For anyone that didn't see the most recent video, this channel used to be 'MajorPrep' and the name just recently changed. I'll stop bugging you guys about it after this but I know I'll still get comments from people who don't watch every video on this channel and didn't know about the change. Also if you enjoyed the 'mathematics used to solve crime' video I did a while back you will definitely enjoy the video coming next!
How can one not watch all your videos? I have to give you credit for picking interesting topics and explaining it well, and even if I know about things, there is always some new angle or insight as well as nice visualisations.
Who even approximates sin(x) as x anymore? That is so yesterday! *Today, we say 1=cos(x)* _No, we do not approximate cos(x) as 1, we approximate 1 as cos(x)_
Calc 1: Derivatives Calc 2: integrals Calc 3: multivariable Calc 1 and 2 (easier than Calc 1 and 2) Calc 4: Calc 3 but with multiple derivatives and integrals. (Pretty tough but much more fun than it sounds.) There, I just told you the next 2 years of math classes. Lol
@@rijulranjan8514 because you do convergence tests and they're not always straightforward. It's a guess and check kind of thing and it can get super annoying. You'll spend like 10 minutes doing a test to see if an equation converges or not and the test will be inconclusive, so you have to try something else. Will some of the convergence tests you just have to get lucky and do the correct test
I feel like you should do a Dear.... for all Calc courses, or just all courses in general like Linear Algebra. That'd help out so much more than you think
9:03 I learned the Taylor approximation for the far-field strength of a dipole in my electricity and magnetism class. That was a particularly frustrating day. Our prof had us all attempt it ourselves and we all failed. Then he showed us the Taylor series approximation. When he started striking off terms that "didn't matter" I just about lost it. I left the class thinking "well yeah, anything is easy if you can just call the hard parts insignificant and strike them out. Let's see how he feels about me doing that on the midterm". As you can tell, I'm so totally over it.
Thats so true. I just lost it when i thought the prof and books were messing up eqn. I thought C'mon how do u say that as equal and not mess up later! I lost interest for all E&M, Antennas, microwave.
As my calc professor put it some years back: "most equations are rude and hard to work with, but Taylor is great at making them well behaved and easy to work with." Or as I tend to paraphrase it: "equations can be assholes that are impossible to work with, but Taylor can kick their ass into place so that they're well behaved."
Student: what are the purpose of the equations if we can't use them? Teacher: yes EXACTLY! They have a purpose, they're not just hanging out in reality for no reason 😂
@pyropulse they always say that, it doesn't matter who tutors them, as long as it is not the teacher they'll always say "if the teacher taught like you did it's be easier."
@pyropulse because it's harder to have engagment during the lecture. When you tutor, you are having them do actual problems. But during the lecture, it's more about delivering the background knowledge required to do the problems. And it could be a long road from there. When time is limited, some just choose the easy path and just say everything they need and leave the rest to you. The students should proactively take notes and think. But for me lectures don't really work sometimes because when you take notes and think on your own, your mind wanders off and miss the next bit of the lecture. Missing any critical bit of information can make the rest of the lecture incomprehensible.
@pyropulse I've also been an assistant in many courses(albeit electrical engineering and computing science) and all I can say is that I despise your look on students. I've had people who would repeatedly ask simple questions, which I would eventually ask to stop asking simply for they'd slow down my tutorial, but I wouldn't come close to saying "I hated being a student because others asked stupid questions". If you think like that you fail to understand the frustration that comes with studying for so many of your fellow students. My problem with studying always was that there is practically *no* engagement; take for example analysis or calculus. These subjects float somewhere in the realm of extreme abstraction without being applied anywhere until way later(usually masters). Due to the modularity of most studies you'd have a course about them in year 1, then one somewhere in year 2 and sometimes another in year 3, without any logical connection between them. So you'd push yourself through just to have to redo most of it again a while later, instead of making sure everything taught is reinforced by applying it after being taught(and no, jump through the hoop I don't consider applying). I'll also vow for designing semesters in a fashion that would allow particular subjects to be analyzed in depth, before moving on, rather than spreading them out over several years. I think the fact that it isn't is a very large offender in the never ending, as you call it "stupid", questions.
@pyropulse can i ask where do you live? If its USA i found that it is so popular to rely on tutors it almost takes away the responsibility off the students to learn and acts more like a good business model for people in academia around colleges. Mass tutoring isnt that popular in my country.
Having just finished my Machine Learning Class last semester, I can say with confidence that Taylor Series, while Hell, are far easier for computers to calculate than doing the "normal" method. And when you have to run more than a million calculations of a particular function even a 1% increase in computational speed/efficiency may save HOURS of computing time (given large enough datasets). Even if you aren't in Computer Science, if you have a friend even tangentially interested in AI, being able to lord over them the gift of Taylor Series is going to be worth it for them.
Thank you. I got a physics undergrad and I've never really understand the Taylor series. The bit at 8:53 where you series expanded the total relativistic energy and turned it into mc^2+.5mv^2 blew my mind.
This was really helpful. No one has explained the context for using Taylor series which made learning how to do them really hard. Appreciate the in-depth vid!
I enjoy going back and reviewing the basics, which I was forced to cram during my semesters of calculus. it's also fun to solve problems using C or python once you have a good intuitive understanding
I love your maths videos because I think it's important to bring the applications of maths closer to those studying it. I wasn't a fan of the subject in school because I simply didn't get why I was being taught something I'd never use. Later on I discovered just how amazing and powerful maths is and by learning about the applications of maths I worked backwards and studied some topics that really got me into it. It's the most interesting field by far but gets such a bad rep in school lol
I absolutely LOVED Calculus II. Despite not being a mathematics major, Calc II has been my favorite class so far. Having a great professor definitely helped.
"So although it doesn't sound professional; being good enough is often what we're after." (11:13) I disagree that it's in any way unprofessional to approximate. I'll agree it's not rigoristic in a mathematical and analytical sense, but that's not the point. Without approximations there's a plethora of things we wouldn't have been able to do technologically in today's society. Having an answer that works with 0,x% error is infinitely more professional than not having an answer at all.
Another great video dude! I appreciate the way you teach people the applications of different mathematical topics. It's a great way of motivating people to learn and appreciate math like i do.
Thank you! And if you haven't seen them already I've done a few in depth videos on fourier and laplace. Fourier (and some laplace): th-cam.com/video/3gjJDuCAEQQ/w-d-xo.html Laplace: th-cam.com/video/n2y7n6jw5d0/w-d-xo.html
At around 9:00 , the equation at the top is not an approximation but in fact the same equation as 'm' is not equal to 'm0’. If you plug in the value of 'm' in terms of 'm0', you will get the same equation. The equation which is arrived at the bottom is the approximated equation as it has 'm0' at both places and is only valid for objects not moving at a speed close to light.
Hey Zach I really appreciate you putting out these awesome videos. People like you are what keep my interest in math and physics mainstream. Also, I really enjoyed your skit videos. Those were Fricking hilarious!
Oh my holly forking COW. I've learned Taylor series many times in various classes till the end of my Master's, but JUST NOW got the true intuition on the Taylor Series. Thanks for the crazily awesome video. This is crazy.
Thank you for doing this As a second year engineering student I had no idea what the point of series was despite getting an A in Calc II, I just thought it was some useless math talk. Now I understand and I have you and this great video to thank for so
Most springs cease to be useful as springs, once you extend them beyond the linear elastic range. The metal deforms permanently, and the spring doesn't return to its original position. With metal springs, Hooke's law is good enough for the entire reversible elasticity domain, and rarely would you need to know a higher order function to model it. For plastic springs, the stress strain function has curvature in this range, so indeed using Hooke's law is simply a linear approximation.
You should put keywords in the description so this would pop up when I'm learning about energy, velocity (kinematics), and electrical fields, since that would make learning all that even more interesting and explain how all those formulas are connected. I never made that connection until rewatching this video. This also is intetesting that we are applying the ideas for alternating series in the electric field example.
I'm glad you made this video - I enjoy the engineering memes, and I was looking for a reason behind calc 2 because it's definitely more than just learning to integrate more functions. Thank you!
As someone teaching differential equations to high school kids, the unit that we have on infinite series always feels weirdly arbitrary to them. This is a great video that really demonstrates a lot of how these things are used in day-to-day calculations. Thank you.
At 4:17 plugging in y(0)=-1 and x=0 into the differential equation to solve for y’’(0) the equation was written incorrectly as y’’(0) = 0 + y(0) + [y(0)]^2 instead of the correct form y’’(0) = 0 + y(0) -[y(0)^2] but it was evaluated correctly to be -2.
10:47 This is why atom appears to be neutral from a distance even though the location of positive charges (nucleus) and negative charges (electrons) are far but not too far from each other.
NICE! Some numerical methods, like Runge-Kutta, are derived from some terms of a Taylor series, also, some differential equations, like the heat conduction and the famous Navier-Stokes, are derived from some terms of a Taylor series.
This was so helpful! I wish my calc professor in undergrad could have explained this as well as you did. Thank you for making this video and sharing it!!
The applications for the equations are left in youtube for us to browse i wish i had math a teacher who could teach me math like this You are doing a hell of a job brother keep going.........😍
I appreciated watching this video very much - in some sense, I gather mathematic is a bit like art. There is a sort of piece where its about intuition, and you make those aproximation and it works in certain cases.
this video is cool as an algorithmic recommendation because while it is squarely in my interest zone, it is completely outside my understanding and competency so it's just jazz to me
Right... Babies never ,"learn English" in order to speak it. All American Babies just yell, "mERica!" as they slide out of the birth canal. Afterwords they immediately realize that America is so great that you don't have to learn any other languages in order to thrive. #DontHateThePlayerHateTheGame
A lot of content electrodynamics course as a physics major dealt with approximations just like that last example. Everything in STEM, minus pure math is approximation.
I would try to reassure the students: you will probably never have to compute the derivative of a complex equation by hand if you program a computer. There is a general trick called "autodiff" that helps you compute the derivative of a function at the same time you compute its value on the fly with a very simple composition rule instead of having to derive the entire closed form derivative equation on paper. On the reason of *why* Taylor, I would probably compare that to other approximations. Taylor being a polynomial makes everything very comfortable for algebraic work. Whereas a Padé approximant or a neural network are a bit of a crap show to deal with for example. I think also a bit of an explanation of successive approximation would be a plus, your video gives the impression that the only way to get a good number is to get a higher degree derivative, while in real life we slide the approximation point and we very often get to the floating point number the closest to the mathematical solution, which is the best you can do. There is no general sloppiness in successive approximation.
This is probably as good a single-use motivation as any. But the viewer should be aware that the Taylor series is at the heart of analytic functions and complex analysis. Those subjects have many profound consequences aside from the ability to approximate. A more accurate title would be: This is ONE REASON you are learning Taylor series.
A mathematician and an engineer were both chaparones at a middle school dance. There was a line of boys, and a line of girls, who started 16 feet apart, and were very shy of one another. Every minute, they halved the distance to each other. From 16 ft to 8 ft, then 8 ft to 4 ft, and so on. The mathematician remarked, "they will never make to each other." The engineer replied, "yeah, but in a few minutes, they will be close enough for all practical purposes".
Calculus courses: we want EXACT answers, anything else is WRONG. Actual applications of calculus in engineering: we want it within +- 5% because exact is impossible when dealing with outside influences.
No shit, I had a Calculus 3 problem that dealt with physics. Having already taken physics I knew how to solve the problem in a single step. I turned the quiz in and got a big X on the problem. I compared answers to everyone else in the classroom and got roughly the same answer to within 3 decimal places. I asked how is mine wrong? The professor said, "you didn't decompose the vector to get an exact answer." (and he had an ME doctorate).
Where are you going that you're asked for only exact answers? Even in calc 3 my professor would tell us to use our calculators and approximate and discouraged us from getting exact answers, which sadly is the only thing I wanted to do.
@@QmcometdudeShardMaster Tarrant County College. They discourage decimal answers to the highest degree. Even to the point they would rather your answer be an 8" equation than a short answer with decimals.
if you use taylor series to solve for PDE's, you are going to either make a super computationally unstable/inefficient algorithm or one that just doesn't work (due to discontinuous boundary conditions or such). The REAL reason you learn Taylor series is so that you can kinda learn a bit of numerical analysis, and THEN you learn the real shit known as "Fourier Series". Fourier Series can be used to solve anything if you have the right spectral resolution and sample rate.
Thank You so much. Please help by answering this question, as average students how do we visualize day-to-day topics of our stream, and find their practical use and how are they applied in the complete process. Just like a regular CS student knows how to implement all the data structures but the actual code used in production is way different than those taught or written in classrooms, how to bridge that gap, and get the actual reality/purpose of the concept.
Hi Zach star Can you also shed some light on why we use the fundamental constants of nature like Boltzmann constant, phank constant etc.. Along with their applications and how they are found if possible. I mean they are also everywhere from quantum mechanics to electromagnetics. Kindly add them in your upcoming video bucket list. Thanks
Yeah e^x just happens to have an infinite radius of convergence, maybe should've been more specific about that but no it's not always a perfect approximation.
I get that we need to approximate things for simpler and possible solutions, but what about when you need 100% accuracy like in aeorspace engineering for example? I assume you'd want pretty much 0 error when you're dealing with rockets, no?
Ha ha. I'm so glad I never have to look at another calculus equation. I struggled (as did almost everyone else) to get through Calc 2. The entire class was graded on a curve, because the textbook did such a horrible job at explaining it. I definitely blame the textbook. Calc 1 was not bad. I earned a real B, which was due to a very experienced and excellent teacher who took the time to really help her students understand the material. Can't say the same for Calc 2. My sympathies to all who are forced to take this class to graduate, even though (like me, a Staff Systems engineer who hasn't seen or used any math at all since graduation) your future career will never require it.
Maybe there is a solution and method, but the math just doesn't exist yet. Add a layer of abstraction. Approximations of pi yield accuracies at 40 digits of approximately 1 proton at a universe level. 42 digits? 2x dimensions?
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For anyone that didn't see the most recent video, this channel used to be 'MajorPrep' and the name just recently changed. I'll stop bugging you guys about it after this but I know I'll still get comments from people who don't watch every video on this channel and didn't know about the change. Also if you enjoyed the 'mathematics used to solve crime' video I did a while back you will definitely enjoy the video coming next!
Second like.
@VeryEvilPettingZoo -- 3 = pi = e.
Oksy, i came to ask what happened to MajorProp that his voice've transferred to a new channel. I see now
How can one not watch all your videos? I have to give you credit for picking interesting topics and explaining it well, and even if I know about things, there is always some new angle or insight as well as nice visualisations.
I liked the old name better, and I admit I was quite confused because I didn't remember subscribing to a channel named Zach Star :]
Who even approximates sin(x) as x anymore? That is so yesterday!
*Today, we say 1=cos(x)*
_No, we do not approximate cos(x) as 1, we approximate 1 as cos(x)_
@@Zack-xz1ph You probably mean -(x^2)/2
I approximate sin x to be tan x
i mean if you do that
1+sin^2(x)=cos^2(x)
which is cool
We dont do that here
As long as x is really small
I should be studying calculus 1 right now but might as well see what the future holds for me haha
Don't mind you gonna fail
Calc 2 is a whole different ballpark lol
Calc 1: Derivatives
Calc 2: integrals
Calc 3: multivariable Calc 1 and 2 (easier than Calc 1 and 2)
Calc 4: Calc 3 but with multiple derivatives and integrals. (Pretty tough but much more fun than it sounds.)
There, I just told you the next 2 years of math classes. Lol
Slim Jim why is Calc 2 considered so hard? I'm also in Calc 1 and it seems like everybody at my school dreads it
@@rijulranjan8514 because you do convergence tests and they're not always straightforward. It's a guess and check kind of thing and it can get super annoying. You'll spend like 10 minutes doing a test to see if an equation converges or not and the test will be inconclusive, so you have to try something else. Will some of the convergence tests you just have to get lucky and do the correct test
why do 1 feel so uncomfortable with seeing "Calculus 2" instead of "Calculus II"
calculus two
Seems like you're fine with writing "why do i" instead of "Why do I" so I think you'll get over it
2(Calculus)
Calculus 1+1
calculus 2rd edition
I feel like you should do a Dear.... for all Calc courses, or just all courses in general like Linear Algebra. That'd help out so much more than you think
"Linear Algebra" ~ Gilbert Strang MIT
th-cam.com/video/7UJ4CFRGd-U/w-d-xo.html
"Linear Algebra" ~ 3Blue1Brown
th-cam.com/video/fNk_zzaMoSs/w-d-xo.html
@@douglasstrother6584 this is obviously not the same format
I can vouch for 3B1B teachings
"Introduction to Linear Algebra" ~ Gilbert Strang
math.mit.edu/~gs/linearalgebra/
9:03 I learned the Taylor approximation for the far-field strength of a dipole in my electricity and magnetism class. That was a particularly frustrating day. Our prof had us all attempt it ourselves and we all failed. Then he showed us the Taylor series approximation. When he started striking off terms that "didn't matter" I just about lost it. I left the class thinking "well yeah, anything is easy if you can just call the hard parts insignificant and strike them out. Let's see how he feels about me doing that on the midterm".
As you can tell, I'm so totally over it.
mellow I’m taking e&m this semester, if I remember, ilyk
Thats so true. I just lost it when i thought the prof and books were messing up eqn. I thought C'mon how do u say that as equal and not mess up later!
I lost interest for all E&M, Antennas, microwave.
Me too... But I still gotta face it
@@plentygolden Yep. E&M and Stat Mech. Did a number on my GPA back in the day.
Ok, I'm two years late but whatever.
When they say "far-field" they always mean in an asymptotic sense.
Look at this engineering propaganda smh. Stay in maths; don’t approximate kids
Lmao
Pure mathmatics do not approximate rather abstract.
Clearly has not studies PNT as the prime counting function is an approximation.
Mathematicians hate engineers because they take math and apply it for something with practical value.
@@Cyberspine An applied mathematician enters the chat.
As my calc professor put it some years back: "most equations are rude and hard to work with, but Taylor is great at making them well behaved and easy to work with."
Or as I tend to paraphrase it: "equations can be assholes that are impossible to work with, but Taylor can kick their ass into place so that they're well behaved."
@pyropulse This one is going in my cringe compilation
im surprised marvel never made a comic about the taylor hero!
@@pewpew9711😂😂😂😂
Student: what are the purpose of the equations if we can't use them?
Teacher: yes EXACTLY! They have a purpose, they're not just hanging out in reality for no reason 😂
@pyropulse they always say that, it doesn't matter who tutors them, as long as it is not the teacher they'll always say "if the teacher taught like you did it's be easier."
@pyropulse because it's harder to have engagment during the lecture. When you tutor, you are having them do actual problems. But during the lecture, it's more about delivering the background knowledge required to do the problems. And it could be a long road from there. When time is limited, some just choose the easy path and just say everything they need and leave the rest to you. The students should proactively take notes and think. But for me lectures don't really work sometimes because when you take notes and think on your own, your mind wanders off and miss the next bit of the lecture. Missing any critical bit of information can make the rest of the lecture incomprehensible.
@pyropulse I've also been an assistant in many courses(albeit electrical engineering and computing science) and all I can say is that I despise your look on students. I've had people who would repeatedly ask simple questions, which I would eventually ask to stop asking simply for they'd slow down my tutorial, but I wouldn't come close to saying "I hated being a student because others asked stupid questions". If you think like that you fail to understand the frustration that comes with studying for so many of your fellow students.
My problem with studying always was that there is practically *no* engagement; take for example analysis or calculus. These subjects float somewhere in the realm of extreme abstraction without being applied anywhere until way later(usually masters). Due to the modularity of most studies you'd have a course about them in year 1, then one somewhere in year 2 and sometimes another in year 3, without any logical connection between them. So you'd push yourself through just to have to redo most of it again a while later, instead of making sure everything taught is reinforced by applying it after being taught(and no, jump through the hoop I don't consider applying). I'll also vow for designing semesters in a fashion that would allow particular subjects to be analyzed in depth, before moving on, rather than spreading them out over several years. I think the fact that it isn't is a very large offender in the never ending, as you call it "stupid", questions.
@pyropulse can i ask where do you live? If its USA i found that it is so popular to rely on tutors it almost takes away the responsibility off the students to learn and acts more like a good business model for people in academia around colleges. Mass tutoring isnt that popular in my country.
@@howardlam6181 true. Never learned anything from school, but when I read the books on my own I understand it perfectly.
We approximate, I didn't mean to round (pi = 3).
Lol, 🤣 🤣 🤣 🤣
=e=(g-1)^.5
355/113
Absolutely hilarious
Guys let it be π.
355/133 seems a bit irrational...
Sounds like an engineering class
Having just finished my Machine Learning Class last semester, I can say with confidence that Taylor Series, while Hell, are far easier for computers to calculate than doing the "normal" method. And when you have to run more than a million calculations of a particular function even a 1% increase in computational speed/efficiency may save HOURS of computing time (given large enough datasets). Even if you aren't in Computer Science, if you have a friend even tangentially interested in AI, being able to lord over them the gift of Taylor Series is going to be worth it for them.
Omg this motivated me
i just started to learn taylor series for computer science
Thank you. I got a physics undergrad and I've never really understand the Taylor series. The bit at 8:53 where you series expanded the total relativistic energy and turned it into mc^2+.5mv^2 blew my mind.
This was really helpful. No one has explained the context for using Taylor series which made learning how to do them really hard. Appreciate the in-depth vid!
I enjoy going back and reviewing the basics, which I was forced to cram during my semesters of calculus. it's also fun to solve problems using C or python once you have a good intuitive understanding
I love your maths videos because I think it's important to bring the applications of maths closer to those studying it. I wasn't a fan of the subject in school because I simply didn't get why I was being taught something I'd never use. Later on I discovered just how amazing and powerful maths is and by learning about the applications of maths I worked backwards and studied some topics that really got me into it. It's the most interesting field by far but gets such a bad rep in school lol
I absolutely LOVED Calculus II. Despite not being a mathematics major, Calc II has been my favorite class so far. Having a great professor definitely helped.
Yeah it was a interesting class.
"So although it doesn't sound professional; being good enough is often what we're after." (11:13)
I disagree that it's in any way unprofessional to approximate. I'll agree it's not rigoristic in a mathematical and analytical sense, but that's not the point. Without approximations there's a plethora of things we wouldn't have been able to do technologically in today's society. Having an answer that works with 0,x% error is infinitely more professional than not having an answer at all.
it's funny because newton actually thought of power series as "decimal places for functions". for newton, taylor approximations WERE rounding.
5:06 "perfect approximation" sounds weird
You upload this a day before I take my first Cal 2 class in the Spring semester. Stop stalking me!
Lol same
This is amazing... My teacher always says that he loves the Taylor series.. now I know partly why.
Another great video dude! I appreciate the way you teach people the applications of different mathematical topics. It's a great way of motivating people to learn and appreciate math like i do.
Wow. Someone finally speaking mathematics. Very much satisfied. Great job expecting more regarding Laplace and Fourier transform
Thank you! And if you haven't seen them already I've done a few in depth videos on fourier and laplace.
Fourier (and some laplace): th-cam.com/video/3gjJDuCAEQQ/w-d-xo.html
Laplace: th-cam.com/video/n2y7n6jw5d0/w-d-xo.html
Great stuff. It's frustrating that the motivation is so often never mentioned.
At around 9:00 , the equation at the top is not an approximation but in fact the same equation as 'm' is not equal to 'm0’. If you plug in the value of 'm' in terms of 'm0', you will get the same equation. The equation which is arrived at the bottom is the approximated equation as it has 'm0' at both places and is only valid for objects not moving at a speed close to light.
Hey Zach I really appreciate you putting out these awesome videos. People like you are what keep my interest in math and physics mainstream. Also, I really enjoyed your skit videos. Those were Fricking hilarious!
Oh my holly forking COW. I've learned Taylor series many times in various classes till the end of my Master's, but JUST NOW got the true intuition on the Taylor Series. Thanks for the crazily awesome video. This is crazy.
Thank you for doing this
As a second year engineering student I had no idea what the point of series was despite getting an A in Calc II, I just thought it was some useless math talk. Now I understand and I have you and this great video to thank for so
Hooke's law of spring force is also a linear approximation of real spring force.
Most springs cease to be useful as springs, once you extend them beyond the linear elastic range. The metal deforms permanently, and the spring doesn't return to its original position. With metal springs, Hooke's law is good enough for the entire reversible elasticity domain, and rarely would you need to know a higher order function to model it.
For plastic springs, the stress strain function has curvature in this range, so indeed using Hooke's law is simply a linear approximation.
I love this, I watched this while talking calc 1 a year ago and today was my last day of calc 2!!! I understand this so much now 😍
I like the Equioscillation theorem from Tschebyschow a bit more then the Taylor Approximation.
You should put keywords in the description so this would pop up when I'm learning about energy, velocity (kinematics), and electrical fields, since that would make learning all that even more interesting and explain how all those formulas are connected. I never made that connection until rewatching this video. This also is intetesting that we are applying the ideas for alternating series in the electric field example.
I'm glad you made this video - I enjoy the engineering memes, and I was looking for a reason behind calc 2 because it's definitely more than just learning to integrate more functions. Thank you!
I needed this video 3 years ago.
As someone teaching differential equations to high school kids, the unit that we have on infinite series always feels weirdly arbitrary to them. This is a great video that really demonstrates a lot of how these things are used in day-to-day calculations. Thank you.
At 4:17 plugging in y(0)=-1 and x=0 into the differential equation to solve for y’’(0) the equation was written incorrectly as
y’’(0) = 0 + y(0) + [y(0)]^2 instead of the correct form y’’(0) = 0 + y(0) -[y(0)^2] but it was evaluated correctly to be -2.
1=v^2 /c^2 and c is speed of light
Thanks for the informative overview, nice. I like and appreciate your videos
Hey hey! Teaching series next month - I'm going to play this video for them then. Thanks!
10:47 This is why atom appears to be neutral from a distance even though the location of positive charges (nucleus) and negative charges (electrons) are far but not too far from each other.
Thanks, perfect motivation to study for the final
What do you mean "doesn't sound professional"? "Good enough" is essentially the definition of "professional" 🤣
NICE!
Some numerical methods, like Runge-Kutta, are derived from some terms of a Taylor series, also, some differential equations, like the heat conduction and the famous Navier-Stokes, are derived from some terms of a Taylor series.
This was so helpful! I wish my calc professor in undergrad could have explained this as well as you did. Thank you for making this video and sharing it!!
The applications for the equations are left in youtube for us to browse i wish i had math a teacher who could teach me math like this
You are doing a hell of a job brother keep going.........😍
I make math content on my channel
I appreciated watching this video very much - in some sense, I gather mathematic is a bit like art. There is a sort of piece where its about intuition, and you make those aproximation and it works in certain cases.
Thanks for this video. I find it really helpful to know what motivates the techniques we learn.
I feel so interesting when you say about it's applications but in classes solving problems by hands makes me de motivate
Love your work pal. I am 57 yrs old and trying my best to understand math and your work is very helpful 😊👍
e=mc^2 itself is an approximation based on the first term of a Taylor series.
As an engineering student, I have to say your videos are very helpful and much easier to understand than 3b1b 😁
this video is cool as an algorithmic recommendation because while it is squarely in my interest zone, it is completely outside my understanding and competency so it's just jazz to me
We talked about Taylor Series in my Numerical Methods class today. I’m glad I found this
Calculus 2?!?
i ALrEaDy hAd ThIs iN eLemEnTeRy ScHoOL!!
Right... Babies never ,"learn English" in order to speak it. All American Babies just yell, "mERica!" as they slide out of the birth canal. Afterwords they immediately realize that America is so great that you don't have to learn any other languages in order to thrive. #DontHateThePlayerHateTheGame
Hey great video. My physics professor was going through a derivation and used these and I was so lost. Now it makes loads of sense, thanks.
I didn’t understand. Maybe because I still lack the prerequisites of this topic but great video though. I understood the essence of this taylor series
Thank you for posting this AFTER I take calc 2 LMFAO
Well done! But, when I solve the world's energy crisis, should I mention Zach Star or Major Prep in my Nobel acceptance speech??
e^x vs e^x (Taylor's Version)
Without Taylor Series, we'll have to go to "Plan B": philosopher, musician, poet, bar bouncer.
What is a bar bouncer?
Bouncer: www.urbandictionary.com/define.php?term=bouncer
We do the Taylor expansion because, at equilibrium, it is quadratic…and that is exactly solvable as simple harmonic oscillators
Taylor approximations can solve problems and simplify some math formulas (eg Taylor series can solve complicated limits better than l’hopitals)
Please make videos on detailed understanding about various techniques on solving differential equations numerically
I remember the estimation unit in 6th grade math. I always estimated wrong, I guess.
Mom: are you studying?
Me:
Thank you
I think It's time to get re admission in UG my university
fascinating! Didn't know maclaurin series is this useful.Thnx for letting us know. :)
Define engineering in two words
Me: 1:07
Wow, and I thought I had forgotten everything I learned in Calculus
Love this! Thank you very much :)
I studied Maclaurin Series recently in Alevel Further Maths. It was good to know the reason for it 😍😍😍
Trinidad?
@@chrisjfox8715 sorry I dont understand what you mean
Congrats on your videos. I wish I had a resource like that 28 years ago, when I was studying calculus.
>the first e^x example
8 years of calculus [40 years ago] and no-one ever told me WHY!
thanks, Zach.
same here , it was just trying to memorize meaningless junk , now 5 yr olds get a better understanding
A lot of content electrodynamics course as a physics major dealt with approximations just like that last example. Everything in STEM, minus pure math is approximation.
I would try to reassure the students: you will probably never have to compute the derivative of a complex equation by hand if you program a computer. There is a general trick called "autodiff" that helps you compute the derivative of a function at the same time you compute its value on the fly with a very simple composition rule instead of having to derive the entire closed form derivative equation on paper.
On the reason of *why* Taylor, I would probably compare that to other approximations. Taylor being a polynomial makes everything very comfortable for algebraic work. Whereas a Padé approximant or a neural network are a bit of a crap show to deal with for example. I think also a bit of an explanation of successive approximation would be a plus, your video gives the impression that the only way to get a good number is to get a higher degree derivative, while in real life we slide the approximation point and we very often get to the floating point number the closest to the mathematical solution, which is the best you can do. There is no general sloppiness in successive approximation.
I always round up pi to 10 so it cancels g out.
"love on a real train" by tangerine dream on the background, huh, nice
This is probably as good a single-use motivation as any. But the viewer should be aware that the Taylor series is at the heart of analytic functions and complex analysis. Those subjects have many profound consequences aside from the ability to approximate. A more accurate title would be: This is ONE REASON you are learning Taylor series.
What is a difference between a mathematician and engineer? Topology and approximation
A mathematician and an engineer were both chaparones at a middle school dance. There was a line of boys, and a line of girls, who started 16 feet apart, and were very shy of one another. Every minute, they halved the distance to each other. From 16 ft to 8 ft, then 8 ft to 4 ft, and so on.
The mathematician remarked, "they will never make to each other."
The engineer replied, "yeah, but in a few minutes, they will be close enough for all practical purposes".
I am an enginering student and I confirm we have studied topology 😀
Calculus courses: we want EXACT answers, anything else is WRONG. Actual applications of calculus in engineering: we want it within +- 5% because exact is impossible when dealing with outside influences.
No shit, I had a Calculus 3 problem that dealt with physics. Having already taken physics I knew how to solve the problem in a single step. I turned the quiz in and got a big X on the problem. I compared answers to everyone else in the classroom and got roughly the same answer to within 3 decimal places. I asked how is mine wrong? The professor said, "you didn't decompose the vector to get an exact answer." (and he had an ME doctorate).
Where are you going that you're asked for only exact answers? Even in calc 3 my professor would tell us to use our calculators and approximate and discouraged us from getting exact answers, which sadly is the only thing I wanted to do.
@@QmcometdudeShardMaster Tarrant County College. They discourage decimal answers to the highest degree. Even to the point they would rather your answer be an 8" equation than a short answer with decimals.
perfection is really is the opposite of good. Aim for perfection and you get nothing, but trying to get it good enough will yield a result.
1:03 You Filthy Engineers
You mean 1:10
113355. now separate: 113 355. flip and divide. 355/113 ≈ π
if you use taylor series to solve for PDE's, you are going to either make a super computationally unstable/inefficient algorithm or one that just doesn't work (due to discontinuous boundary conditions or such). The REAL reason you learn Taylor series is so that you can kinda learn a bit of numerical analysis, and THEN you learn the real shit known as "Fourier Series". Fourier Series can be used to solve anything if you have the right spectral resolution and sample rate.
Well Thanks For Great Introduction For Series 😘
Gonna Learn Them Next Year 🔥
Thank You so much.
Please help by answering this question, as average students how do we visualize day-to-day topics of our stream, and find their practical use and how are they applied in the complete process.
Just like a regular CS student knows how to implement all the data structures but the actual code used in production is way different than those taught or written in classrooms, how to bridge that gap, and get the actual reality/purpose of the concept.
Hi Zach star
Can you also shed some light on why we use the fundamental constants of nature like Boltzmann constant, phank constant etc.. Along with their applications and how they are found if possible.
I mean they are also everywhere from quantum mechanics to electromagnetics.
Kindly add them in your upcoming video bucket list.
Thanks
2:41 WAIT that only works with some functions there are a bunch of them that aren't equal to it's Taylor series.
@pyropulse precisely
@pyropulse yes
Yeah e^x just happens to have an infinite radius of convergence, maybe should've been more specific about that but no it's not always a perfect approximation.
zach star himself is a genius.
I get that we need to approximate things for simpler and possible solutions, but what about when you need 100% accuracy like in aeorspace engineering for example? I assume you'd want pretty much 0 error when you're dealing with rockets, no?
Ha ha. I'm so glad I never have to look at another calculus equation. I struggled (as did almost everyone else) to get through Calc 2. The entire class was graded on a curve, because the textbook did such a horrible job at explaining it. I definitely blame the textbook. Calc 1 was not bad. I earned a real B, which was due to a very experienced and excellent teacher who took the time to really help her students understand the material. Can't say the same for Calc 2. My sympathies to all who are forced to take this class to graduate, even though (like me, a Staff Systems engineer who hasn't seen or used any math at all since graduation) your future career will never require it.
Maybe there is a solution and method, but the math just doesn't exist yet. Add a layer of abstraction. Approximations of pi yield accuracies at 40 digits of approximately 1 proton at a universe level. 42 digits? 2x dimensions?
9:30 Me having JEE flashbacks...
Lol
So true!
Lol I’ve been there 😂
Well dang u could have told me this before I'd completed calc 2 in December
@pyropulse oh dude I was joking lol, I understood the application I always do that for my math classes cuz it's interesting
@pyropulse eh it's alright I couldn't sense certain jokes sometimes as well
Watching videos like these makes me wish I actually tried harder in high school calculus
Really liked the aplication videos
Crazy that I watched this video in class today knowing that the other channel exists
Thanks
Awesome video
Could you do a video on quantum computing and what majors/minors needed in your undergrad to get into the field?