Exercises: Trend and periodicity

Look at the graph of the following function which describes the position of a pendulum over time. Determine its period $T$ and, knowing that the pattern repeats indefinitely, calculate the value of the function at the instant $x = 22$.

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 the evolution of the temperature of a freshly poured cup of coffee over time. What value does the temperature tend to as time advances 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 when the independent variable $x$ tends to $+\infty$?

We take a can of soda out of the fridge at $4^\circ\text{C}$ and leave it on the dining table, where the room temperature remains constant at $23^\circ\text{C}$. What temperature will the soda tend to 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 throughout their life.

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

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

• The cycle of the phases of the Moon.

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 the value of $f(14)$, explaining your reasoning.

The time $t$ (in hours) it takes to empty the irrigation tank on 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 to if the number of pumps we use tends to infinity?

The famous "London Eye" Ferris wheel has an approximate diameter of $120\text{ m}$. Passengers board the capsules at the lowest part, which is practically at ground level ($0\text{ m}$). The wheel rotates at a constant speed and takes exactly $30\text{ minutes}$ to complete one full revolution. What is the period of the function that describes the height of a capsule with respect to time? At what height will a passenger be $45\text{ minutes}$ after boarding if they have not gotten off?

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

The following graph shows a simplified electrocardiogram (ECG) of a resting patient. 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 of a fishing port indicates that the first high tide of the day occurs at 04:00 in the morning and the next low tide occurs at 10:15 in the morning. Assuming that the 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 sneaker factory has daily fixed costs of $\$800\text{ \$}$ (rent, electricity, insurance) and manufacturing each pair of shoes has a materials cost of $\$5\text{ \$}$. The average cost per shoe when manufacturing $x$ pairs a day is given by the function $C_m(x) = \frac{800 + 5x}{x}$. Calculate the average cost if $10$, $100$ and $5000$ shoes are manufactured in a day. What value does the average cost tend to if the production volume tends to infinity?

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

A maritime signaling lighthouse emits an intense flash of light that lasts exactly $2\text{ seconds}$, followed by a period of total darkness of $7\text{ seconds}$, repeating this cycle uninterruptedly. If the lighthouse turns 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 representing the water level (in feet) 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 cord. Analyze 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 to as time tends to infinity, and what does this value represent physically?

The monthly profits of a start-up company (in thousands of dollars) since its creation are given by the rational function $B(x) = \frac{ax + 50}{x + 2}$, where $x$ is the number of months elapsed. If we know that in the initial month ($x=0$) the profit was $\$25\text{ mil \$}$ and that, in the long term, market research indicates that profits tend to stabilize at $\$120\text{ mil \$}$ 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 stop every $45\text{ minutes}$ and machine B requires a stop every $60\text{ minutes}$. If both machines receive simultaneous maintenance at 08:00 in the morning, at what time will they exactly coincide in maintenance again? 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?

An initially empty tank of $500\text{ L}$ is filled by a faucet that supplies $20\text{ L/min}$ for $15\text{ minutes}$. At exactly that moment, the faucet is closed and a drain valve is opened, emptying the tank at a constant rate of $50\text{ L/min}$ until it is completely empty, at which point the valve closes, the faucet opens, and the cycle starts again. Mentally draw the graph of the volume of water against time. What is the complete period of this process? What exact volume of water will be in the tank at minute $100$?

Let $f(x)$ be a function defined in 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 an operating industrial motor (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 it was turned on. What is the temperature at the moment the motor is started? What maximum temperature does it tend to if the motor runs uninterruptedly for weeks? How much exact time must elapse for the internal temperature to reach $85^\circ\text{C}$?

A weather satellite orbits the Earth such that it passes exactly 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 in total will it pass over the station throughout the entire week (until Sunday at 23:59)? On what day and at what exact time will the last pass be recorded that week?

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 limit values do the sales of each product tend to in the long term? From which day $t$ do the sales of product B definitively exceed those of product A?