textbook-theory · English
Difference Equations to Differential Equations — Section 1.1: Calculus: Areas And Tangents
Section 1.1 of Dan Sloughter's open calculus textbook 'Difference Equations to Differential Equations', introducing the two fundamental problems of calculus — computing areas and finding tangent lines — through the method of sequences and limits. The section uses inscribed regular polygons to approximate the area of a unit circle (arriving at pi) and secant-line slopes to approximate the tangent to the parabola y = x² at (1, 1), demonstrating that both problems reduce to evaluating limits of sequences. Eight problem sets reinforce the concepts with additional curves, polygon side counts, and numerical experiments.
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