Taylor series approximations work because the derivatives of the taylor series expansion and the function it is approximating are the same (and the values at the given point). Likewise, the derivative of e^x is e^x because the taylor expansion of e^x is 1 + x/1! + x^2/2! + x^3/3!.. which becomes 0 + 1 + x/1! + x^2/2! + x^3/3!.. when the derivative is taken. So the derivative of the taylor expansion of e^x is itself.
Differential equations. Had the worst math teacher ever for LA and DE. I memorized enough to pass the class but did not really understand diff eq. Next semester, we used diff eq in my physics classes with a great professor who explained them beautifully. What form do you think the solution takes? Using an algebraic formula for that form, solve for the unknowns. Tada! If there is a solution, you’ve just solved a differential equation!
It’s = it is
Could of could have should of should have
This was explained to me here on Lemmy last month.
Trigonometry. My high school math teacher was a literal math genius and would always go deep into proofs and theory, sometimes not even getting to our homework stuff until the last 5 minutes of our 50 minute class. As a result I went from the “gifted” math group to nearly failing.
When I went to college I had to take a math placement test and ended up in Math 99 (below college level math).
It was there I was finally taught SohCahToa and everything clicked. I actually use simple trig a lot in my job now.
I was very bright when I was young, but unless I was given practical application of knowledge, it just leaked out of my ear.
I was exactly the same with trigonometry, I couldn’t understand it or why were even learning it.
As soon as I started to get into programming and I wanted to have a gun with a bullet that had a certain speed, and it was going at a certain angle, and I needed to break down the horizontal and vertical components of the motion, all of a sudden it felt like I had invented trigonometry myself.
I found that true of so many different things especially with math. No matter how much it was explain to me theoretically, it never made sense until I had a practical application and then it was just obvious to me.
I wish more of my education was that way instead of just learning theory.
Exponentiation. I don’t think it was ever really explained before, instead it was treated like something I should’ve known.
One day I watched a YouTube video that made the world of difference, then I got it.
The number of math epiphanies I’ve had on youtube is way too high. Good math teachers are a rare breed.
Sometimes a specific explanation works for you, but school has to be generic enough for explanation to work for most. There are probably a million videos with explanations that are utter shit.
There are, unfortunately, also millions of teachers who are utter shit. I appreciate every single one of the good ones I’ve met along the way.
For sure! The same can probably be said for science teachers too.
How instances work.
When doing a convolution of two curves in time, you flip one of them backwards.
Our shit statistics teacher made it so complicated.
And then one day, next semester, in a lab for signals class, the TA casually said “flip one, so they both start at zero seconds” and half the class started convulsing as an entire semester worth of misunderstood math magically snapped into place.
Lots of engineering teachers, have no business being teachers.
