Math: Calculus Video #10

3Blue1Brown Intro To Calculus Series Summary

Preface ✨

Hello everyone! My intentions for writing these articles are:

• Explain technical knowledge about geology in simple terms (to the public)
• Document my journey to getting an AD-T Math degree in one year
• Store information and habits for my future self and others (in <7 minutes)

Coolio? Sweet. Enjoy the series :-)

notes as i watched the video 📹

p.s. — the video will be linked at the bottom of the article. cheers.

in life, problems do not have one root cause, or one layer.

it’s occasional that these demons are convoluted into a labyrinth of issues that stand impossible to eradicate.

math is no exception.

sometimes, we must take other derivatives, especially when we encounter constriants… like physics.

physics is math with one constraint: reality.

and when reality hits, you may need to dig deep into the layers.

given a function labeled “f(x)”,

the derivative (df/dx) can be interpreted as:

the slope of its graph above a specific point on the function “f(x)”.

but what happens when we need to know how quick the slope “accelerates & deaccelerates”?

use the 2nd derivative.

the 2nd derivative tells us how the slope* is changing (in value) along the graph.

* [slope is synonymous with “the 1st derivative”]

the purple is where the 2nd derivative exists if it’s positive value

when the curve of “f(x)” looks like a bucket of water (the purple line, aka concave up), the value of the 2nd derivative is positive.

why?

the yellow line is the slope (or the first derivative).

if the line begins to shift from pointing down to pointing up, our slope is increasing in value.

but if the slope looks like it’s pointing down (far right end of the yellow stick), our slope is decreasing.

see? the far right end of the stick is pointing down.

and as long as the slope (yellow line) stays on the orange part of the curve,

the slope will begin its transition from positive to negative slope value.

and the steeper the curvature for the “f(x)” graph, (whether positive or negative)…

the value of the 2nd derivative will be higher.

this a slope steep. the slope value will increase dramatically, as it transitions from negative to positive.

this slope not steep. but still is transitioning from negative to positive values, so the slope still increases.

this slope don’t transition from negative to positive values. or vice versa. so the 2nd derivative is zero.

and the mathematicians have used these symbols to describe the 2nd derivative:

d²f/dx²

but wtf does that supposed to mean?

why not this?

f.u.²/fx²

explanation ✏️

example time.

let’s say u have an input function of “f(x)”…

and let’s nudge the graph two steps to the right.

these nudges need names, both of them with “dx” will be confusing lol

so!

the first nudge will be “df1”, and the second one will be “df2”.

but how do we know how much the graph changed because of the two extra nudges?

answer:

• put them side-to-side.
• the difference between the 2nd nudge (bigger one) minus the 1st nudge (smaller one) is an approximation…
• …of the 2nd derivative.

in grant’s video, he labeled it as “d(df)”,

which i like to think of it as:

[the difference]* of the [difference in the original function]**

* 1st derivative

** 2nd derivative

but “d(df)” isn’t “d²f”…

so what does “d(df)” mean?

well, it means this…

d(df) ≈ (a constant) * (dx)²

and the value of the 2nd derivative must be proportional to the first derivative (dx)…

here’s what this means.

the word “proportional” can also approximately mean “equivalent”.

if “dx” had a value of 0.01, then (dx)² would need to be a value that’s proportional to the “dx” value (0.01)…

btw, if we plug 0.01 into (dx)²… we get:

before: (0.01)² ||| after: 0.0001

dx² is similar to dx, just 100 times smaller.

AND THEREFORE…

the 2nd derivative is the size of the change between the two steps,

then divided by the two nudges of “dx” … which “morphs” into (dx)².

it’s the equation up above, btw.

d(df) / (dx)² ≈ (a constant)

and because d(df) is the two changes of the function, we can label it as:

df² / dx² ≈ (a constant)

but this “df² / dx²” ratio in english means this:

what happens as as the “dx’s” approach zero? (aka the nudges never happened)?

and this is when physics & its restraints on the real world can play an application.

to visualize the 2nd derivative in real life, picture a car.

it will travel at a certain speed, and speed is always positive.

but if u wanna know what’ going on at every point in time, we need to narrow in on those points…

and to narrow in on that point… use the derivative.

the 1st derivative for the speed is velocity.

the 1st derivative tells us the change of speed at a specific point in time.

velocity can have a negative value (slowing down), or positive value (speeding up).

and the 2nd derivative for the speed (and the derivative for velocity) is acceleration.

the 2nd derivative describes the rate of change for the velocity at a specific point in time.

ahh, the wonder of the real world…

physics is applied mathematics, with reality sinking in…

and mathematics is the wonder servant to explain it all.

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