Fundamental Theorem of Calculus

Single-variable calculus from first principles

This is the theorem that ties the entire course together. Derivatives and integrals, slopes and areas, look like two separate worlds. The Fundamental Theorem of Calculus (FTC) shows they are exact inverses of each other. Differentiating undoes integrating, and vice versa.

Define an area function A(x) = ∫ₐˣ f(t) dt, the running area under f from a fixed start up to x. Part 1 says: the rate at which that area grows is exactly the height of the curve at the right edge:

Intuitively: when you nudge the right edge a tiny bit, the new sliver of area you add is (height)×(tiny width) = f(x)·dx. So area accumulates at rate f(x). The figure shows the area filling and its growth rate tracking the curve's height.

Where this lives in MLThe FTC is why we can move freely between densities and cumulative probabilities. A probability density function (PDF) is the derivative of a cumulative distribution function (CDF), and the CDF is the integral of the PDF: that's Part 1 and Part 2 at work. Computing P(a ≤ X ≤ b) = CDF(b) − CDF(a) is literally FTC Part 2. Every time a model converts a density to a probability, it's using this…
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