Trend and periodicity

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.

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?

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?

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.

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.

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.

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.

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.

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 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?

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 $f(100)$.

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 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?

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.