Calculus 1 — 8.2: Computing Tangent Lines via Limits скачать в хорошем качестве

Calculus 1 — 8.2: Computing Tangent Lines via Limits

Learn how to calculate the exact slope of a tangent line using the limit definition of the derivative. This lesson walks through two complete worked examples — a polynomial (y = x²) and a rational function (y = 3/x) — demonstrating the distinct algebra strategies each function type requires. Key concepts covered: • The limit definition of slope: m = lim(h→0) [f(x₀+h) − f(x₀)] / h • Why direct substitution yields the indeterminate form 0/0 and what that means • Substituting (x₀ + h) as a single input — and the common mistake of adding h separately • Polynomial technique: expanding (1+h)², canceling terms, factoring out h • Rational function technique: finding a common denominator, simplifying a complex fraction, then canceling h • Writing the final tangent line equation using point-slope form y − y₁ = m(x − x₁) • Worked result: tangent to y = x² at (1,1) is y = 2x − 1 with slope 2 • Worked result: tangent to y = 3/x at (3,1) is y = −(1/3)x + 2 with slope −1/3 • A four-step checklist for finding any tangent line from the limit definition ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from    • Calculus 1 Lecture 1.5:  Slope of a Curve,...