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?

An oceanographic buoy located off the coast of Galway records wave height using a periodic function with period $T = 6\text{ seconds}$. In the first cycle, between $t = 0\text{ s}$ and $t = 6\text{ s}$, the recorded heights are the following: $f(1) = 1.8\text{ m}$; $f(2) = 2.5\text{ m}$; $f(3) = 1.2\text{ m}$; $f(4) = 0.4\text{ m}$; $f(5) = 0.8\text{ m}$. The trainee technician, Darragh, needs to find out the wave height at the instant $t = 81\text{ s}$. To do so, he divides $81$ by $6$ using a calculator and gets $13.5$. Darragh concludes in his report: «Since the result is $13.5$, the buoy is at an intermediate value, so I calculate the arithmetic mean between $f(13)$ and $f(5)$».\na) Identify the conceptual error Darragh makes when interpreting the decimal quotient from the calculator in the context of a periodic function.\nb) Correct the reasoning by applying integer Euclidean division (quotient and remainder) to determine which specific instant of the fundamental cycle $[0, 6)$ the second $t = 81\text{ s}$ exactly corresponds to.\nc) From your deduction, state the exact value of the water height at $t = 81\text{ s}$.

In a quality control laboratory in Cork, a freshly quenched metal part is placed inside a cooling chamber. The temperature of the part (in $^\circ\text{C}$) as a function of elapsed time $t$ (in hours) follows a model of the type:\n$$T(t) = A \times (0.65)^t + C$$\nIt is known that at the instant the part is inserted ($t = 0\text{ h}$) its temperature is $320^\circ\text{C}$, and that, if it were left inside the chamber indefinitely ($t \to \infty$), its temperature would tend to stabilize at the resting temperature of the refrigerator, which is kept constant at $18^\circ\text{C}$.\na) Reasonably deduce the value of the constants $A$ and $C$ from the data provided about the trend and the initial value.\nb) Write the final formula for $T(t)$ and determine the equation of the horizontal asymptote of the function, explaining the physical meaning of that line in the industrial process.

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 $800\text{ \euro}$ (rent, electricity, insurance) and manufacturing each pair of shoes has a materials cost of $5\text{ \euro}$. 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 one day. What value does the average cost tend towards 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 metres) 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 water level in the Vigo estuary oscillates periodically due to tides, completing a cycle every $12\text{ hours}$. At high tide, at $t = 0\text{ h}$ and at $t = 12\text{ h}$, the water reaches its maximum elevation of $4.5\text{ m}$. At low tide, halfway through the cycle ($t = 6\text{ h}$), the level drops to its minimum elevation of $1.5\text{ m}$.\na) Without using complex formulas, make a reasoned estimate of between which two values the tide height will be at $t = 27\text{ hours}$ from the start, and determine whether at that instant the water will be rising or falling. Justify your estimate by deducing how many full cycles will have elapsed and at which point in the cycle that instant is located.\nb) Knowing that the height is modeled by the function $h(t) = 3 + 1.5 \times \cos\left(\dfrac{\pi}{6} t\right)$ (with the angle in radians), calculate the exact value of $h(27)$ and check whether your initial estimate was accurate and consistent with the numerical result.

A report from the Central Statistics Office (CSO) analyzes depopulation in a small rural municipality in the Burren in County Clare. Starting from the year 2020 ($t = 0$), the number of registered inhabitants fits the following function with a trend:\n$$P(t) = 540 \times (0.8)^t + 360$$\nwhere $t$ is the elapsed time in years and $P(t)$ represents the total population.\na) What was the population of the municipality in the year 2020 and how many inhabitants are predicted for the year 2023?\nb) Analyze the population trend in the very long term ($t \to \infty$). Justify mathematically what value the population stabilizes toward, relying on the behavior of the base $0.8$ when raised to increasingly larger exponents, and state the equation of the horizontal asymptote.\nc) A local newspaper headline claims: «If the demographic trend does not change, this municipality will end up completely deserted with $0$ inhabitants». Mathematically refute this statement based on the concept of the model's horizontal asymptote.

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 euros) 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 $25\text{ mil \euro}$ and that, in the long term, market studies indicate that profits tend to stabilise at $120\text{ mil \euro}$ 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?

A sensor installed on a wind turbine in a wind farm in Albacete records the vibrations of a blade using a periodic function with period $T = 1.25\text{ seconds}$. The maintenance team only has the data table for the first cycle, corresponding to the interval $[0; 1.25)\text{ seconds}$, and needs to evaluate the condition of the blade at exactly the instant $t = 253.5\text{ seconds}$. Two analysts propose different methods:\n- Method A (Direct decimal quotient): Divide $253.5$ by $1.25$ on the calculator ($253.5 : 1.25 = 202.8$), take the decimal part $0.8$, and multiply it by the period ($0.8 \times 1.25 = 1.0\text{ s}$) to locate the point within the first cycle.\n- Method B (Grouping cycles into exact integers): Note that the period is fractional ($T = \dfrac{5}{4}\text{ s}$), so every $4$ full periods exactly $4 \times 1.25 = 5\text{ seconds}$ elapse. Decompose $253.5\text{ s}$ by separating the largest integer multiple of $5\text{ s}$ to drastically reduce the value before performing the final division.\na) Develop both methods step by step. Do you obtain the same equivalent instant within $[0; 1.25)$ in both cases? What is it?\nb) Now imagine that the sensor is replaced by another one whose exact period is $T = \dfrac{4}{3}\text{ seconds}$ ($1.3333\dots\text{ s}$). Reason which of the two methods would be risky due to decimal rounding on the calculator and justify which one you should choose to ensure maximum mathematical precision.

In the design department of a car manufacturer in Cork, a vehicle's shock absorption is tested when driving over a speed bump. The vertical displacement of the body relative to the chassis, $y(t)$ (in centimeters), oscillates up and down, reaching successive crests every $0.5\text{ seconds}$, but the height of these crests decreases rapidly until the car returns to its resting position ($y = 0$).\na) A trainee engineer states: «Since the car goes up and down regularly every $0.5\text{ seconds}$, the function describing the movement is periodic with period $T = 0.5\text{ s}$». Use the rigorous mathematical definition of a periodic function, $f(t+T) = f(t)$ for all $t$ in its domain, to refute their statement and prove that a damped oscillation can never be strictly periodic.\nb) Construct a function $y(t)$ that mathematically models this behavior by meeting the following three simultaneous conditions:\n 1. At the moment of impact ($t = 0$), the initial displacement is $y(0) = 8\text{ cm}$.\n 2. It contains an oscillatory factor of the form $\cos(k \times t)$ whose period is $0.5\text{ seconds}$ (remember that the period of $\cos(kt)$ is $\dfrac{2\pi}{k}$).\n 3. Its amplitude decays toward the equilibrium position through a factor of the form $b^t$ with $0 < b < 1$, such that the function tends to $0$ as $t \to \infty$.\nCheck algebraically that the formula you designed satisfies $y(0) = 8$ and that its limit as $t \to \infty$ is indeed $0$.