Variable Dependence: When One Quantity Controls Another
Textbook Reference
| Primary source | OpenStax Calculus Volume 1, Section 1.1: “Review of Functions” |
| Direct link | https://openstax.org/books/calculus-volume-1/pages/1-1-review-of-functions |
| Supplementary | OpenStax Precalculus 2e, Section 1.1: “Functions and Function Notation” |
| Supplementary link | https://openstax.org/books/precalculus-2e/pages/1-1-functions-and-function-notation |
Both sources are free and openly licensed.
Try This First
A water tank is being filled from a hose. The table below shows the water level at different times.
| Time (minutes) | Water level (centimeters) |
|---|---|
| 0 | 4 |
| 1 | 7 |
| 2 | 10 |
| 3 | 13 |
Before reading further:
- Which quantity is changing because of the other?
- If you wanted to know the water level, what information would you need first?
- Could you predict the water level at 4 minutes? What does that prediction depend on?
Keep your answers. The ideas below name what you just noticed.
Key Idea
Two quantities covary when changing one causes (or is associated with) a change in the other. In the tank example, the water level changes because time has passed. Time does not change because the water level changed. One quantity drives the other.
The quantity that is controlled or chosen is the input (also called the independent variable). The quantity whose value is determined by the input is the output (also called the dependent variable).
Recognizing which quantity depends on which is the first step in building a mathematical model of any situation. Every function, every derivative, every integral you will meet in calculus is about a relationship between two covarying quantities. Getting clear on which is which matters.
Two Connected Representations
The relationship “water level depends on time” can be shown two ways:
Table:
| Time (min) | Level (cm) |
|---|---|
| 0 | 4 |
| 1 | 7 |
| 2 | 10 |
Graph: plot time on the horizontal axis, level on the vertical. The horizontal axis holds the input; the vertical holds the output.
Translation prompt: In the graph, time runs left to right. What does moving one unit to the right correspond to in the table? What does moving upward correspond to?
Worked Example
Identify the input and output in each situation.
(a) The cost of a taxi ride depends on how many kilometers are driven.
Predict first: which quantity controls which? The driver chooses the destination; distance determines cost.
Check: the distance driven (km) is the input. The cost (dollars) is the output. Cost depends on distance; distance does not depend on cost.
(b) The temperature outdoors at a fixed location is measured every hour.
Predict first: can we “choose” the temperature? No. We choose the time to take a measurement, and the temperature is what we find.
Check: time (hours) is the input. Temperature (degrees) is the output. Temperature depends on the time of day, not the other way around.
(c) A ball is dropped from a building. The height of the ball above the ground is recorded.
Predict first: what was set in motion by the drop? The ball falls; time passes; height decreases.
Check: time (seconds) is the input. Height (meters) is the output. Height depends on time.
the quantity that “sounds more important” is the output. Students sometimes put the quantity that seems more interesting or consequential on the output side, regardless of which actually controls which. In situation (c) above, “height” might seem like the main thing to track, so a reader might write “time depends on height.” But the ball’s height at 2 seconds is determined by the fact that 2 seconds have elapsed, not by the height itself. Ask: which quantity is fixed or chosen first, and which is then determined? That first quantity is the input.
A Non-Example
Not every pair of changing quantities forms a clear dependence relationship. Consider:
- The number of hours studied and the score on a test.
Does the score depend on hours studied, or do hours studied depend on the score? In reality the score does not cause the hours, and the hours do not guarantee a specific score. There is a relationship -- more hours often leads to higher scores -- but it is not the clean input-output dependence of the tank problem. Recognizing this limits what we can model and reminds us that dependence in mathematics means something precise: for each input, exactly one output.
Predict-Then-Check Practice
Problem 1 (Low floor): A recipe uses 2 cups of flour for every dozen cookies. What is the input? What is the output?
Check your answer
The number of dozens of cookies is the input (the baker chooses how many to make). The cups of flour is the output (that amount is determined once the batch size is fixed).
Predict: if you make 3 dozen cookies, how many cups of flour do you need? Predict before computing: probably between 4 and 8 cups.
Compute: $3 \times 2 = 6$ cups. Did your estimate bracket the answer?
Problem 2: A park charges an entry fee per car. As more cars enter, the total revenue collected increases. Identify the two quantities and which depends on which.
Check your answer
The two quantities are: the number of cars that enter, and the total revenue.
Revenue depends on the number of cars. Revenue is the output; number of cars is the input.
Problem 3 (High ceiling): Consider the perimeter $P$ and area $A$ of a square. Both change as the side length $s$ changes. Can we say that $P$ depends on $A$, or that $A$ depends on $P$, or that both depend on $s$? Is there more than one valid answer?
Check your answer
All three of $s$, $P = 4s$, and $A = s^2$ covary. The most natural input is the side length $s$, with both $P$ and $A$ as outputs.
However, since $P$ determines $s$ (namely $s = P/4$), and $s$ then determines $A$, one can also say that $A$ depends on $P$, writing $A = (P/4)^2 = P^2/16$.
There is more than one valid framing. The model you build depends on which quantity you have the freedom to choose. Why might a builder prefer to work with side length rather than perimeter? Why might a fencing contractor prefer perimeter?
Where this shows up
Every rate-of-change question -- “how fast is the water level rising?” “how fast is the cost increasing?” “how fast is the population growing?” -- requires knowing which quantity is the input (usually time) and which is the output (usually what you are measuring). The derivative is defined as how much the output changes per unit change in the input. Without knowing which is which, the derivative has no meaning.
This first step, identifying the two quantities and which depends on which, is the foundation of everything in Chapters 2 and 3.