Geology: Calculus Video #5
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.
in the real world, a lot of things can multiply into the masses.
like the flu, bee population, or air particles in a specific radius.
and they’re not constant.
they can go exponential.
like 2^x, 3^x, 7^x, u name it.
let’s use 2^x as the exponential function.
and then let’s say we’re looking how many water molecules will form if we add oxygen & hydrogen gas into a glas tube for 15 seconds.
x = 1 second
ik it says “t” instead of “x”, but imma stick with “x”.
and yes, if we replace “x” with 1, we get 2 molecules ||| (2^x → 2¹ = 2).
if we do 5, we get 32 molecules ||| (2^x → 2⁵ = 32).
BUT!
what happens when we change something by a miniscule amount in an exponential function?
what happens now?
the equation would look like this:
we nudge it by a new amount, and subtract it from the original exponential function, and then divide it by the value (number) of the nudge.
and the closer the nudge was to zero (aka nothing happened), we’ll find out how many water molecules formed during that time.
the derivative (change) of 2^x is whatever the picture above, approaches as the “dt” reaches zero.
ok, but how do we go from this to finding out what 2^x really is?
like this:
and then this:
but now, let’s add a value for “dt”.
grant got me & u covered hahaha
“0.693147…" is what he means by “this value” btw
so to find out what the derivative of any expoential function is, use this formula:
p.s. - just remove the “2” and use “x” and plug in whatever number u’re using to solve ur current math problem…
so apparently, u need a constant to multiply it with 2^t to find out its derivative (or nudge).
but is there a place where we don’t need to worry about that?
and YES THERE IS!
the letter “e”!
“e” in math is Euler’s number, or approximately 2.71828
let’s plug in “e” instead for “2” on this problem, everything else stays the same.
so it would be [(e^.00001 -1) / 0.00001]…
and we get 1 as our answer!
the constant is 1, damn!
it got rid of all the heavy load of trying to find a six-decimal number!
that means no matter what number we have in the exponent, we can pull it down and have it muliplied by one, and still have e^t intact.
the change (dt) is equal to one.
replace everything u see with “2” with “e”, and let’s say u plug in “0.0000001” for “dt”, u see how it will approach one.
that is the beauty of euler’s number.
the nudge (dt) is equal to one.
this is why professors in math class will teach me & u how to convert things like 2^t to e^t & vice versa.
example, if we’re trying to go from (e^t) to (2^t), what must be done to get there?
well, we need the constant, or the “dt” of (2^t).
the value of that “dt’ was 0.69314…
and that constant (c) must be added to the e^t to get to 2^t.
and the final answer will look like this:
and where anything exists, there’s an inverse.
good & evil.
gas & trash.
winners & losers.
exponential functions & logarithms.
exponential functions mean this in english: “‘e’ to the power of ‘0.69314718056…’ gives me what?”
logarithms in english mean: “‘e’ to the what power gives me two?”
if u do it in a calculator, u find out that the constant is 0.69314718056 lmao
instead of writing all that out, we can just use “ln 2”, which means the natural logarithm, only use that when “e” is the base.
one more example.
what is the derivative of the function 3^t?
well, i do not wanna do a long-ass equation, so imma use the exponential function.
3^t becomes this:
and when we find the derivative, we have the unfair advantage of the constant of “e” equals one.
so nothing will change in terms of the base.
the exponent, will be dropped down “ln(3)t”, the “t” will be eliminated due to the finding of the change in value, so all we’re left is with “ln(3)”.
and ln3 is 1.09861228867, or 1.098 or short, it’s on the logarithm graph too, no concidence there.
and this is the general formula anyone can use to find the derivative when going from exponent to logarithm answer in a jiffy:
when u go with the smooth, u’re in for a calm ride.
so use exponentials “e”, than the constant, plain, distateful numbers like 2, 3, 4, 9.
math gunna luv it when u go with the flow lol
CJ
© 2023–2100 by Carlos Manuel Jarquín Sánchez. All Rights Reserved.
