Introduction of the hyperbolic trig functions cosh and sinh as functions parameterizing the unit hyperbola by (twice) the area of the hyperbolic sector. Derivation of derivatives of cosh and sinh and expressions in terms of the exponential function e^x.
**Correction** Johann Lambert proved the irrationality of pi. The transcendence of e was proved by Hermite.
This is episode 7 in my series Tricky Parts of Calculus, a series that covers the subtle and difficult parts of calculus that often get glossed over in a calculus class. See the whole playlist here: https://www.youtube.com/playlist?list=PLw3pvR_YJeRcMaubDZvkjayqDJT4Tx47A
Also check out my channel for general advice and opinions about math, as well as the Daniel Rubin Show podcast for interesting conversations about math and other topics: https://www.youtube.com/channel/UCYLOc-m8WuQaBHJ56lEw4VQ
Reference:
Janet Barnett, Enter, stage center: the early drama of the hyperbolic functions https://www.maa.org/sites/default/files/321922717729.pdf.bannered.pdf