenance simultaneously at 08:00 in the morning, at what time will they next undergo maintenance together? If we define a mathematical function that equals $1$ when both machines are stopped at the same time and $0$ the rest of the time, what is the period of this function?

A tank of $500\text{ L}$ initially empty is filled with a tap providing $20\text{ L/min}$ for $15\text{ minutes}$. At that exact moment, the tap is turned off and a drain valve opens, emptying the tank at a constant rate of $50\text{ L/min}$ until it is completely empty, at which point the valve closes, the tap opens and the cycle begins again. Picture the graph of water volume against time. What is the full period of this process? What exact volume of water will there be in the tank at minute $100$?

You can do it on paper and send a photo with your answer.

Let $f(x)$ be a function defined on the interval $[0, 4)$ by the algebraic expression $f(x) = x^2 - 2x$. We also know that the function is periodic with period $T = 4$ for any value of $x \ge 0$, satisfying $f(x+4) = f(x)$. Based on these properties, calculate step by step the exact value of $f(2019)$.

The internal temperature of a running industrial engine (measured in $^\circ\text{C}$) follows the mathematical law $T(t) = 90 - \frac{120}{t + 2}$, where $t$ is the time elapsed in hours since turning it on. What is the temperature at the moment the engine starts? What maximum temperature does it tend towards if the engine runs continuously for weeks? How much time must elapse for the internal temperature to reach $85^\circ\text{C}$?

A weather satellite orbits the Earth such that it passes directly over a ground tracking station every $4\text{ hours and } 15\text{ minutes}$. If the first pass of the week over the station is recorded on a Monday at 02:30 in the morning, how many times will it pass over the station in total throughout the whole week (up to Sunday at 23:59)? On what day and at what exact time will the last recorded pass of that week occur?

The estimated daily sales of two competing products, A and B, follow these trend functions as the days $t$ pass: $V_A(t) = 500 - \frac{1000}{t+1}$ and $V_B(t) = 600 - \frac{3000}{t+1}$. What limiting values do the sales of each product tend towards in the long term? From which day $t$ onwards do the sales of product B permanently exceed those of product A?