Calculus 3 Lecture 13.8: Finding Extrema of Functions of 2 Variables (Max and Min)

[@Alexander_444.20]

Born too early to explore the Universe, born too late to explore the world. But born at the perfect time to explore calculus 3 with Professor Leonard.

[@stevenjames5874]

I like that way of thinking.

[@LoneWolfAndKub]

Great comment

[@luie5598]

I mean technically only 42% of the world has been explored

[@krishnapatel6463]

Amen

[@lugia8888]

His examples are too easy. Barely any trig and doesn’t write out theorems completely

[@janam7474]

0:00 Introduction 32:32 Second Derivative Test 46:50 Example 1 1:04:22 Tricky Systems 1:25:14 Example 2 1:51:37 Example 3 2:02:44 Important Note 2:05:54 Absolute Max and Min 2:10:00 Example 4 2:53:55 Example 5 3:07:57 Example 6 3:25:40 Example 7 (Optimization)

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Einstein said if you couldn't explain something clearly, you didn't really understand it. You sir are a brilliant teacher. And you make the world a more educated place.

[@ProfessorLeonard]

I TRULY appreciate your comment. Thank you so much for taking the time. A more educated world is what this is all about.

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True. If you can't explain anything clearly, that just makes you stupid

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That wasn't Einstein, it was Feynman

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I get it now why Professor's lectures are so effective and why people really understand them. For example, a lecture is 3 Hours Long. He would spend 1 Hour 'Defining things', 1 Hour 'Doing Examples', and 1 Hour 'Repeating' the important points. This leaves no room for confusion. A solid 3 in 1 Package with a lot of mathematical humor. Thank You very much Sir for your Efforts.

[@kanivakil198]

3:38 critical points. 5:53 inflection points = concavity. 19:44 saddle point. 22:07 inflection point example. Summary 25:38. 29:01 connection between calculus 1 and 2. 32:32 second derivative test. How many second partial derivatives: 34:38.at 37:15 four possible outcomes. Example 1 at 46:55. 57:59 fxy. 1:04 "Tricky Systems."

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@@Danny-zb1ze BROO I said the same thing , God blessss you

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[@Salamanca-joro]

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[@iceverything2000]

Absolute MAX and MIN starts at 2:05:53

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[@wildchild1161]

mine did 1 example that took 15minutes and that was it " it simple just follow formula "

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My math teacher from last year just stood and talked in a monotonous voice the whole time, and it was extremely obvious that it was all pure routine for him. It felt almost as if he was talking to himself.

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The fact that he can not only teach people the necessary things they need to know in succeeding, but to also put a smile on our faces while doing it shows so much character. Thank you so much for being there when people are in dire need of learning and getting an education. You truly are a great man. I truly look up to you sir.

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[@ueharaplas6]

Professor Leonard, you are really amazing. I was learning this in class but only memorizing the various functions etc with your explanation I finally understand WHY we are doing what we are doing. If you are looking for understanding then watch these videos! They may be long but they absolutely pay off in the end. Hope you continue making videos Professor! :)

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Hey Professor Leonard, I don't know if you read these or not, but I really admire your passion for teaching and you are REALLY good at it (& you're funny as hell too (albeit in a corny way no offense)). I know you'll never get to know me, but you've helped me so much in this class (as I am sure you did with thousands of others) and I hope you realize the impact you are making on students across the world. Thank you man!

[@ProfessorLeonard]

Thanks very much for taking the time to write this!! I do read them, and they mean a lot to me :) I'm certainly happy you have found value in these videos!

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There are those who know what they teach, and then there are those of a much rarer breed, those who can ' teach' what they know. Professor Leonard is in the latter category, and we should al be truly thankful for it.

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I just find it so refreshing how he gets more energized when he notices his students are dozing off. Instead of getting angry and getting a serious, even MORE boring tone than usual professors do. Altho Professor Leonard seems to always have a decent voice for explaining. ...OR maybe I just have mediocre faculty at my college?

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Wow, I wish I found these early. This has definitely helped with my Calc III course. Our classes are usually one section a lecture, with only a little over an hour a lecture. Whereas this is basically a week on one lecture in my class! Normally we just cover the equations quickly and one or two quick examples. This helps with conceptualizing a lot better. Thanks a lot for sharing these!

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Professor Leonard's lecture structures are excellent; he starts with explaining how something works, and then immediately gives a bunch of example problems, with lots of communication with the students, and detailed solution steps. This is _exactly_ how you are supposed to teach math.

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[@ProfessorLeonard]

:) thanks!! I appreciate it!

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3:38 hour :P Popcorn+one liter coffee => play the video at speed 1,5x => enjoy your time

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just wanted to comment to show my appreciation for these videos. I have not gotten lower than a 96 on any calculus 3 exams at my university and it's mainly due to these lectures since my professors decide that it is okay to skip over harder problems and the reasoning behind them. i also ended calc 2 with a 99 because of your videos. literally CARRYING me through math classes. thank you so much :)

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[@augustoallignani3260]

Lo escribo en español porque me la re banco. Profesor Leonard sus clases han de ser lo mejor que hay de calculo en todo internet. Le estoy muy agradecido por su aporte y su buena disposicion para los videos. Saludos y espero se encuentre bien mi pana.

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[@Peter_1986]

@@lunaeaqua4339 Actually now when I think about it, Michel Van Biezen is probably even better. That guy has LOTS of physics videos on several different topics, and he really helped me out a lot when I studied physics topics such as Thermodynamics and Interference/Diffraction, among other things.

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professor your lectures are remarkably well and you discuss every concept in a very good way and if you add subtitles in your video then your videos become the best math videos on youtube.

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Cant thank you enough for not rushing the class and not chasing quantity over quality

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3:26:00 "60% of the time, it works all the time" killed me on the Anchorman reference

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[@AdreanalineRush]

At 3:22:30 where you are restricted by a circle you can also substitude for x = cos t and y= sin t between -pi

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This style of lecturing could teach multivariate calculus to highschool students. Really nice and thorough presentation!

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Before you watch his whole lifesaving video, please go to 1:33:23, then back to the start and enjoy!

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Just wanted to express my thanks in the span of a month i learned calc 1 2 and 3 in this video here, and managed to pass my math test, that i thought was super hard, easily with a big score .Life saver, big thanks for your work prof !!

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This lecture alone is better than my actual college teacher's career. May God bless you!

[@elam812]

بعد سبعة سنوات اتيت للتعلم من تلك المحاضرات العظيمة والمفيدة ، شكرا لمجهودك

[@videosformybrain3606]

Leonard: today we will be looking at double integrals, first, do you guys remember how to add? alright, let's do a quick review.

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i love it i even stopped watching my zoom math class and only follow his courses to save me somr time and headaches and frustration

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For me, that's the best part. I'm taking calc 3 eight years after I took calc 2, so the review of the basics really helps refresh and solidify!

[@GoScience123]

@@Freedomind901 same I just go to class to mute it and start watching leonard

[@Freedomind901]

@@GoScience123 lmao same .Here is a technic that worked for me so far, i mute myself and record teachers class. then i watch prof leonard class.After that i do the exercices that my teacher showed in class before he does them to see if i understood well. saves me tons of time

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You sir, are a critical component in humanity. I appreciate your brain.

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I love this! I am currently a high schooler taking the equivalent of Calc 1. I clicked on the video out of curiosity and I found it incredibly engaging!

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Calc used to be something I worried about day and night, now it's something I look forward to. I wouldn't have passed calc 2 without these videos, and I wouldn't have the A in Calc 3 that I have now. Thank you so much Professor Leonard!

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bruh i was so confused about the regions bounded by triangles and whatnot, but now i fully understand just because you went step by step and slowly!!!!!!! just overjoyed rn, thank you for enlightening me.

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1:48:42 But wait Leonard that doesn't give you a substitution...."Yea it do.......listen! Yea it do" lmao had me cracking up in the library hahahaha. Thanks once again professor Leonard there's no way I would be able to survive my calc 3 course without you!

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Thanks for the incredible work! It helped me a lot! I got an A in Calculus! I'm from Brazil and I don't speak english fluently, but I could understand almost everything, due to your didactic way to explain.

[@Peter_1986]

You seem proficient enough at English to make yourself very clearly understood.

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i wish i could pay YOU my tuition fees instead of my school

[@ihgnmah]

me too

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do it

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Not the 50 cent dance haha. Professor Leonard you are so awesome! Blessings to you and your family!

[@arianarice4911]

Thanks for these full length lectures. You are extremely easy to follow and thorough, which I appreciate in a math class

[@Dangineering]

I have watched literal days worth of these lectures and I have never heard him inflect his voice this way. This was weirdly entertaining. 3:01:30

[@georgesadler7830]

Professor Leonard, thank you for an exceptional video/lecture on Finding Extrema of Functions of Two Variables. This lengthy lecture is a tremendous amount of mathematical tools and information for any student to absorb in Multivariable Calculus. From watching this lecture, I found this topic highly problematic, however, I will rewatch this video for a deep analysis and understanding. This information was also part of Calculus One that is, finding Max and Min values.

[@tanakabrandonchikuku]

one of the best you make calculus a hobby keep up the good work sir I truly appreciate what you are doing

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Wasn’t planning on watching the whole thing to the end but those first 30 mins really got me hooked! Wow you’re a great professor ❤

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CAN NOT GET ANY EASIER THAN THIS. THANK YOU PROFESSOR LEONARD. YOU ARE A HERO.

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I am literally ready to watch all the advertisements in the world just so you can make money of these lectures. You are too good my man!

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carried me through calc 1, calc 2, and now calc 3. you deserve all my tuition

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Great teacher! You explain much more clear than my professor.

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This is the perfect video for my double major in Dance and Mathematics! /s

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So great so great to see you still going at it! Great videos, wish I could've had you as a teacher.

[@gateclub218]

Best lecture on this topic being an teacher i used to watch professor leonard lectures and teaching techniques too thanks alot professor Leonard

[@Listania]

Are you going to be recording a class on differential equations?

[@Peter_1986]

He now has lots of videos on Differential Equations available. :)

[@trevor6711]

@@Peter_1986 do they cover the entire course?

[@Peter_1986]

@@trevor6711 Not sure, but apparently he is still working on them, even though he already has a bunch of long videos uploaded, so they should cover quite a lot. I definitely know that they cover all the most common ODEs, such as separation of variables, linear ODEs, substitution methods, Bernoulli's ODE and so on.

[@mimortal1000]

@@trevor6711 kzbin.info/www/bejne/i6KwhqKhe7VnaZo, on this channel there is a 44 episode series on diff equations that are very good

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@@Peter_1986 The originial comment was 5 year ago, i think this person has graduated by now XD

[@Peter_1986]

You know how you typically get bored very easily when you watch math lectures? Well, that doesn't happen to me when I watch these videos by professor Leonard - his videos feel more like almost watching an exciting television series.

[@melisaalfonso471]

literally. I watch them before going to be and I do not get sleepy, In fact I cant sleep after watching them lol

[@Peter_1986]

@@melisaalfonso471 I think one important quality of these videos is that Professor Leonard seems to have excellent chemistry with his students, and this makes these videos feel very "alive". Sure, his students can seem quite shy a lot of the time, but they definitely like him as a teacher.

[@mathewkyle8698]

When people ask me who my math professor is I just say professor Leonard. When I pass calculus 3 it will be because of this man.

[@chornnysmathforuni.6402]

I really love your teaching !! Thank you so much professor !!

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A BIG THANKS TO YOU SIR .HONESTLY WITHOUT YOU ME AND MY MATES!!! WE NOTHING TO CALCULUS

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You don’t know how great you are. Thank you so much

[@problemstudios]

'There's this class called calc 1' I lost it lol You sir are my hero.

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Thank you Professor Leonard. This is going to save my Calc 3 grade. I really appreciate these videos!

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I can't describe how much your videos have helped me. Thank you so much sir :)

[@Hustler628]

i love u man I just sat my calc exam and thanks to ur videos, I think I did amazing (which is new for me cause I have always been horrible at maths)

[@naveednaiemi3979]

Thank you Mr Professor....:) God bless you....:)

[@tomctutor]

@3:08:00 Wolfram Alpha can do it in a blink of the eye: type this into Wα query box.. _Extrema {xy-x^2} given y=x^2, y=4_ but don't tell Prof Leonard, it might upset him!

[@rafayimran5732]

Professor Leonard. You the GOAT!

[@bhavana9618]

I wanna hear the concavity song! lol

[@martinong5937]

Thanks sir! Ive been confused about this for a long time and you literally cleared up all my confusions :)

[@SteamPunkLV]

Never thought I'd see the day when Leonard would teach algebra in Calculus Nonetheless, an amazing lecture

[@Imperialogan]

Watched through the whole video, great work. Hope you will continue stay passionate and motivated with your teaching. :)

[@PLP0321]

Nodded my head the whole time. Thanks!

[@ferkahmathiasgyinantwi1106]

Professor Leonard is the best!!!

[@roshinis9986]

at 2:18:00, I solved for the CP at (0,0) and it comes out to be a minima, and now if suppose there were to be a minima at the boundary, there would have had to be a maxima (a turning point of concavity) somewhere between (0,0) and the boundary but since we only got 1 CP, we know there are no other max or min CPs, hence the boundary can't be a minima, and hence (0,0) must be the absolute minima , also the absolute maxima must occur somewhere on the boundary. Can we say this? Is my line of thought right? Also, at 3:03:00 I found the CP along the diagonal line of the triangle by finding the directional derivative and setting it equal to 0 and solving for its intersection with the equation of the actual line joining the two points.

[@sertralydian]

professor leonard is singlehandedly rescuing my junior year math grade

[@hk_ballers]

This is true meaning of maths thank you prof.

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Sir i appreciate your hard work and hope you keep on teaching like this👍

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bro looks like superman fr. A GREAT TEACHER as well

[@jiangminzhu1263]

thank you professor Leonard, you teach so well and so careful that I can understand all the things. you are so interesting. my major is mathematic, you really help me a lot. thanks again. if you can do linear algebra, that could be better, haha. you must the best teacher to teach math.

[@sandlertossone1813]

I have a question about the 3rd tricky system example on 1:14:00. For the fy partial, couldn't you solve for x by adding 2xy to both sides yielding 2xy=-y^2+3y. From there dividing both sides by 2y which gives x=(-y+3)/2 and then subbing that in for x for the fx partial? When I do that I get y=-3 and y=-1 which is different from your values of y.

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Leonard is an absolute BEAST

[@esraamohammed930]

Thank you , for these videos you are so brilliant

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professor Arnold is my immortal hero

[@blb]

My teacher couldn't teach this lesson worth a damn. I was utterly lost. I completely understand the section thanks now, thank you for posting your lectures Prof.

[@ProfessorLeonard]

You're absolutely welcome!! I'm very glad that they helped you to understand

[@Peter_1986]

+Professor Leonard Your lectures flow so well that they don't even feel like lectures, they feel more like exciting adventures.

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I am learning machine learning maths.. and so watching your videos.

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We love you prof Leonard

[@aliatruman]

Jesus, I wish I had any professors in any class who taught this simply. My professors' lectures are almost entirely comprised of proofs. Yay, universityyy.