Geology: Calculus Video #1
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 Geology 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.
use the area of a circle to understand calculus by going deep into the geometry:
the geometry is about the area of a circle.
A = Area
π = pi, 3.14 (approximate numerical value)
r = radius (r² = the value of the radius, but squared)
if r = 4, then 4² = 16… then multiply 16 by 3.14 to get the area.
but why tho?
the circle has been used in the past to identify the fundamental pieces of what makes calculus, well, calculus…
- Derivatives
- Integrals
- Both of them are opposities, the yin & yang
let’s use that circle from above.
and we make it slightly bigger, just a bit more.
that tiny change we make to the size…
is going to need a name.
in the video, grant called it “dr”.
“dr” means: a derivative (change) in the radius size.
“dx” & “dy” can also be thought as this:
- “dx” : a derivative (change) in the value of “x”
- “dy” : a derivative (change) in the value of “y”
now let’s get a graph that doesn’t bend in uniform (equal on both sides).
if i asked you to find the area under the blue color, could you?
maybe not, it bends a lot.
but what you can do is use the red bar color and chop it up into smaller, potato-like thin pieces to make it exactly cover from the bottom to the top of the blue curve.
and yes, it gotta be smaller than the red color u see on the graph lol.
like this:
but can u see the yellow bar?
yup, very thin.
that change, or derivative (“dr”) is what u can use to find out the area under the curve in the blue color graph.
the smaller the thickness* of one sliver of area, the better the approximation of the area under the curve.
* that thickness depends on how small the change of area u added to the original graph, (the derivative, “dr”).
ok, but how tf did u understand it, carlos?
not all the time, will we have an area given to us.
so let’s do the example.
let’s give “x” the value 3.
and the change or added amount to the graph is 0.001 extra.
and for area, let’s use “A(x)” as the legend.
so the original value of “x” (“A(original-x)”) is 3.
and the final value of “x” after the added area (“A(final-x)”) is 3.001
to find out the graph’s “y” value (or f(x)), do the following:
- subtract the final “x” value from the original “x” value
- and then divide it by the added “x” value into the area
in practice, it looks like this:
(3.001 - 3) / 0.001 ≈ 3² (or nine)
[ A(final-x) - A(original-x) ] / extra change or nudge
3², or 9, is the “y” or “f(x)” value for that graph.
next, to find the change in area (dA) because of that 0.001 we added:
- multiply the “f(x)” value by the added change to the “x” value (or “dx”)
dA ≈ f(x) * dx
dA ≈ 9 * 0.001
this gives us 0.009 as our added change in area (or dA)
and to make sure that nine is our original area (before we added 0.001 to the graph), do this:
- divide the “dA” by “dx”
dA / dx ≈ f(x)
0.009 / 0.001 ≈ 9
and the approximation of our true area gets closer to the truth as the “dx” value (0.001) gets closer to zero,
(but never actually gets to zero, otherwise it’s undefined)
and u see what we did in that last equation?
dA / dx ≈ f(x)
this is what a derivative means:
it is the ratio (number) as “dx” gets smaller and smaller.
remember, “dx” is what we add (or remove) from the original area.
and the smaller and smaller we add (or remove) to discover the area, we can get closer to what the true original area is.
and as grant melodically said:
math has the tendency to reward u with no mental breakdowns when u respect it’s symmetry (or its order).
order breeds progress.
CJ
© 2023–2100 by Carlos Manuel Jarquín Sánchez. All Rights Reserved.
