Change, one small step at a time

Average change is a ratio

For the function $f(x)=x^2$, moving from $x=1$ to $x=2$ changes the output from 1 to 4. The average change per input unit is $(4-1)/(2-1)=3$. On the graph, it is the slope of the line connecting the two points: a secant line.

What if we want the rate at the starting point itself? Bring the second point closer.

Let a step shrink

Starting at any $x$ and taking a nonzero step $h$ gives

As $h$ approaches zero, the slope approaches $2x$. We define

We do not substitute $h=0$ into the fraction. We examine a limit of values with nonzero $h$. At $x=1$, the tangent slope is 2. For a small change $\Delta x$, the output change is approximately $2\Delta x$.

The approximation is local. At $x=1$, a step of $0.1$ predicts a change of $0.2$; the actual change is $1.1^2-1=0.21$. The missing $0.01$ is the quadratic term.

Several inputs, one at a time

For $f(x,y)=x^2+3y$, a partial derivative changes one coordinate while holding the others fixed:

At $(1,2)$, the first rate is 2 and the second is 3. Collecting these rates produces the gradient $\nabla f=(2x,3)^\top$. Module 5 will develop the geometry and formal rules; here the important idea is “one input direction at a time.”

Not every function has a derivative everywhere. The graph of $|x|$ has a corner at zero: slopes from the left approach $-1$, while slopes from the right approach 1.

Add small contributions

Switch the experiment to Area. Divide $[0,2]$ into intervals of width $\Delta x=2/n$. Each rectangle uses the function value at the midpoint of its interval. Add the rectangle areas:

For $f(x)=x^2$, as the rectangles become narrower, this approaches

The integral is a limit of accumulated contributions. For a nonnegative function, it gives the area under the curve. More generally it is signed accumulation: portions below the axis contribute negatively. Integrating a speed over time gives distance; integrating a probability density over an interval gives probability.