Calculus 1 — 8.4: Instantaneous Velocity скачать в хорошем качестве

Calculus 1 — 8.4: Instantaneous Velocity

How can a speedometer show 65 mph at a single instant when zero time means zero distance — and zero divided by zero is undefined? This video resolves that paradox by building the concept of the derivative from scratch, using the falling-rock problem s(t) = 500 − 16t² to show how shrinking secant lines converge to a tangent line through the power of limits. Key concepts covered: • The zero-over-zero paradox of measuring speed at a single instant • Average velocity as the slope of a secant line between two points on a position curve • How secant lines rotate toward a tangent line as the time interval shrinks • The formal limit definition: v(t₀) = lim as h→0 of [s(t₀+h) − s(t₀)] / h • Full worked example computing instantaneous velocity at t = 5 for s(t) = 500 − 16t² • Deriving the general velocity formula v(t) = −32t by keeping t as a variable • The distinction between velocity (directional, can be negative) and speed (magnitude, always positive) • Why more negative velocity means falling faster, not slowing down • The unifying insight: tangent-line slope, instantaneous velocity, and instantaneous rate of change are all the same limit operation — the derivative ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from    • Calculus 1 Lecture 1.5:  Slope of a Curve,...