FOUNDATIONS 08
Mathematical Change
Calculus studies how quantities change. Before derivatives, we need a precise language for finite changes, direction and rates of change.
The main idea
A value tells us where a quantity is. A change tells us how far and in which direction it moved.
\[\boxed{\text{change}=\text{final value}-\text{initial value}}\]This simple subtraction becomes the starting point for average rate of change and, later, the derivative.
1. The symbol \(\Delta\)
The Greek capital letter \(\Delta\), read “delta”, denotes a finite change.
If \(x\) changes from \(x_1\) to \(x_2\),
\[\Delta x=x_2-x_1.\]2. Positive and negative change
If the final value is larger, the change is positive. If it is smaller, the change is negative.
Thus the sign records direction.
\(8\to3:\quad\Delta x=3-8=-5\)
3. Change is different from the final value
If a temperature moves from \(12^\circ\) to \(17^\circ\), the final temperature is \(17^\circ\), but the change is only \(+5^\circ\).
This distinction is important throughout calculus.
final value: \(17\)
change: \(17-12=5\)
4. Comparing two changes gives a rate
If both \(x\) and \(y\) change,
\[\frac{\Delta y}{\Delta x}\]measures how much \(y\) changes for each unit change in \(x\).
5. Average rate of change of a function
For \(y=f(x)\), between \(x_1\) and \(x_2\),
\[\boxed{\frac{\Delta y}{\Delta x}=\frac{f(x_2)-f(x_1)}{x_2-x_1}}.\]This is the slope of the secant line joining the two points on the graph.
6. Worked example
For \(f(x)=x^2\), between \(x=1\) and \(x=3\),
\[\Delta x=3-1=2,\qquad\Delta y=9-1=8.\]So the average rate of change is
\[\frac{\Delta y}{\Delta x}=\frac82=4.\]\(f(3)=9\)
\(\Delta y=8\)
\(\Delta x=2\)
average rate \(=4\)
7. Units matter
A rate combines the units of two changing quantities.
If distance is measured in miles and time in hours, then
\[\frac{\Delta\text{distance}}{\Delta\text{time}}\]has units miles per hour.
time: hours
rate: miles/hour
8. Why average rate may hide what happens inside an interval
A car travelling 120 miles in 2 hours has average speed 60 mph, but it may have travelled at many different speeds during the journey.
The average rate compresses all internal variation into one number.
9. Making the interval smaller
To learn about the rate near one point, compare values over a smaller interval.
As \(\Delta x\) becomes smaller, the secant line follows the curve more closely near the chosen point.
smaller \(\Delta x\): more local average
\(\Delta x\to0\): instantaneous rate
10. From finite change to differentiation
\(\Delta x\) is a genuine finite change. Differentiation studies the limiting behaviour of the ratio
\[\frac{\Delta y}{\Delta x}\]as the interval shrinks toward zero:
\[\boxed{\frac{\Delta y}{\Delta x}\longrightarrow\frac{dy}{dx}}.\]The derivative is therefore built from the familiar idea of change, refined through a limit.
\(\frac{\Delta y}{\Delta x}\)
↓ shrink the interval
instantaneous change
\(\frac{dy}{dx}\)
\(\Delta\) denotes a finite change. Its sign records direction. Dividing one change by another creates a rate, and the average rate of change of a function is the slope of a secant line. Calculus begins when that interval is allowed to shrink toward zero.
Why this matters for calculus
The derivative is not a completely new idea introduced from nowhere. It grows directly from finite changes and average rates. Understanding \(\Delta x\), \(\Delta y\), units and secant slopes makes limits and derivatives much more intuitive.