Calculus 1 — 18.4: Combining First and Second Derivative Tests скачать в хорошем качестве

Calculus 1 — 18.4: Combining First and Second Derivative Tests

Your curve sketches have the right peaks and valleys but look like stick figures? Learn how combining the first and second derivative tests on a single number line gives you everything you need to sketch smooth, accurate curves — no calculator required. This video walks through a complete curve-sketching example for f(x) = x³ − 3x² − 24x + 28, showing how to merge increasing/decreasing information with concavity data to produce four distinct curve shapes. You'll build a sketch segment by segment, left to right, and see it match the computer-generated graph perfectly. Key concepts covered: • First derivative test review: sign of f'(x) determines increasing vs. decreasing intervals and locates relative extrema • Second derivative test review: sign of f''(x) determines concavity (concave up vs. concave down) and locates inflection points • Merging both derivative sign charts into a single combined number line • The four shape combinations: rising arch, rising scoop, falling arch, falling scoop • Why inflection points do NOT require f'(x) = 0 — the function can be increasing or decreasing through an inflection point • Step-by-step progressive curve sketching using only three strategic points and derivative signs • Practice challenge: narrating a new combined number line left to right ORIGINAL SOURCE This video is based on standard topics in single-variable calculus covering curve sketching with first and second derivative analysis.