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 intervals.

Change10–12% of exam

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Context

What this topic is and why it exists

Calculus begins with a radical idea: change can be measured at a single instant.

The mechanism that makes this possible is the limit.

Limits let you determine what value a function approaches as the input nears a specific point, even when you can't directly plug in that point.

When the independent variable's change is zero, the average rate of change becomes undefined.

Limits bridge this gap by conceptualizing the instantaneous rate of change as the limit of average rates over shrinking intervals.

This is not intuitive because you're used to thinking of change as something that happens over time or space, not at a single point.

The trap is assuming that if a function isn't defined at a point, you can't say anything about it there.

Limits show that you can.

Misunderstanding this can lead to errors in reasoning about derivatives and integrals.

Mastering limits early sets the foundation for everything else in calculus.

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