Introducing Calculus: Can Change Occur at an Instant?

Calculus uses limits to define instantaneous rates of change as the limit of average rates of change over increasingly small intervals.

Change10–12% of exam

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Context

What this topic is and why it exists

Calculus starts with the idea of instantaneous change, which seems counterintuitive because change typically implies a duration.

The key is limits.

You're used to average rates of change: ∆y/∆x.

But what happens when ∆x approaches zero?

The average rate becomes undefined, yet that's where limits step in.

By considering what the average rate approaches as ∆x shrinks, you define an instantaneous rate of change.

This is the derivative.

The cognitive trap here is thinking of rates as needing a time span.

Instead, focus on what happens as the interval vanishes.

Limits let you make sense of a change at a precise moment, even if it seems like a paradox.

Recognize the difference between evaluating a function at a point and determining what it approaches.

Misunderstanding this distinction means missing the essence of calculus.

Continuity is another layer: when a function's limit and value match at a point, it's continuous.

Be precise with this; every major theorem in calculus depends on it.

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