Chapter 1: What Is Change?
Understanding "change" through everyday examples
Introduction
1.1 Change in Everyday Life
The world around us is full of change.
- The temperature changes over the course of a day
- A savings balance grows with interest
- A thrown ball changes its position
- A car's speed changes
Differentiation is a tool for describing this kind of change mathematically.
The figure shows how the temperature changes over time. It is low in the morning, rises towards midday, and falls again at night.
1.2 Position and Velocity
To understand change, let us start with the most familiar example: velocity.
Example: A Moving Car
A car travels at a constant speed along a straight road (the starting point is $0$ km).
- After 1 hour: at the 60 km mark
- After 2 hours: at the 120 km mark
- After 3 hours: at the 180 km mark
In this graph:
- the horizontal axis is time (the independent variable, or input);
- the vertical axis is position (the distance along the road from the start; the dependent variable, the quantity determined by the time).
The graph is a straight line because the car travels at a constant speed.
Slope of the graph = velocity
The slope of a position-time graph gives the velocity of the object.
In this example the car moves in one direction, so its velocity and its speed coincide. If it turned back, its position would decrease and its velocity would be negative, while its speed (distance travelled per unit time) would stay positive.
1.3 Average Rate of Change
We now introduce the term "rate of change".
For a function $y = f(x)$, when $x$ changes from $a$ to $b$ ($a \neq b$), the average rate of change is
$$\text{average rate of change} = \dfrac{f(b) - f(a)}{b - a} = \dfrac{\Delta y}{\Delta x}$$Here $\Delta$ (delta) is the symbol for "change in".
A Worked Example
Example: For $f(x) = x^2$, find the average rate of change as $x$ changes from $1$ to $3$.
Step 1: Evaluate the function at each point.
$$f(1) = 1^2 = 1$$ $$f(3) = 3^2 = 9$$Step 2: Find the changes.
$$\Delta x = 3 - 1 = 2$$ $$\Delta y = f(3) - f(1) = 9 - 1 = 8$$Step 3: Compute the average rate of change.
$$\text{average rate of change} = \dfrac{\Delta y}{\Delta x} = \dfrac{8}{2} = 4$$Note: for readability, this figure and the one in Section 1.4 compress the vertical axis relative to the horizontal axis. The apparent steepness of a line therefore does not match its numerical slope.
What does this result mean?
While $x$ changes from $1$ to $3$, on average $y$ increases by $4$ for each increase of $1$ in $x$.
Caution: the average rate of change says nothing about what happens in between
It is tempting to think that "if the average rate of change is $0$, nothing changed on the interval". This is not true.
For $f(x) = x^2$, as $x$ changes from $-1$ to $1$ we have $f(-1) = 1$ and $f(1) = 1$, so the average rate of change is $\dfrac{1 - 1}{1 - (-1)} = 0$. Yet along the way $y$ falls from $1$ to $0$ and then rises back to $1$.
The three functions in the figure all take the value $1$ at $x = -1$ and at $x = 1$, so their average rates of change are all $0$, and the line joining the endpoints (dashed line) is the same for all three. In between, however, $y = x^2$ falls and then rises, $y = 2 - x^2$ rises and then falls, and $y = x^3 - x + 1$ rises, falls and rises again. The average rate of change alone cannot tell these apart.
The average rate of change reflects only the difference between the start and the end. To see how the quantity changes in between, we must split the interval into smaller pieces and look at each one.
1.4 The Rate of Change Is Not Constant
Look closely at the parabola $y = x^2$, however, and you will see that its slope (steepness) differs from place to place.
Let us compute the average rate of change over different intervals.
| Interval | Δx | f(start) | f(end) | Δy | Average rate of change |
|---|---|---|---|---|---|
| $[0, 1]$ | 1 | 0 | 1 | 1 | 1 |
| $[1, 2]$ | 1 | 1 | 4 | 3 | 3 |
| $[2, 3]$ | 1 | 4 | 9 | 5 | 5 |
| $[3, 4]$ | 1 | 9 | 16 | 7 | 7 |
What if we want the rate of change at a single point (the instantaneous rate of change) rather than an average?
This is the central question that leads to the derivative.
1.5 Towards the Instantaneous Rate of Change
Think of a car's speedometer. It shows the speed "right now, at this instant".
That is not an "average speed" but an "instantaneous speed".
Mathematically, we obtain the instantaneous rate of change by making the interval smaller and smaller without limit.
For example, consider the instantaneous rate of change of $f(x) = x^2$ at $x = 2$. In the table of Section 1.4, the intervals $[1, 2]$ and $[2, 3]$ on either side of $x = 2$ have average rates of change $3$ and $5$. Shrinking them to $[1.9, 2]$ and $[2, 2.1]$ gives $3.9$ and $4.1$, which close in on $4$.
In general, the instantaneous rate of change at $x = a$ is the value that the average rate of change $\dfrac{f(a+h) - f(a)}{h}$ approaches as $h$ approaches $0$. In symbols:
$$f'(a) = \lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$$This value is defined only when the limit on the right exists. The precise meaning of a limit, and examples where the limit does not exist, are covered in the following chapters.
This idea is the heart of limits and derivatives.
The idea of an instantaneous rate of change is not confined to mathematics. It underlies many fields that deal with change: velocity and acceleration in physics, marginal cost in economics, and the gradient that tells a machine learning model which way to adjust its parameters to reduce the loss.
Chapter Summary
- Change is everywhere in daily life
- The slope of a graph represents how fast something changes (its rate of change)
- Average rate of change $= \dfrac{\Delta y}{\Delta x} = \dfrac{f(b) - f(a)}{b - a}$
- On a curve, the rate of change differs from place to place
- The goal of differentiation is to find the instantaneous rate of change
Frequently Asked Questions
Q1. What is the difference between "rate of change" and "velocity"?
Velocity is a special case: the rate of change of position with respect to time. Rate of change is a more general idea that describes how much any quantity $y$ changes relative to another quantity $x$. The rate at which temperature changes over time and the rate of change of a share price over time are also examples of rates of change.
Q2. How do $\Delta x$ and $dx$ differ?
$\Delta x$ (delta x) denotes a finite change, and $dx$ does not mean the same thing. In elementary calculus, $\dfrac{dy}{dx}$ is Leibniz notation for the derivative $f'(x)$, understood as the limit of $\dfrac{\Delta y}{\Delta x}$ as $\Delta x \to 0$. There are also treatments that give $dy$ and $dx$ meanings of their own (as differentials), but those are more advanced topics.
Q3. Is the average rate of change always positive?
No. If the function is decreasing, it is negative. When $f(a) > f(b)$ with $a < b$, we have $\Delta y = f(b) - f(a) < 0$, so the average rate of change is negative. For example, during the hours from afternoon until early morning, when the temperature falls, the rate of change is negative. On an interval where the value does not change overall, it is $0$.
Practice Problems
Problem 1 (basic): For $f(x) = x^2$, find the average rate of change as $x$ changes from $2$ to $5$.
Hint
Compute $\dfrac{f(5)-f(2)}{5-2}$, using $f(2)=4$ and $f(5)=25$.
Problem 2 (standard): For $f(x) = 2x^2 - x$, find the average rate of change as $x$ changes from $0$ to $2$.
Hint
$f(0)=0$ and $f(2)=2\cdot4-2=6$, so the average rate of change is $\dfrac{6-0}{2-0} = 3$.
Problem 3 (advanced): For $f(x) = x^2$, express the average rate of change as $x$ changes from $a$ to $a+h$ in terms of $a$ and $h$.
Hint
Expanding $(a+h)^2 = a^2 + 2ah + h^2$ gives an average rate of change of $2a + h$ (for $h \neq 0$). What happens as $h \to 0$?