Geology: Calculus Video #3
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.
from the first two publications, we now know this:
tiny changes or nudges to a function are the core of a derivative.
yet, why do calculus students do some weird-ass abstract functions than something more real-world as practice?
because the weird-ass abstract (pure) functions are the ones used to model graphs and plots.
some of them include:
- polynomials
- trigonometric functions
- exponentials
one of grant’s previous examples included:
f(x) = x²
from the previous article, we now know the change to the function (aka derivative), will be:
f’(x) = 2x * dx
the “dx” is the extra nudge we added to the original function (the derivative).
and let’s say “x” was 9 , and “dx” was 0.002…
our answer would be 0.036
and the reason it works, not just because of finding the nudge, is also the power rule:
as u take the exponent down…
one power must be removed because it will be “used’ for the derivative.
confused?
this might help (with kudos to grant):
on the left, that is the polynomial.
on the right, focus on the numbers with red & black. it’s the only piece that matters.
the rest in gray can be discarded, because the change (derivative) to small to even notice.
don’t believe?
ok.
if “dx” was 0.001 meters, what will “(dx)²” be?
0.000001
one to the millionth of a meter.
now imagine if it was “(dx)⁵” lmao…
the original function was x².
now we “used” on of the “x”’s for the derivative (change in area):
x² → 2x * dx
x³ → 3x² * dx
x⁴ → 4x³ * dx
u get the idea.
plus, there must be one “dx” in the equation.
yes, that means no … dx + …. dx² …. + dx³. only “dx”
why?
it’s used to represent the nudge or change in the graph / figure of the original function.
now, as grant says, time to have some fun.
what is the derivative of: f(x) = 1/x
1/x means this: “‘x’ times what other number will give me 1?”
kinda like a rectangle problem.
the length of the rectangle is “x”.
the width of the rectangle is “1/x”.
now let’s say the length is 2, then the width has to be 1/2 to make it equal to 1.
2 * 1/2 = 1 OR 2 * 0.5 = 1
and now, what happens if we add some extra width to “1/x”?
what’s gunna happen to “x”?
well, if we nudge the length of the rectangle by “1/x + dx”, then…
the length of the rectangle “x”, must shrink by the same amount as the derivative “dx”.
the d(1/x) is that small negative value nudge, because it’s decreasing the height (length) of this rectangle, and adding more width.
and finally, using the multiplication rule of derivatives, we find out why it works, and not just using the plug-in power rule.
here’s the summary:
the multiplication rule (step one) can be distilled into a step zero:
Step Zero: (x)’ * (1/x) + (x) * (1/x)’ = 1
the “x” is our length, and because the length was reduced, we need to know its original length + the new length after the nudge was made.
then the width, we need to know its original width + the new width after the added width (nudge) was made.
the area was one, but when u take the derivative, constants go to zero.
now, u know where all the pieces come from.
CJ
© 2023–2100 by Carlos Manuel Jarquín Sánchez. All Rights Reserved.
