Introducing Calculus: Can Change Occur at an Instant? AB & BC

Average rate of change, $\frac{f(b) - f(a)}{b - a}$, is a statement about an interval: it is the slope of the secant joining the endpoints. Setting $b = a$ collapses it to $\frac{0}{0}$, which is why a rate at a single instant seems impossible. Calculus resolves this by never evaluating at the instant, instead asking what the average rate approaches as the interval shrinks toward it. That limiting value is the instantaneous rate, and geometrically it is the slope of the tangent line the secants settle into.

Two things go wrong here. The first is rejecting the idea outright, insisting that nothing changes in an instant so the rate must be zero or undefined, which treats the limit as a dodge rather than a definition. The second is sloppiness in the approach itself: sampling too coarsely to see where the values are heading, coming in from one side only, or reading the value plotted at the point instead of the trend the nearby values establish.