FTC Part 1 (Differentiation)

Real Analysis · Axiom Academy

LESSON FTC Part 1 (Differentiation) Accumulate area under a curve, then ask how fast that area grows — the answer is the height of the curve itself. In words: the derivative of the accumulation function equals the original function. As x moves right, F(x) is the running area swept out under f — and the rate at which that area grows is exactly f(x) . Increase x by a small amount . The extra accumulated area is approximately a thin rectangle: width , height f(x) . The new area added is about , so Dividing by and letting gives F'(x) = f(x) . To find F'(x) we form the difference quotient. The numerator F(x+h) - F(x) is precisely the area of the thin slice between x and x+h : As the slice thins and its average height approaches f(x) . Geometrically: the slope of F at x equals the height of f at x — watch the two readouts stay locked together. We now prove the theorem using the Mean Value Theorem for Integrals : the area over a slice equals the height at some interior point c times the width. 5. A Worked Example, and Why It Matters Take the very curve in these animations, , and accumulate from a = 0 . Integrating gives the accumulation function, and differentiating it hands back f : At the slope of F is , which is exactly the height of f there — the slope-equals-height law, in numbers. Integrate then differentiate and you recover the original function — the two undo each other. is itself an antiderivative of f — a built-in construction. This makes the evaluation formula possible.

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