Session #3

発表 / Presentation

#	Presenter	Topic	Language	Duration
1	YDF	Limits, Continuity, and Trigonometric Limits	自由	40min + 10min Q&A

Timeline

Time	Activity
21:00-21:05	Intro / 开场
21:05-21:45	Limits, Continuity, and Trigonometric Limits (YDF) — 40min
21:45-21:55	Q&A
21:55-22:00	Closing / 结尾

Links

• Google Meet: https://meet.google.com/nbj-ioxw-tda
• Meeting Notes (by Gemini): https://docs.google.com/document/d/1SuZ054eGJnkdOXxs1K14qdLYOtO1Ir-eto3YGxfk9c0/edit?usp=drivesdk

内容概要 / Summary

YDF — Limits, Continuity, and Trigonometric Limits (MIT OCW 18.01 Lecture 2)

• Derivative definition — geometrically as the limit of the difference quotient Δy/Δx as Δx→0; shifts from average to instantaneous rate of change
• Three physical applications — speed = ds/dt, electric current = dQ/dt, temperature gradient = dT/dt (temperature only, not entropy)
• GPS example — measurement error ΔH affects ground location L via ΔL/ΔH (sensitivity analysis)
• Easy vs hard limits — lim(x→3)(x²+x)/(x+1)=3 by plugging in; derivative limits are hard since Δx→0 zeroes the denominator
• Continuity definition — lim(x→x0) f(x) = f(x0)
• Four types of discontinuities — removable / jump / infinite (1/x) / oscillating (“ugly”)
• Two trigonometric limits — lim(sinθ/θ)=1 and lim((1−cosθ)/θ)=0, geometric proof with unit circle (θ in radians)
• Theorem: Differentiable ⇒ Continuous — elegant limit-manipulation proof
• Radians vs degrees — radian = arc/radius, π = 180°; all trig in radians from now on

Feedback

👍 Good

• Clear geometric explanations (difference quotient, unit circle proofs)
• Real-world motivation (GPS sensitivity) made the derivative concept tangible
• Nice exchange on “ugly discontinuity is beautiful”

💡 Improvements

• Visuals (diagrams) would help the unit-circle proofs be easier to follow live
• Some terms were hard to catch via auto-transcript; writing key formulas on the board beforehand could help

📝 Notes

• Radians/degrees discussion; group agreed to use radians for all trigonometry
• Q&A: temperature gradient vs entropy (Cheese leopard), meaning of radians (GG)

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