Calculus 1 — 16.1: Related Rates: Core Idea скачать в хорошем качестве

Calculus 1 — 16.1: Related Rates: Core Idea

How fast is the volume of a draining cone changing at a specific moment? You already know the volume formula — the key step is differentiating it with respect to time. This video walks through the core idea behind every related rates problem: implicit differentiation with respect to t, the chain rule generating rate terms, and why you must always differentiate before substituting values. Key concepts covered: • Building the cone volume formula from circle area to cylinder to cone: V = (1/3)πr²h • Why V, r, and h are all implicit functions of time during drainage • Implicit differentiation with respect to t using the chain rule and product rule • How the chain rule adds dr/dt and dh/dt factors (comparing d/dr vs. d/dt) • A simple worked example: differentiating y = x³ with respect to time • The most common mistake — substituting values before differentiating (and why it gives zero instead of the correct answer) • Full numerical cone calculation yielding dV/dt = −13π ≈ −40.84 cm³/s • The three-step framework for every related rates problem: Identify → Differentiate → Substitute ───────────────────────────── ORIGINAL SOURCE ───────────────────────────── This video is based on content from:    • Calculus 1 Lecture 2.8:  Related Rates   All credit to the original creator for the source material.