So glad to hear, Vikrant!

Hey, from which university?

@@abhisheksoni9774 St. Xavier's College, Ahmedabad

@@PhysicswithElliot Hello Elliot, why do we need to express Taylor Series into something with exp(ε*d/dx), if we still take sum of derivations for solving Taylor Series?

Me who has nothing to do with this still this video came when i searched for taylor swifts song love story.

Hello everyone I'm just tuning into this channel trying to get ready for my MCAT exam and boy let me tell you that I'm a little lost😢

Same here. I am 62 as of this writing. BSChE in the 80's. Hope to get a degree in physics when I retire in 21 months... so I'm trying to get a head start. I will need it. Great videos Elliot. Thank you. Wish we had KZfaq when I was in college.

​@@sirwinston2368 best of luck❤️🙏✌️

Glad it helped Joel!

Every physics class I had from undergrad to grad had a brief overview of Taylor Series somewhere along the way

We had 2-3 lectures on Taylor and McLaurin series when we started off. We were asked to remember all the things said and the method to use them till our de@th.

I'm so glad it's helping, Rick!

Thanks Shadow!

Einstein was not responsible for the formula, in fact all his papers on mass equivalence were wrong.

Glad it helped Sietse!

Thanks Dwayne! Excited for you!

Thanks Ankido!

Wow that's a great analogy, thanks !

Interesting!

yes and Abraham Seidenberg proved the Indian scientists knew the Pythagorean Theorem thousands of years before Pythagoras! You gotta make those chariot wheels precise. haha.

intuitive to me would be like the Fourier series - the more extensions you add then the more precise it fits the geometric function.

Thank you Charles! I love how technology makes it possible to teach things in new ways!

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Thanks Nash!

Glad it helped Joseph!

This one was really a 2-for-1!

Thank you Nick!

So glad, Jatin!

When dealing with real valued functions, it only works on very nice functions. Just about every real valued function doesn’t obey this rule, but just about every function you encounter in school is a very nice function (also known as an analytic function). There exists functions where at any point on the function you can take any derivative you want, but the Taylor series of that function at any given point doesn’t match the original function anywhere near the starting point except at the starting point itself. You can find a description of such a function in the wikipedia article titled “Non-analytic smooth function” under the section “A smooth function which is nowhere real analytic”.

Imagine you want to figure out the position of an object rolling up a hill at the time t = 1 second. If you know nothing about the position of the object, then we can approximate the position of the object with f(t) = 0, the simplest polynomial. If, however, we are given the position of the object at t = 0, say f(0) = 3 meters, we can assume the object will not have had much time to move in a single second, so we’ll approximate our function with f(t) = 3. Our guess for the object’s position went from 0 (a random guess really) to 3 (slightly better guess). The issue with this guess is that it assumes the object isn’t moving. If we knew both the objects position and how fast that object was moving at t = 0, we could better predict the objects position at t = 1. If the speed of the object at t = 0 is 0.5 m/s, then our new guess as to what the position function is f(t) = 0.5t + 3. This gives us the approximation f(1) = 3.5 meters. This function agrees with our functions position and speed at t = 0, but is the simplest function to do so. The issue with this approximation is that it assumes the object isn’t accelerating or decelerating at t = 0. If we knew the position, speed, and acceleration of the object at t = 0, then our approximation should be even more accurate. If the acceleration of the object at t = 0 is -2 m/(s^2), then our new approximation of the function becomes f(t) = -t^2 + 0.5t + 3. At t = 1, our new approximation is f(1) = 2.5 m/(s^2). This function agrees with the position, speed, and acceleration of our function at t = 0, and is the simplest function to do so. The issue with this approximation is that it assumes the object’s acceleration never changes. If we knew the jerk (i.e. how fast the acceleration is changing) of the function at t = 0, our guess as to what the object’s position will be at t = 1 should be more accurate. Tldr the idea is that the more information you know about the behavior of an object’s movement and position at some starting point, the better you are able to predict where that object will be in the future (and where it was in the past). If you knew everything about the behavior of the object’s movement and position at some starting point, you should be able to perfectly predict the behavior of the object’s movement and position at any given time (at least within a close enough region to the starting point). In my previous post (above), I show you where to find an example where this intuition collapses, and for just about every real function out there, this intuition collapses. However, for just about every function you encounter in school, this intuition works perfectly.

Thanks LJ!

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Thanks Brian!

Glad it helped Robin!

E = hv was found experimentally by shining light on metals.

@@raspian1019 Maybe there is possibility to find it out pure theoretically these days...?

Glad you liked it Noah!

Yeah, should've said "analytic"

Probably you mean something like e^-1/x^2, since e^-1/x isn't smooth, but yes, I'm looking here at the well-behaved functions that we typically need for physics purposes

@@PhysicswithElliotYou are absolutely right! I was too lazy to write out the whole piecewise-defined function as I thought I had made my point clear. I apologise for not completing the definition. Here is a more precise one of the function I had in mind: It is identically zero for x0. People use e^-(1/x^2) most often when giving an example of a smooth non-analytic function, so I wanted to give a lesser-known one which decays slower than that one. Great video btw! Keep 'em coming!

@@NoNTr1v1aL Ah yes that makes more sense!

Dive into single-variable calculus (derivatives & integrals of elementary functions), and Vector Algebra (addition/subtraction, dot products & cross products). These are some of the first mathematics featured in Introductory Physics. Learning to apply physical principles to a situation to formulate a reasonable mathematical representation (model), and conjecturing how the system should behave based on the physical principles are the "hardest part" of Physics. Learning the mathematical techniques to analyze the implications of the model is the other "hard part".