Lupen
@0xLupenn • 1,858 subscribers
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This man teaches at a community college in California. His salary: around $800,000 a year. The engineers who passed calculus because of him: $1,800,000 to start. He has more calculus students than Harvard, MIT, and Stanford combined. This is Professor Leonard's Calculus 2, Lecture 6.2. Free on YouTube. Professor Leonard has taught calculus on YouTube for over a decade. His channel has millions of subscribers across 150 countries. Every major university has students who watch him the night before their exam. Then the concept. An inverse function is a machine that undoes another machine. If a function takes 2 and gives you 8, the inverse takes 8 and gives you back 2. Finding an inverse means switching every x and y in the equation and solving for y again. The graph flips across the line y = x like a mirror. Then the problem. Sometimes it is easy to find the inverse. Sometimes it is impossible to write it explicitly. A function like 3π sin x + sin x cannot be solved for x with algebra. You have to think. What angle makes the whole thing equal to 1? You work backwards through the unit circle until the answer appears. Then the shortcut. If you want the derivative of an inverse at a point, you do not need the inverse itself. You only need the derivative of the original function. The formula: the derivative of the inverse at a point equals 1 divided by the derivative of the original function evaluated at the switched point. The inverse flips the coordinates, so you flip where you plug in. Watch the moment he shows why G prime of 8 equals 1/12 without ever writing the inverse function. Every engineering student memorizes the derivative rules. Professor Leonard's lecture is the one that shows why the inverse derivative formula is just those same rules run backwards. A software engineer at a semiconductor company in Austin said Professor Leonard's channel is the reason she passed Calculus 2 on her second attempt. She graduated, joined the company, and now makes $165,000 a year. Bookmark this and watch later - after this lecture every inverse problem on your exam will feel like a question you already answered.
Lupen650,822 次观看 • 1 天前
A Japanese mathematician published a result in 1944 that nobody understood for twenty years. Today it runs inside every options desk on Wall Street. Goldman pays $400K to quants who can derive it from scratch and explain why classical calculus gives the wrong answer without it. His name is Choongbum Lee. MIT, 18.S096, Topics in Mathematics with Applications in Finance. The course that Wall Street watches. This is lecture 17. It derives Ito's Lemma from scratch. He opens with the problem nobody in classical calculus can solve. Then the foundation. Brownian motion is the limit of a random walk taken to infinity. Each trade pushes a price up or down by a tiny amount. A million trades a day. The limit of that process is Brownian motion. Einstein proved this for pollen particles in 1905. The finance world borrowed the math fifty years later. Then three properties that make no sense until you see them derived. Brownian motion crosses zero infinitely often. It never escapes to infinity. And it is nowhere differentiable - with probability one, every path is continuous but has no slope at any point. That last property is why classical calculus breaks completely. Then quadratic variation. For any smooth function, chop an interval into n pieces, square the increments, sum them - the result goes to zero. For Brownian motion it goes to T. The increments are too wild to vanish. That single fact is why Ito's Lemma has a second term that classical calculus does not. Watch the moment he derives it. Taylor expansion applied to a function of Brownian motion. The first term is what you expect. The second term appears precisely because the squared increment does not vanish. Without it, options pricing gives wrong answers. With it, you have Black-Scholes. A quant I know sends this lecture to junior analysts who cannot explain why their pricing model drifts. Says it fixes in ninety minutes what two years of finance courses left open. Free on YouTube, MIT OpenCourseWare, 18.S096. bookmark this and watch later - the math behind every options desk on Wall Street fits on one blackboard, and this is the lecture that shows you why
Lupen65,255 次观看 • 5 天前
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