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Antiderivatives

Undoing a derivative — and why +C matters · BC Calculus · greeting

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Owl · Picture a hillside covered with trails. Every trail has the same steepness at each horizontal position — the same slope field of rates — yet hikers starting at different elevations end on different paths. That is the heart of antidifferentiation. Given a derivative, many functions share it. They differ only by a vertical shift. Suppose a sensor reports that the height of a rising balloon changes at rate h′(t) = 4t meters per minute. What functions for height h(t) could produce that rate? Before we name formulas, guess one candidate and check it by differentiating. Then ask: is yours the only possible height function?

Larry’s gold answer — Show text to read it out loud.

Larry · gold answer · One guess: h(t) = 2t². Differentiating gives 4t, so it works. But h(t) = 2t² + 5 also differentiates to 4t.

Owl: Picture a hillside covered with trails. Every trail has the same steepness at each horizontal position — the same slope field of rates — yet hikers starting at different elevations end on different paths. That is the heart of antidifferentiation. Given a derivative, many functions share it. They differ only by a vertical shift. Suppose a sensor reports that the height of a rising balloon changes at rate h′(t) = 4t meters per minute. What functions for height h(t) could produce that rate? Before we name formulas, guess one candidate and check it by differentiating. Then ask: is yours the only possible height function?

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