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Average Value

The constant height matching signed area · BC Calculus · greeting

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Owl · A greenhouse thermometer logs temperature all afternoon. The graph wiggles — cool after lunch, warmer near 4 p.m. The gardener asks for one number: "What was the average temperature from 1 to 5?" They don't mean "add the endpoints and divide by two." They mean the constant height whose rectangle on [1, 5] has the same signed area as the actual temperature curve. That constant is the average value of the function. It equals (1/(b−a)) ∫ₐᵇ f. What constant temperature would produce the same "degree-hours" as a varying day on that interval?

Owl: A greenhouse thermometer logs temperature all afternoon. The graph wiggles — cool after lunch, warmer near 4 p.m. The gardener asks for one number: "What was the average temperature from 1 to 5?" They don't mean "add the endpoints and divide by two." They mean the constant height whose rectangle on [1, 5] has the same signed area as the actual temperature curve. That constant is the average value of the function. It equals (1/(b−a)) ∫ₐᵇ f. What constant temperature would produce the same "degree-hours" as a varying day on that interval?

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