Question You've estimated a derivative at one time. Now imagine a weather balloon's altitude graph for an entire afternoon. At each moment where the path is smooth, there's a vertical velocity — up positive, down negative. Collect those velocities into a second graph. That second graph is f': a function whose input is time and whose output is slope of altitude.
Consider
Interpret f' as the function of slopes of f Match qualitative graphs of f and f' Read a sign chart of f' for increase/decrease of f Avoid confusing zeros of f with zeros of f'
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Altitude f(t) rise → crest → fall Invent velocity f'(t) f' > 0 rising f' = 0 at crest? f' < 0 falling
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