AP Calculus AB · Unit 1: Limits & Continuity

1.1 Can Change Occur at an Instant?

Average vs. instantaneous

Calculus is the study of change. The average rate of change of f over [a, b] is (f(b) − f(a))/(b − a) — the slope of a secant line. The instantaneous rate of change is the rate at a single point — the slope of the tangent line.

Why limits

You can't compute a rate 'at an instant' with ordinary division (that would be 0/0). Calculus makes it precise using limits: shrink the interval toward zero. The two big ideas built this way are the derivative (instantaneous rate) and the integral (accumulation).

Key terms

Average rate of change: (f(b) − f(a))/(b − a); the slope of a secant line.
Instantaneous rate of change: The rate at a single point; the slope of the tangent line.
Tangent line: A line touching the curve at one point; its slope is the instantaneous rate.
Secant line: A line through two points of a curve; its slope is the average rate of change.
Why limits are needed: A rate 'at an instant' is 0/0 by division, so calculus defines it with a limit.

Try an AP-style question

A particle's position is s(t) = t² meters. What is its average velocity from t = 1 to t = 3 seconds?

8 m/s
3 m/s
6 m/s
4 m/s

Show the answer

D. Average velocity = [s(3) − s(1)] / (3 − 1) = (9 − 1)/2 = 4 m/s. (8 is the change in position; 6 is the instantaneous velocity at t = 3.)

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