Trends and periodicity

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 $+\infty$?

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?

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?

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 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?

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?

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 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?

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.

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 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?