I am awestruck. If only my professors taught me even half of this it would have made a huge difference in my life now. Anyways it is always better late than never. My huge thanks to you Professor, I will never forget this lecture. Thanks MIT and TH-cam.
William Gilbert Strang (born November 27, 1934 in Chicago[1]), usually known as simply Gilbert Strang or Gil Strang, is an American mathematician, with contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra. He has made many contributions to mathematics education, including publishing seven mathematics textbooks and one monograph. Strang is the MathWorks Professor of Mathematics at the Massachusetts Institute of Technology
Prof Strang is the very best teacher I've ever encountered, perhaps because he knows the subject so deeply and thoroughly that he can spot where noobs might go down the wrong path and point out the danger. Plus, his enthusiasm motivates me.
Like you said proffessor, the texts don't give enough focus on the key points & their implications. Instead they jump to huge exercises intended to have the seeker make mistakes & only then realize whats missing in his understanding. Thank you so much for dispelling any misconceptions as to what the concepts of limits & continuity are trying to achieve. Your use of simple language & following the rabbit all the way inside its hole was a fun trip. Bless you
Thanks to Dr. Strang and the MIT OCW Team for helping me to develop "mathematical eyes". OCW is on my list to donate to as monies become available. ..loved the Socrates/Plato analogy story.
If anybody new to calculus like me wants the best explanation of limits, I would advise you to look elsewhere. But fortunately for those people, unlike with other essential calculus concepts, there are tons of great, precise videos on limits out there, even on TH-cam! I would recommend this video as a REVIEW once you have covered limits already. Then it will be much better!
I think it depends on one's learning style as to whether this is best as primary or review instruction (e.g., some seem to learn "top-down" while others learn "bottom-up, where the latter goes more from rote to concept, and the former from idea or application to the tactics of doing). As a student (of 60+ years), I often struggle with the discipline (and attention span) to learn many things with the bottom-up approach, yet given an interesting problem and a 'big picture', I can move right along and actually internalize these kinds of important tools to my thinking and doing.
I will never forget about this lecture! I always wondered the dangers about such seemingly continuous functions, the clear importance of limits, and its applications in real world. This lecture provided the best explanation! I wish I had the opportunity to be taught by the one and only professor! Thank you!
@TreyNizzle What its scary about it? G.Strang is great teacher/mathematician with worldwide renown.I dont find it sad either.Teachers are not responsible for poor USA education.And he teachs great minds of all around a world not only in USA(nor America) with these youtube lectures and im grateful for it!
A function is continuous on "a" only if the left a+ and right a- derivative at that point are the same. e.g. the function f(x) = x if x >=0 and f(x) = -x if x
Its not his fault, he is a very honest, intelligent mathematician, but the whole system needed an overhaul ever since Russel dared to speak out. An overhaul is a huge endeavor & govs are only interested in solutions to specific problems, its the American way.
The Prof. Said that the function x^1/2 's slope cant be defined as x approaches 0 BUT the function is continuous at 0.Heres my question if that so is it correct if we say that the function of the line parallels to the y-axis also continuous? Im confused since according to the theorem the function is continuous if the limit of f(x) as x approach es a exists = f(a).
Hi, the line parallels to the y-axis can't be defined as functions. Remember that an application, to be a function, need to "relate" every x with an unique y.
While we are fixing the redirect, you can find the course at: ocw.mit.edu/courses/res-18-005-highlights-of-calculus-spring-2010/. Best wishes on your studies!
@@mmwapec Yes, although curve look continuous, at x=1 the y value is undefined because you are dividing sin(pi*x) with zero. sin(pi*1)/(1-1). So you might say that at x=1 there is a "hole" in the curve :-) but you can still approach x=1 from both sides.
@@mmwapec You can keep on going closer and closer to a number without ever reaching it. That is in the nature of limits. In the curve for the function sin(pi*x)/(x^2-1) the point at exactly (1,1) is still missing, so the curve is discontinuous although we can't see is it :-)
He is very confusing and he ist jumping around between the topics. If you want to learn all these things in a systematic way this clip isn't the right one.
I have read and seen many explanations of limits and continuity and this was without question the absolute worst. prof. strang may be a brilliant and very nice man but my subjective opinion is that his pedagogical skills are atrocious. this topic can be better explained by most high school math teachers.
That my friend is the difference between college professors and high school teachers. There are two main reasons. High school teachers are required to take Education classes in order to learn how to teach where as professors are not. In high school they will give you opportunities for extra help, extra credit, homework that they actually check and grade, and in class activities. In college all they do is spend an entire period lecturing and then you're expected to pass the next test. Professors are not paid to give you extra help so it's your responsibility to find a tutor or a study partner. Also in high school they have a math lab that you can visit during lunch. All my college had was a writing lab to go over essays, but no math lab.
David DeMar bcz he is teaching logic and theory. he wants you to use your brain, to think, to inderstand and spot trend; not just plug into a formula. it's a prestigious college attended by motivated top notch students. it's not HS, a jr. college or a state college.
I am awestruck. If only my professors taught me even half of this it would have made a huge difference in my life now. Anyways it is always better late than never.
My huge thanks to you Professor, I will never forget this lecture. Thanks MIT and TH-cam.
William Gilbert Strang (born November 27, 1934 in Chicago[1]), usually known as simply Gilbert Strang or Gil Strang, is an American mathematician, with contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra. He has made many contributions to mathematics education, including publishing seven mathematics textbooks and one monograph. Strang is the MathWorks Professor of Mathematics at the Massachusetts Institute of Technology
What a legend. God bless him!
Prof Strang is the very best teacher I've ever encountered, perhaps because he knows the subject so deeply and thoroughly that he can spot where noobs might go down the wrong path and point out the danger. Plus, his enthusiasm motivates me.
Like you said proffessor, the texts don't give enough focus on the key points & their implications. Instead they jump to huge exercises intended to have the seeker make mistakes & only then realize whats missing in his understanding. Thank you so much for dispelling any misconceptions as to what the concepts of limits & continuity are trying to achieve.
Your use of simple language & following the rabbit all the way inside its hole was a fun trip. Bless you
He explains where ¨things¨ come from. My teacher simply gave me formulas without explaining where they come from. But he explains really good.
Dr. Strang, thank you for your wonderful approach to the teaching of mathematics. I am finding your lectures delightful and insightful.
Thanks to Dr. Strang and the MIT OCW Team for helping me to develop "mathematical eyes". OCW is on my list to donate to as monies become available. ..loved the Socrates/Plato analogy story.
These lectures are very helpful for calculus and it's development. Dr. Strang is the grandfather of calculus.
If anybody new to calculus like me wants the best explanation of limits, I would advise you to look elsewhere. But fortunately for those people, unlike with other essential calculus concepts, there are tons of great, precise videos on limits out there, even on TH-cam!
I would recommend this video as a REVIEW once you have covered limits already. Then it will be much better!
What other videos if uh can specify
I think it depends on one's learning style as to whether this is best as primary or review instruction (e.g., some seem to learn "top-down" while others learn "bottom-up, where the latter goes more from rote to concept, and the former from idea or application to the tactics of doing). As a student (of 60+ years), I often struggle with the discipline (and attention span) to learn many things with the bottom-up approach, yet given an interesting problem and a 'big picture', I can move right along and actually internalize these kinds of important tools to my thinking and doing.
"even on TH-cam": TH-cam has great resources, no need for the "even".
I will never forget about this lecture! I always wondered the dangers about such seemingly continuous functions, the clear importance of limits, and its applications in real world. This lecture provided the best explanation! I wish I had the opportunity to be taught by the one and only professor! Thank you!
Beautiful Strang. Thank you from across the globe
first time i understand what is meaning of limit thank Dr.Strang and Mit
I'm watching the video in Sept 2023, this is pure brilliance. So grateful for this video.
Greatest Ever Explanation!
I'm watching this lecture in 2024 thank you ❤ for solving my all queries which I couldn't find on any channel ❤❤
Looking at MIT OCW videos, makes me realise, how much I want to be a part of that enchanting atmosphere at the great great great mit !!! ^_^
I couldnt really understand Li Hopital's rule had to do with calculus till that few minutes blew my mind. Big thanks..
@TreyNizzle What its scary about it? G.Strang is great teacher/mathematician with worldwide renown.I dont find it sad either.Teachers are not responsible for poor USA education.And he teachs great minds of all around a world not only in USA(nor America) with these youtube lectures and im grateful for it!
Soory for 31 guys may u love maths in next lifetime!! The teacher is living ❤️maths
Now this is what i call true class
You make everything you teach so clear to the students ..Sir .. you're amazing .. all the videos are very helpful !
Gilbert you are a delight! Straight talk much appreciated!
I love this guy. I won't watch any Calc tutorials by anyone else.
for some reason you make many topics very interesting.
A function is continuous on "a" only if the left a+ and right a- derivative at that point are the same. e.g. the function f(x) = x if x >=0 and f(x) = -x if x
Best explanation ever!
Good Job Professor
Excelente presentación! Clase Magistral! Gracias Dr. Strang. :)
Muy bueno cierto ?, es un gran profesor.
best teacher ever
Thank you Mr. Strang, thank you MIT.
you are the best teacher doctor
Thanks professor, greatings from Istanbul.
"7^4, whatever that may be... 49^2... 2401 or something." Yeah right "or something".
lol yeah i immediately open the calculator and was shocked lol. dafuq. how is he so fast?
i think he just did 2500-99 but still was good lol
continuous functions starts at 26:44
and suddenly it all makes sense! much thanks
These clips helping me a lot.keep uploading and thankyou sir for the lecture..
Honestly, with the mathematics ability he had, he could have made it more interesting and valuable.
excellent explanation by Dr.strang.....
Many thanks for this useful course.
@14:17 isn't 1^infinity is one?how is equal to e?
Its tending to 1 not 1.
Try it in a calculator or would approach e
I want one of those blackboards...I'd make one but I don't have the space :(
Finally I understand L'Hopital rule.
Wow all this while i though continuous functions are functions you can draw without lifting your pen. I love all his lectures
hes such a cutieee!!!! i just want to hug him
maybe so, but he's also one of the most intellectually intimidating people in MIT
Its not his fault, he is a very honest, intelligent mathematician, but the whole system needed an overhaul ever since Russel dared to speak out.
An overhaul is a huge endeavor & govs are only interested in solutions to specific problems, its the American way.
The Prof. Said that the function x^1/2 's slope cant be defined as x approaches 0 BUT the function is continuous at 0.Heres my question if that so is it correct if we say that the function of the line parallels to the y-axis also continuous? Im confused since according to the theorem the function is continuous if the limit of f(x) as x approach es a exists = f(a).
Hi, the line parallels to the y-axis can't be defined as functions. Remember that an application, to be a function, need to "relate" every x with an unique y.
I like the way he teaches.
Excellent lecture on continuity ! The best demonstration and insight explanation by Prof Strang. 👍🏻👍🏻
Did anyone notice how he just casually squared 49 to get 2401 in his head in like half a second?
Squaring numbers when you have been doing math for years becomes routine
Sir, could you please provide us with some challenging problems of function, limit and continuity
Muchas gracias, una clase magistral!!! :)
Je 5
He is GREAT!
can anybody tell me why add x to sin(1/x) can take it down : xsin(1/x)?
you are a great professor!!!
Link is broken in the description.
While we are fixing the redirect, you can find the course at: ocw.mit.edu/courses/res-18-005-highlights-of-calculus-spring-2010/. Best wishes on your studies!
Try this "continuous function" sin(pi*x)/(x^2-1) and make a tangent in x=1 ;-)
can you still explain it after 3 long years 😶
@@mmwapec Yes, although curve look continuous, at x=1 the y value is undefined because you are dividing sin(pi*x) with zero. sin(pi*1)/(1-1). So you might say that at x=1 there is a "hole" in the curve :-) but you can still approach x=1 from both sides.
@@hklausen so it's a removable discontinuity... btw thanks ❤️❤️
@@mmwapec You can keep on going closer and closer to a number without ever reaching it. That is in the nature of limits. In the curve for the function sin(pi*x)/(x^2-1) the point at exactly (1,1) is still missing, so the curve is discontinuous although we can't see is it :-)
@@hklausen thanks for tge explanation 😇😇
i seriously love you so much
Que pronunciación tan exquisita
Wow he pulled that 49 squared out his head like a calculator, genius
it's 50^2 - (50 + 49) = 2401
Nice, but what about profs of A+B and A*B.
Can someone tell me what kind of chalk he is using, its so much easier to read than what my prof uses.
difference of n and n^2 below or I don;t understand that a part then ? 0.939946202892171673454322405826552150808296781656085087913...
5:35 oh you mathematicians daredevils :) haha
Did Socrates truly discussed this issue with Plato, or just Dr. Strang made up a story simply to make the lecture more interesting?
gilbert is awesome!
keep doing its a great work doing
Favourite part is at 10:39
minute 35ish, a good theory as to why Socrates drank the hemlock :-), Plato was such a nag! Great joke and lecture!
what. is. behind. the. prof. ??
More than great,
Seriously? This is about continuity....?
thank you
49 square 2401...wow..
Sir I am preparing iit so may you tell me any idea please sir
just try to match the speed of his approximation of 49^2
i really didn't fully understand the second part of the lecture
Sir,are u using f (x) for representing the y-axis or depicting it as a function pllzzz reply sir...
@alisaffah indeed
best fucking explanation. thanks a lot!
Teachers are partly responsible for poor US education, but that has nothing to do with Strang and this video.
A little Strang-e, but I like it.
what is scary about it? that plato and socrates are already dead!
He is very confusing and he ist jumping around between the topics. If you want to learn all these things in a systematic way this clip isn't the right one.
Totes love chino charlie
Hospital rule))
Türklere Selam :D
100th comment....
Lol
I have read and seen many explanations of limits and continuity and this was without question the absolute worst. prof. strang may be a brilliant and very nice man but my subjective opinion is that his pedagogical skills are atrocious. this topic can be better explained by most high school math teachers.
That my friend is the difference between college professors and high school teachers. There are two main reasons. High school teachers are required to take Education classes in order to learn how to teach where as professors are not. In high school they will give you opportunities for extra help, extra credit, homework that they actually check and grade, and in class activities. In college all they do is spend an entire period lecturing and then you're expected to pass the next test. Professors are not paid to give you extra help so it's your responsibility to find a tutor or a study partner. Also in high school they have a math lab that you can visit during lunch. All my college had was a writing lab to go over essays, but no math lab.
then this system is totally fucked up
David DeMar bcz he is teaching logic and theory. he wants you to use your brain, to think, to inderstand and spot trend; not just plug into a formula. it's a prestigious college attended by motivated top notch students. it's not HS, a jr. college or a state college.
It would be more helpful if you indicated what you don't like
His lecture is the most insightful lecture i ever saw on limits
AmAZING, Thanks