Exercises: Trends and periodicity

Observe the following graph representing the oscillation of a pendulum over time $t$ (in seconds), measuring its displacement $d$ (in centimetres) from the equilibrium position.

Determine the period of the periodic function shown, its amplitude, and the maximum and minimum values it reaches.

Observe the following graph representing the water level in a tank (in metres) over time (in seconds). Determine the period of the function shown and calculate what the water level will be at $t = 45$ and at $t = 50$.

An industrial packaging machine emits a warning beep every $45$ seconds at a constant rate. If the first beep occurs at time $t = 0$, determine whether the machine will emit a beep at $t = 300$ seconds and at $t = 360$ seconds.

Given the rational function $f(x) = \frac{3x + 1}{x + 2}$, calculate the values of $f(10)$, $f(100)$ and $f(1000)$. From these results, deduce what value the function approaches as $x$ becomes infinitely large.

A function $f(x)$ is strictly periodic with a period of $T = 6$. The following values in its first cycle are known: $f(2) = 5$, $f(4) = -1$ and $f(6) = 3$. Calculate the exact values of $f(26)$ and $f(40)$.

The temperature of a cup of freshly poured coffee is $80^\circ\text{C}$. When left in a room, its temperature decreases over time, modelled by a function $T(t)$ that has a horizontal asymptote at $y = 22$. What physical meaning does this value of $22$ have in the context of the problem?

State the exact period of the basic trigonometric functions $f(x) = \text{sen}(x)$ and $g(x) = \text{cos}(x)$. If $\text{sen}(\frac{\pi}{6}) = 0,5$, what will be the value of $\text{sen}(\frac{13\pi}{6})$?

The light of a lighthouse rotates at a constant speed, making one full revolution every $12$ seconds. If at time $t = 3$ seconds the beam of light directly illuminates a nearby ship, at what other exact times during the first minute ($60$ seconds) will it illuminate it again?

Look at the graph of the function $T(t)$, which represents the temperature variation of a server's electronic component over operating time. What value does the temperature approach in the long term?

A traffic light in downtown Valencia has a programmed phase cycle that repeats periodically every $80\text{ s}$ ($T = 80\text{ s}$). In each cycle, the traffic light stays green during the first $40\text{ s}$ ($0 \le t < 40$), amber during the next $5\text{ s}$ ($40 \le t < 45$), and red during the last $35\text{ s}$ ($45 \le t < 80$). To find out what colour the traffic light will show exactly at $t = 345\text{ s}$ after the sequence starts, Mateo divides $345 : 80 = 4,3125$ and states: “Since the result starts with $4$, the traffic light is at second $4$ of its cycle, so the light will be green.” a) Point out precisely what the error in Mateo's reasoning is and explain what the integer $4$ and the decimal part $0,3125$ actually represent in this context. b) Correct the calculation by determining how many full cycles have been completed and at what exact second of the current cycle the traffic light is at $t = 345\text{ s}$. What colour does it actually show?

At fast-charging stations for electric cars in Spain, the charging speed decreases as the battery fills up to protect the lithium cells. The charge percentage $C(t)$ of a vehicle's battery after being plugged in for $t$ minutes is given by: $C(t) = 100 - 85 \cdot 0,96^t$ (for $t \ge 0$). a) Estimate: without performing detailed operations on your calculator, observe the structure of the function and indicate what percentage the charge tends toward if the car were left plugged in for a very long time ($t \to \infty$). Justify your estimate by analyzing what happens to the power $0,96^t$ when repeatedly multiplying a number less than $1$. b) Check: identify the horizontal asymptote of the function ($y = L$) and calculate the actual charge percentage for $t = 10\text{ min}$, $t = 30\text{ min}$, and $t = 60\text{ min}$ (rounding to the nearest integer). c) Critically assess whether your initial estimate is consistent with the results obtained and reflect on why many users prefer to unplug the car after half an hour instead of waiting a full hour.

A function $f(x)$ is periodic with a period of $T = 8$. In the interval $[0, 8)$, the function is defined piecewise as follows: $f(x) = 2x$ if $0 \le x < 4$, and $f(x) = 12 - x$ if $4 \le x < 8$. Calculate the exact values of $f(45)$ and $f(100)$.

The population of a bird species introduced to an island follows the mathematical model $P(t) = \frac{1500t + 200}{3t + 1}$, where $t$ represents the time elapsed in years since its introduction. Determine the initial bird population and calculate what value the population approaches in the long term as $t$ becomes very large.

An electrocardiogram records the electrical activity of the human heart, showing a pattern that repeats cyclically. If the time interval between two consecutive R peaks (a full cardiac cycle) is $0,8$ seconds, how many beats per minute does the resting patient have? If an exact peak is recorded at $t = 1,2$ seconds, at what time will the $50$th peak after that moment be recorded?

The water level in a sea port varies due to tides according to a periodic function. Given that throughout a day the maximum water level (high tide) is reached at exactly 02:00 and 14:30, determine the exact period of the tides in this port. If this pattern continues unchanged, at what time will the first maximum level be reached the following day?

The economic value of an industrial machine depreciates over time according to the exponential function $V(t) = 50000 \cdot 0,85^t + 2000$, where $t$ is the years of use and $V(t)$ is the value in euros. What is the value of the machine when bought new ($t=0$)? What is the trend of the machine's value as the time of use becomes infinitely large? Explain the economic meaning of this limit.

Determine the exact period of the following combined trigonometric functions, giving reasons for your answer using the properties of the sine and cosine functions: $f(x) = \text{sen}(4x)$ and $g(x) = \text{cos}\left(\frac{x}{3}\right)$.

A Ferris wheel at an amusement park has a diameter of $40$ metres and its central axle is at a height of $22$ metres above the ground. The Ferris wheel rotates at a constant speed and takes exactly $4$ minutes to make one full revolution. If a person gets into the carriage at the lowest point at time $t = 0$, determine at what exact height above the ground they will be in $t = 11$ minutes.

A function $f(x)$ is strictly periodic with a period of $T = 5$ and also exhibits odd symmetry. Given that $f(1) = 4$ and $f(2) = -3$, determine with justification the values of $f(-4)$ and $f(13)$.

Tide tables from Puertos del Estado in Santander (Cantabria) show that the water height follows a periodic function with an approximate period of $T = 12\text{ hours}$. During the first $12$ hours of a spring tide day, the height $h(t)$ (in metres) relative to the chart datum changes as follows: it falls to a low tide of $1,2\text{ m}$ at $t = 3\text{ h}$ and rises to a high tide of $4,6\text{ m}$ at $t = 9\text{ h}$. To calculate the tide height at $t = 195\text{ hours}$ from the beginning of the month, two port technicians propose different methods: Method 1 (Iterative subtraction) consists of repeatedly subtracting $12$ from $195$ as many times as necessary ($195 - 12 = 183$; $183 - 12 = 171$; ...) until obtaining a number within the interval $[0, 12)$; Method 2 (Euclidean division) consists of dividing $195$ by $12$, taking the integer remainder of the division, and locating that remainder in the initial cycle table. a) Explain why both methods are mathematically valid based on the property of periodic functions: $f(t) = f(t + k \cdot T)$ with $k \in \mathbb{Z}$. b) Evaluate the advantages and disadvantages of each method. Which one would you choose if you needed to predict the tide in one year ($t = 8\,760\text{ h}$)? Justify your answer in terms of speed and error prevention. c) Apply the most efficient method to determine which point of the base cycle corresponds to $t = 195\text{ h}$. Based on that result, indicate whether it coincides with low tide or high tide, and what the height of the tide is.

A demographic report analyzes the evolution of a rural municipality in Teruel affected by depopulation. The population $P(t)$ (in number of inhabitants) $t$ years after the year 2010 is approximated by the model: $P(t) = 640 \cdot 0,85^t + 180$. a) Determine the population of the municipality in the year 2010 ($t = 0$) and calculate it for the year 2020 ($t = 10$). b) Analyze mathematically the long-term trend ($t \to \infty$): what is the horizontal asymptote line of the function? Justify in detail why the term $640 \cdot 0,85^t$ tends to 0 as the years go by. c) A local newspaper headline states: “At the current rate, this municipality in Teruel will become completely uninhabited (0 inhabitants) within fifty years.” Using the calculated asymptote and the concept of resting or stabilization value, give a reasoned argument explaining why the headline makes a mathematical interpretation error. d) If the regional medical services require at least $200$ registered inhabitants to keep the local clinic open, calculate whether in the year 2025 ($t = 15$) the clinic faces an immediate risk of closure and what will happen in the very long term according to this model.

The concentration of a medication in the bloodstream (measured in mg/L) $t$ hours after its intravenous administration is given by the function $C(t) = \frac{50t}{t^2 + 9}$. Analyse the long-term trend of this concentration. Is there a time in the future when the concentration stabilises at a non-zero constant value, or will it eventually disappear completely from the body? Justify your answer mathematically.

Two coastal lighthouses, A and B, emit flashes of light to guide ships. Lighthouse A flashes every $15$ seconds and Lighthouse B flashes every $24$ seconds. If at 22:00:00 both lighthouses flash completely simultaneously, how many times will their flashes coincide again over the next hour? At what exact time (hours, minutes and seconds) will the tenth coincidence occur after 22:00:00?

The position of a mass hanging from a spring oscillating with friction is modelled by the damped function $x(t) = 10 \cdot 0,9^t \cdot \text{cos}(\pi t)$, with $t$ measured in seconds and $x(t)$ in centimetres. Look at its graph. Is this function strictly periodic? What value does the position $x(t)$ approach as $t$ becomes very large, and what does this limit mean physically?

Let $f(x)$ be a periodic function with a period of $T = 4$, which in the interval $[0, 4)$ is defined by the quadratic expression $f(x) = -x^2 + 4x$. Prove algebraically whether $f(x)$ is symmetrical about the vertical line $x = 2$ within its first period. Next, calculate the exact value of $f(2023)$.

Determine the exact value of the parameter $k$ so that the rational function $f(x) = \frac{kx^2 - 3x + 1}{2x^2 + 5}$ tends to $4$ as $x$ becomes infinitely large. Once that value of $k$ is found, calculate the y-intercept of the function.

Consider the mathematical function $f(x) = x - [x]$, where the symbol $[x]$ represents the floor function (or integer part) of $x$ (that is, the greatest integer less than or equal to $x$). Prove with reasons that $f(x)$ is a periodic function and find its period. Finally, calculate $f(14,75)$ and $f(-3,2)$.

An observation satellite orbits the Earth at low altitude, passing directly over the equator every $90$ minutes. Each time it completes an orbit, the Earth has rotated beneath it. Given that the Earth takes $24$ hours to make one full rotation on its axis, what is the minimum time that must elapse for the satellite to pass directly over the exact same geographical coordinates on the equator again?

The trigonometric functions $f(x) = \text{sen}(ax)$ and $g(x) = \text{cos}(bx)$ have fundamental periods $T_1 = 4\pi$ and $T_2 = 6\pi$ respectively, where $a$ and $b$ are positive real numbers. Determine the period of the sum function $h(x) = f(x) + g(x)$ and find the numerical values of the parameters $a$ and $b$.

A pharmaceutical distribution centre in Seville stores vaccines that require continuous cold storage. Due to an electrical breakdown in summer, the cold storage room loses power and the internal temperature $T(t)$ (in $^\circ\text{C}$) begins to rise as a function of the elapsed hours $t$ according to a model of the type $T(t) = A \cdot b^t + C$ (with $0 < b < 1$). The technical team does not have the complete formula, but they have the following data from the safety protocol: 1) The outside temperature of the warehouse is a constant $35\,^\circ\text{C}$; due to thermal transfer, if the breakdown is not repaired, the chamber tends to stabilize at that outside temperature in the long term ($t \to \infty$). 2) At the exact moment the power failed ($t = 0\text{ h}$), the chamber was at its optimal temperature of $3\,^\circ\text{C}$. 3) At $t = 2\text{ hours}$ after the blackout, the digital thermometer recorded $19\,^\circ\text{C}$. a) Deduce the value of the horizontal asymptote and use it to justify the value of parameter $C$. b) From the data at $t = 0\text{ h}$, set up the corresponding equation and obtain the value of $A$. c) Using the temperature record at $t = 2\text{ h}$, set up the equation to solve for the base $b$ and write the final analytical expression of $T(t)$. d) If the vaccines lose their biological properties if the temperature exceeds $25\,^\circ\text{C}$, calculate with reasoning what the temperature will be at $t = 4\text{ hours}$ after the power outage and determine whether the batch of vaccines can still be saved or has spoiled.

At a wind farm in Albacete, the blades of a wind turbine rotate at a constant speed driven by the wind. The height $h(t)$ (in metres above the ground) of the tip of one of its blades after $t$ seconds of observation is described by the periodic function: $h(t) = 45 \cdot \sin(k \cdot t) + 80$. It is known that the blade takes exactly $6\text{ seconds}$ to complete one full revolution ($T = 6\text{ s}$). a) Recall that the periodic function $\sin(kx)$ has a period $T = \dfrac{2\pi}{k}$. Calculate the exact value of the constant $k$ for this wind turbine and write the final expression of $h(t)$. b) Taking into account that the values of the sine function always oscillate between $-1$ and $1$, calculate the minimum height and the maximum height reached by the tip of the blade during the rotation. From these results, deduce the hub height (the central axis of rotation at the top of the tower) and the length of the blade. c) To check for material fatigue wear, a telematic sensor records the position of the blade tip at time $t = 249\text{ seconds}$. Without simulating all the preceding revolutions, explain how to use the properties of periodicity to determine which second of the base cycle $[0, 6)$ corresponds to that instant. d) Calculate the exact height $h(249)$ knowing that $\sin\left(\dfrac{3\pi}{2}\right) = -1$ and $\sin\left(\dfrac{\pi}{2}\right) = 1$. Is the blade at its highest point, at its lowest point, or in an intermediate position?

Topic challenge. How does our country consume electricity? Become an energy analyst by investigating the electricity demand curve on the Red Eléctrica de España (REE) website.

1. Data collection: Go to demanda.ree.es and record the actual peninsular demand ($y$, in thousands of MW, for example $26,4$) in a spreadsheet every two hours over two consecutive days ($x \in [0, 48]$ h).

2. Plotting: Generate the graph by connecting the points using GeoGebra or a spreadsheet. Describe how the information is presented to us (table, graph, and verbal description).

3. Analytical report: Mathematically analyze the function by determining its domain, range, and output for key hours; continuity; intervals of increase and decrease; maxima and minima (peak and off-peak hours); symmetry; daily periodicity ($24$ h cycles), and general trend.

4. Reasoned proposal: Design an efficient consumption recommendation for your home, indicating the time slots in which it is best to use appliances based on the analyzed data.

Assessment criteria:

• Rigorous collection of real data from the official source and proper graphical representation using digital tools.

• Correct identification and justification of functional properties (domain, extrema, continuity, and periodicity).

• Consistency and practical utility of saving proposals based on the mathematical analysis performed.