Calculus 1 — 8.2: Computing Tangent Lines via Limits
Автор: Ludium
Загружено: 2026-03-11
Просмотров: 9
Описание:
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
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SOURCE MATERIALS
The source materials for this video are from • Calculus 1 Lecture 1.5: Slope of a Curve,...
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