A mug of hot cocoa sits on the counter in a $70\,^\circ\text{F}$ kitchen. It cools fast at first, then slower and slower, but it never drops below $70\,^\circ\text{F}$. Meanwhile, the Ferris wheel at Navy Pier in Chicago repeats the same loop on every ride. These are two very different long-term behaviors: a function can level off or it can repeat. In this section you will learn to predict both, and to answer questions like "Where will the cocoa's temperature end up?" and "Where will the rider be at minute 50?"
Leveling Off: Horizontal Asymptotes
What does a function do when $x$ gets bigger and bigger? Some functions grow without bound. Others settle toward a single value. We call this the end behavior of the function.
The easiest way to find a horizontal asymptote from a formula or a table is to plug in bigger and bigger values of $x$ and watch the outputs.
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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…
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 $+\infty$?
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…
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?
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…
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?
Many real situations give a fraction with $x$ in both the top and the bottom: an average cost, a concentration, a rate. To find where it levels off, ask which parts of the fraction matter when $x$ is huge.
The same idea works for any fraction of polynomials. Compare the highest powers of $x$ on top and on bottom.
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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{…
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 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…
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 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 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 $8$.
A horizontal asymptote tells you the limit, but you often also want to know when a function reaches some value before the limit. For that, set the formula equal to the value and solve.
Sometimes you compare two functions and want to know when one is bigger. Write an inequality and solve it. Because $t + 1$ is positive for $t \ge 0$, you can multiply both sides by it without flipping the inequality sign.
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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…
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 $85^\circ\text{C}$?
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…
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?
Why does the cocoa settle at room temperature? Anything left in a room warms or cools toward the room's temperature. The model is an exponential decay plus a constant.
Some growth models also level off. The denominator can contain a decaying power.
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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…
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?
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…
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?
Now the other behavior: repeating. The minute hand of a clock, the phases of the Moon, and the seasons all follow patterns that return to the start and go around again.
To find the output at a far-away input, remove whole periods until you land inside a cycle you already know. Divide $x$ by $T$ and keep the remainder: if $x = nT + r$ with $0 \le r < T$, then $f(x) = f(r)$. The remainder tells you where you are inside the cycle. The quotient only counts how many full cycles you skipped.
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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…
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.
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?
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?
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 $T = 4$…
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 $f(2019)$.
To find a period from a graph, pick a point on the pattern and find the next point where the pattern is at the same spot going the same direction. The horizontal distance between them is $T$. Peak to peak and trough to trough both work.
If you only know a half cycle, such as high tide to low tide, double it.
Often you also need a rate: how many cycles happen in one unit of time. Since one cycle takes $T$, the number of cycles per unit of time is $\dfrac{1}{T}$.
When the repeating fades
A bungee jumper bounces up and down. Each bounce is a bit lower than the one before, and eventually the jumper hangs still.
The time between bounces is roughly the same, but the heights shrink, so the graph never repeats exactly. This function is not strictly periodic. As time goes on, the height tends to the resting height: the height where the cord's pull balances the jumper's weight.
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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…
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 $x = 22$.
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…
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 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…
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?
A repeating schedule is a periodic function made of pieces: an "on" part and an "off" part, or a filling phase and a draining phase. The period is the total length of one full cycle, and the remainder tells you where you are inside it.
When two things repeat on different schedules, they line up again at the least common multiple (LCM) of their periods.
To count how many times a repeating event happens in a time span, count how many whole periods fit, then add $1$ for the very first event.
For a cycle with different phases, add up the phases to get the period, then find where you are in the cycle.
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A maritime signaling lighthouse emits an intense flash of light that lasts exactly $2\text{ seconds}$, followed by a period of total darkness of $7\text{ seconds}$,…
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?
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…
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…
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 $100$?
Back to the start: the cocoa follows $T(t) = 90 \cdot 0.8^t + 70$, so it levels off at the horizontal asymptote $y = 70$, the temperature of the kitchen. The Ferris wheel's height is periodic: after each period it repeats exactly, so to find the rider's position at minute $50$ you only need the remainder when you divide by the period.