AP® Calculus BC · Unit 1 · Topic 1.1
Introducing Calculus: Change at an Instant
Instantaneous speed is not distance over zero time. It is the value that average speeds approach as the time interval around an instant shrinks toward zero.
Average speed over an interval is easy: distance traveled divided by time elapsed. Instantaneous speed sounds impossible, because at one exact instant no time has passed and no distance has been covered either, so you seem stuck with 0/0. Calculus resolves this by refusing to plug in zero at all. Instead, it looks at average speeds over shorter and shorter intervals starting (or centered) at that instant, and watches what number those averages get closer and closer to. That approached number, not a 0/0 calculation, is defined to be the instantaneous speed.
Picture a car's position graphed against time. The average rate of change between two times is the slope of the secant line connecting those two points on the graph. Pick a second point closer and closer to the first, and the secant line rotates, settling into one fixed line: the tangent line at that single point. The slope of that tangent line is the instantaneous rate of change. Nothing about this process ever divides by zero; it studies a trend as the interval shrinks and gives a name to the value that trend is heading toward.
This shrinking-interval idea is the entire reason limits get introduced right alongside it. The next step formalizes 'gets closer and closer to' with precise limit language, so that 'instantaneous rate of change' can be written as a limit of average rates of change. That limit, once made precise, becomes the derivative, the engine for the rest of differential calculus.
Key terms
- average rate of change
- The change in a function's output divided by the change in its input over an interval.
- secant line
- A line through two points on a curve, whose slope equals the average rate of change between them.
- tangent line
- The line that a sequence of secant lines approaches as the second point slides toward the first.
- instantaneous rate of change
- The limiting value of average rates of change as the interval shrinks to a single point.
Watch it explained
- How it worksHow does shrinking the time interval turn an average speed into a speed at an instant?
- How it worksWalk me through how a whole chain of secant lines settles onto one tangent line.
- Worked exampleFind the average rate of change of f(x) = x2 on [2, 2.1] and then on [2, 2.01].
- Common mistakeWhy isn't instantaneous speed just distance divided by zero elapsed time?
Check yourself
End of topic.