Geology: Calculus Video #7
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.
u know how there’s formal and informal ways to dress, behave, & say certain words?
apparently, in math, it’s just the same.
the derivative is the informal way/definition (“wut up gang?” - informal)
and the formal way/definition…
that’s the limit. (“how are you today sir?” - formal)
the limit is like the derivative’s older brother:
- more eloquent
- better dressed & more transparent
“approaching something” but not arriving there exactly is what defines the word “limit”.
ok, see how we have a function: f(x) = x³
the df/dx ratio, as we know, is like the rise over run slope between the original point & the new, nudged point.
but a limit says:
yes… but what happens to the slope as the “dx” approaches zero?
what will the value become?
“dx” was that nudge we made on the x-axis…
“df” is the original function + the extra nudge minus the original function.
to write a limit formally, it will look something like this:
but y do we replace “dx” with “h”? (on the side with the words “lim”?
idk tbh… it was one of those trends back in the day.
but either way…
u see the “d”’s in “df/dx”?
the derivative expression “d” in “df” & “dx” is an abbreviation of the word “limit”… & its long ass formulas.
“h” is the same thing as “dx”…
… a nudge to the input of “f” in “df” of a very small number (i.e 0.0001)
using limits (instead of derivatives) allow us to avoid talking about infinitely small changes…
instead, we can talk about what happens as the size of some change to our variable approaches zero.
variables: “dx” or “h”
but why does the limit help?
how does it help thinking that what happens as it approaches zero?
think about it.
if a line is curved, and we go down to a miniscule level, the graph no longer looks like a curvy mess…
rather, the line looks… straight??
we zoomed in on (0,1) point.
sure enough, i like to think ppl made calculus so that curvy things can become (or look) straight…
and save us so many headaches.
(ε, δ) definitions - limits ➕
since when did the greek alphabet join the party?
always, but not it’s time to introduce them…
the (ε, δ) definitions of limits is:
when u analyze whether or not you’re restricted in how much the output range (“df”) shrinks as we narrow in on a target point (on the graph) and the input range shrinks (“dx”).
ye, that a mouthful.
so here, let’s use this limit & convert it into a graph.
and because the “dx” or “h” gets closer to zero, we’ll plug in “h” with “0”, and see what happens on the graph in real time.
huh, there’s a hole in the graph, when “x” is equal to zero.
but what is the “y” value?
well, let’s approach the hole, both on the x-axis, and then the y-axis, (since the y-axis has the line, & the y-coordinate of the graph.
so we can replace “h” with a value that is close to zero, but not exactly zero.
notice how the values are getting closer to twelve as the “y-value”?
so that means as the limit approaches the “x-value” zero (which is the hole), the y-value is going to be twelve.
but it doesn’t work everywhere, where there’s a hole.
take this example.
as you approach h=0 from the right, the function approaches 2.
but as you come at h=0 from the left, it approaches 1.
that not good.
we don’t have a singular point.
there’s no clear point to connect it as “h” approaches zero.
so the function cannot exist, because we cannot have inexactness in a limit.
but y, u ask?
as you shrink the input range “x-value”, the corresponding outputs don’t narrow in on any specific value.
instead those outputs “y-values” straddle a range that never shrinks smaller than 1, no matter how small your input range.
another way to view it is: because there’s no “graph line”, it can’t shrink more.
the green is the “x-values”, the blue is the “y-values”.
shrinking the input range “x-values” around the hole, or specific point, and seeing if we can narrow the output range “y-values” around the hole, is what “(ε, δ) definitions of limits” are.
when a limit exists: we can make the output range as small as i want.
when the limit does not exist, the output range cannot get smaller than some value…
… no matter how much i shrink the input range around the limiting input.
ε (epsilon) is used to narrow in on the values from top to bottom (or “y-axis”).
and the epsilon number means how far away we are from the limiting number/point.
δ (delta) is used to narrow in on the values from left to right (or “x-axis”).
and the delta number means how far away we are from the limiting number/point.
and that small rectangle box in the middle, is where we can find points that get us closer & closer to the hole…
because there is also the graph line that keeps us on the right track.
when the graph leads us to a point from both sides (from the left & from the right), we got a limit.
but when it breaks into two pieces, then we have too many output possibilities.
that’s epsilon-delta: narrow in on inputs & outputs around a hole in a graph.
l’hôpital’s rule 📐
this rule can be summarized like this:
if u ever come across a limit that will equal 0/0 (zero divided by zero) or a denominator equal to zero…
take the derivative of the numerator & the denominator of that limit.
u will get an answer, and plug in the number that corresponds to “h”
try this one:
if u plug in zero for “x”, that surely will give u 0/0, right…
now, the trick is, take the derivative of sin(x) & x…
if we do that, we get:
- cos(x) *1 for the numerator
- 1 for the denominator
now let’s plug in the number 0 inside of “cos(x)”
the cosine of zero “cos(0)”, is equal to one.
one divided by one is … one.
see? wee beat the system.
ok, one more, to prove it wasn’t a fluke.
this answer is if we decide to take a number that is close to one, like 1.000001
but because this equation will give us 0 in the denominator.
so how can we get by this?
take the derivative of sin(πx) & (x² — 1).
the answers are:
- cos(πx) * π ||| (numerator)
- 2x ||| denominator
now, we will plug in 1 wherever there’s an “x”.
we will get:
- cos(1*π) * π
cosine of π [cos(π)] is equal to -1… multiply by π and we get… -π.
2 by 1 is two.
divide it, we get… -π/2.
-π/2 is equal to -1.57079632679…
that’s creativity at it’s finest.
when there seems no way out, creativity finds a way to break through.
and calculus, or math in general, is a fairly popular way to excercise that muscle.
© 2023–2100 by Carlos Manuel Jarquín Sánchez. All Rights Reserved.