References: Why area equals slope

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Calculus, Chapter 12: "What does area have to do with slope?"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/area-and-slope
  Series index: https://www.3blue1brown.com/?topic=calculus
  License: copyright Grant Sanderson; videos published on his site and YouTube
Clawdemy's lessons are original prose that follows the pedagogical arc of this
series. We do not reproduce or transcribe the videos; we cite them as the
recommended companion. All rights to the original videos remain with the creator.

Watch this next

• What does area have to do with slope? (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. Watching the accumulated area grow by one thin sliver as the right edge slides, and seeing that the sliver’s height is the curve, is the single image that makes the fundamental theorem feel inevitable rather than memorized. About thirteen minutes.

Going deeper

• Essence of Calculus (full series) by 3Blue1Brown. The series this track follows. The previous chapter stated the fundamental theorem and showed how to use it; this one explains why it holds. The next chapter (Higher order derivatives) returns to differentiation, taking derivatives of derivatives.

• Khan Academy: Calculus for a slower, exercise-driven treatment of the fundamental theorem and the area-function argument, with practice problems and immediate feedback.

Adjacent topics

Where this sits in the track and the wider curriculum.

• The essence of calculus (the first lesson) and integration (the previous lesson). This lesson is the bookend to the track’s opening. The first lesson observed, on a circle, that the rate of the accumulated area is the circumference being accumulated; the integration lesson stated the fundamental theorem as a tool; this lesson proves it geometrically, with A'(x) = f(x) for any curve.

• Probability densities and cumulative distributions (across the AI tracks). The relationship F'(x) = f(x) between a cumulative distribution function and its probability density is exactly this lesson’s A'(x) = f(x). Every running-total-versus-its-rate pairing in machine learning, including reading a cumulative loss against the current loss, is the fundamental theorem at work.