0m_ethoc

Math student at Harvard. Yes, I'll explain it again.

[0m_ethoc @0m_ethoc]

@WealthLens_ Right? Jerison only mentions it in passing, but the idea is real: if prices can jump, it matters which side you're standing on. Left is the past, right is the future.

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People pay tens of thousands of dollars a year for lectures like this.

MIT put them on YouTube for free. Here's what one lecture teaches 🧵

The setup, simplified: an 80-meter drop, no air resistance.

Height over time: h = 80 − 5t²

At t = 4 seconds, 5 × 16 = 80. The pumpkin is on the ground.

Average speed is easy:

80 meters ÷ 4 seconds = 20 m/s.

But nobody watching cares about the average. They care how fast it's moving at the instant of impact.

That instant speed is the derivative.

dh/dt = −10t

At t = 4 that's −40 m/s (the minus just means downward). About 90 mph. Twice the average.

Now throw one off the tallest building on campus.

y = 400 − 16t² (feet)

Its derivative is −32t. At t = 5 seconds, that's −160 ft/s, over 100 mph.

Same idea, bigger drop.

Here's the catch.

A derivative is a fraction: change in height ÷ change in time. Shrink the time gap to zero and you get 0/0.

Plug in the point and you get nonsense. That's why calculus needs limits.

Some limits are easy. Just plug in the number.

(x² + x)/(x + 1) as x → 3 gives 12/4 = 3. Done.

A derivative limit is never that easy. The denominator Δx = 0 isn't allowed, so you need some cancellation first.

Limits come from two sides: the right and the left.

They don't have to agree.

That matters for the definition of continuity. A function is continuous at a point when the limit as x → x₀ equals f(x₀).

No jumps. No holes.

The professor then tours the "zoo" of discontinuities:

Removable. Jump. Infinite. And the ugly ones.

Each one breaks continuity in a different way.

Jump: the left and right limits both exist, but they're not equal.

Infinite: take 1/x. From the right it shoots to +∞. From the left it plunges to −∞.

He points out that +∞ is infinite money and −∞ is infinite debt. They're not the same.

Then there are the ugly ones.

A function like sin(1/x) oscillates wildly as x → 0. It doesn't even head toward ±∞.

In that case we say the limit does not exist.

Why does left vs. right matter in real life?

The professor mentions that Bob Merton, who won a Nobel Prize in economics, studied whether stock prices are continuous from the left or from the right in his models.

Left is the past. Right is the future.

A surprising fact about derivatives:

The graph of the derivative doesn't have to look like the original function.

1/x has two branches. Its derivative, −1/x², is negative everywhere.

And an odd function always has an even derivative.

Now the two trig limits, the ones every derivative of sine and cosine depends on.

sin θ / θ → 1 as θ → 0
(1 − cos θ) / θ → 0 as θ → 0
Big rule: θ must be in radians, NOT degrees.

Why is sin θ / θ → 1? A picture proves it.

Draw a circle of radius 1. The arc for angle θ has length exactly θ.

As θ shrinks, the arc and the straight vertical piece, sin θ, become nearly the same length. Their ratio approaches 1.

And (1 − cos θ) / θ → 0?

Same picture. The horizontal gap between the triangle edge and the circle is 1 − cos θ.

As θ shrinks, that gap gets much smaller than the arc length θ. So the ratio goes to 0.

One last theorem, proven in one line:

If a function is differentiable at a point, it's continuous there.

Write f(x) − f(x₀) as [f(x) − f(x₀)] / (x − x₀) times (x − x₀). The first factor tends to f′(x₀). The second tends to 0. Product: 0.

But isn't multiplying and dividing by (x − x₀) illegal? It looks like dividing by 0.

No. A limit never uses x = x₀. It only gets close.

Small, but never zero. That's the trick behind all of calculus.

Calculus isn't about memorizing rules.

It's about asking what happens as you get closer and closer, and knowing exactly where things break.

Source: MIT OpenCourseWare 18.01, lecture 2 by Prof. David Jerison (Creative Commons).

Follow for more math explained through real things. 🔁

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@_GlobalBrief_ Yes. Newton and Leibniz leaned on that 0/0 idea before anyone could justify it, and it took until the 1800s to make limits rigorous. The "problem" was the discovery.

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Down 3-0 at halftime. Still won the trophy.
May 25, 2005. Champions League final, Istanbul. AC Milan destroy Liverpool in the first half — Maldini scores in 50 seconds, Crespo doubles it, then makes it 3-0 before the break. Game over, right?
Six minutes into the second half, everything changes. Gerrard heads one back. Smicer strikes from distance. Then Xabi Alonso scores a rebounded penalty. 3-3. In six minutes.
Extra time solves nothing. It all comes down to penalties — and Liverpool's keeper Jerzy Dudek becomes the hero, saving the decisive spot-kick. Liverpool lift the trophy after being dead and buried at halftime.
Still called the greatest comeback in football history. No team has come back from 3-0 down in a CL final since — or before.
Ever seen a collapse this big get reversed? Drop the match below.

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@premierleague

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@nicksyshottie X100 to fast