Key Theorems About (Learning) Calculus
Six calculus theorems rewritten as truths about learning calculus: FTC, IVT, ε-δ limits, the chain rule, the Mean Value Theorem and the Extreme Value Theorem.
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Cartoon by Ben Orlin, Math with Bad Drawings. https://cartoons.mathwithbaddrawings.com/2024-10-15-key-theorems-about/ (CC BY-NC 4.0)Transcript
[Post subtitle] "This generalizes, of course, to the Theorem of How You Learn Stokes' Theorem."
[1] The Fundamental Theorem of Learning Calculus: What You Know (End of Course) − What You Know (Start of Course) = ∫ Learning dt
[2] The Intermediate Understanding Theorem: If f(t₀) = "This is impossible!" and f(t₁) = "This is easy," then at some time t where t₀ < t < t₁, f(t) = "This is hard… but I think I'm getting it."
[3] Definition of a Cognitive Limit: For all ε > 0, there exists δ > 0 such that if how weird and non-intuitive ε-δ proofs seem to you < δ, then your sense of what it feels like for a human being to learn calculus < ε.
[4] The Chain Rule of Difficulties: Challenge of Learning Calculus That Involves Lots of Algebra = Challenge of Calculus × Challenge of Algebra
[5] The "Meh" Value Theorem: There will be a moment during the year when calculus is not going better than usual, or worse than usual, but just exactly as usual.
[6] Extreme Value Theorem: In a bounded calculus course, there is a part of calculus that's as cool as it gets (and another part that's as dull as it gets) (but in a course of unbounded coolness, all bets are off)