Calculus 2 Lecture 10.4: Using Polar Coordinates and Polar Equations
Пікірлер: 243
@keldonchase44926 жыл бұрын
Day 1: Introduction to Polar Coordinates: 0:00 First Example (Graphing Polar Coordinates, positive r length): 7:32 Second Example (Graphing Polar Coordinates, positive r length): 10:36 First Example (Graphing Polar Coordinates, negative r length): 13:10 Second Example (Graphing Polar Coordinates, negative r length): 15:16 Brief Review of Graphing Polar Coordinates: 16:33 Introduction to Translating Between Polar Coordinates and Rectangular Coordinates: 17:10 First Example (Translating From Polar to Rectangular Coordinates): 20:28 Second Example (Translating From Polar to Rectangular Coordinates): 24:03 First Example (Translating From Rectangular to Polar Coordinates): 27:35 Second Example (Translating From Rectangular to Polar Coordinates): 33:41 First Example (Converting a Polar Equation into a Rectangular Equation): 39:41 Day 2: Introduction to Symmetry: 46:52 First Example (Checking for Symmetry): 48:42 First Example (Graphing a Polar Graph Symmetric about Polar Axis): 52:54 Second Example (Graphing a Polar Graph Symmetric about Polar Axis and Theta = pi/2) : 1:03:22 Third Example (Graphing a Rectangular Graph Symmetric about the X-Axis and the Y-Axis): 1:26:22 Day 3: Introduction to Finding Tangent Lines of Polar Graphs: 1:37:35 First Example (Finding the Slope of a Tangent Line): 1:46:47 Finding Horizontal and Vertical Tangent Lines to Polar Graphs: 1:59:16
@aubreeswart22066 жыл бұрын
Keldon Chase Thank you!
@noelforde2375 жыл бұрын
Thanks
@kingarthurthe5th5 жыл бұрын
Many thank -last minute crammer
@juanangel78645 жыл бұрын
That's very thoughtful. Thank you!
@Wize2k4 жыл бұрын
god
@opelgtman738 жыл бұрын
I heard China is doing a lot in the way of cloning these days. I nominate this guy to clone himself and replace all math teachers in the U.S. Seriously, if I had this guy for my CALC 1 & 2 classes in college, I would have passed with flying colors. I'm currently studying the night before my CALC 2 final in college right now and this guy has explained exactly what I need. Thank You so much for being an awesome teacher and explaining everything AMAZINGLY!
@himesharora44175 жыл бұрын
Dats a damn good idea
@dominusfons44554 жыл бұрын
Studying the day of the test lol #onlineclasses
@alexnastev24302 жыл бұрын
did you pass LMAO wondering how youre doing in life 5 years later hahah
@EatonArrsenik2 жыл бұрын
@@alexnastev2430 His dead... RIP
@jacklu16112 жыл бұрын
help
@danieljones61323 жыл бұрын
at 1:11:00 I believe the correct trig identity is cos(a-b)= cos(a)cos(b)+sin(a)sin(b), not minus. But the answer is the same. I love these videos and Professor Leonard has given me such a better understanding of math than my previous classes. Thank you sir!
@annika4755 Жыл бұрын
Professor Leonard, because of you I passed Calc 2 and ended with an A-. I know I am one of the many comments that say that but I am truly grateful for you. At the beginning of the semester, I did badly on my first exam and felt that I would never get through calc 2. Then, I started watching your videos and I began to actually understand everything. I had a B- at first and thought I would have to settle for a B, but I was able to get a 95 on the final and never in the world thought I would get an A-!! It could not have been without you! You are very good at teaching and I aspire to teach like you one day. Keep up the humor in the videos too! Thank you for the free amazing content!!!!
@ZnNlove8 жыл бұрын
Didn't know superman was a part time calculus professor ^^
@Bulayla14266 ай бұрын
Superman found a flaw in newton's ideas of gravity which let him fly Why you assuming he wouldn't have mastered the tool to physics
@sirling91267 жыл бұрын
Thank you, Superman. You helped a lot, not all the professors are like you, so we have to find a guy like you to save our math grade.
@Kyleser1BlogspotCa10 жыл бұрын
These are great videos! Though I would like to make a suggestion. Seeing as your videos are pretty long it would be a great idea to put topics and time stamps in the description so that if someone wanted to watch a specific category they could find it easily. Other than that, these are amazing. Thanks.
@ishtiaqhussain77664 жыл бұрын
At 1:08:45 I suppose it’s meant to be a + sign rather than negative (check the identity for cos(a-b)). But it doesn’t make a difference in the result. Lmk if I am mistaken. Prof Leonard! I admire your dedication to teaching. If I ever lecture/teach, I will try to do in your way. Edit: It has been noted in comments by others I realise.
@mohamedmarghine41133 жыл бұрын
YEAH, I was about to comment about that small error. fortunately, it doesn't affect the result.
@gulyashigabutdinova21162 жыл бұрын
Yes, I just wrote that comment too! I'll delete mine :)
@harafos2 жыл бұрын
Thank you for this comment, I thought I was going crazy.
@heyysan9860 Жыл бұрын
Nice catch i was scratching my head & thought I was going crazy!
@ProfessorLeonard9 жыл бұрын
kinkehpanda Good Catch!! I meant + but I must have missed that. Thank you. For everyone else, make a note.
@amre32379 жыл бұрын
+Professor Leonard I've been watching your videos since I was in calc 1, and you're the reason why I've passed that class and why I'm gonna get a good grade in calc 2. thing is, my major requires me to take calc 3 and I was hoping you're gonna make calc 3 videos anytime soon!
@cloudboysmusic52236 жыл бұрын
1:09:20 the cosine identity he used should have a + not a -
@Ali-dn9tl7 жыл бұрын
Amazing videos! Although super long, they're clear, detailed, and super easy to understand. I now watch your video before my professor teaches the section and im killin calc 2. Thank You So Much!
@harmsyamial539214 күн бұрын
the best tutor in the history of the world has ever known, He helped me enormously to get through calculus with a deep and full understanding of the material whilst other teachers and professors are lacking in easing the concept. I found a lot of helpful things in Professor Leonard that I have not seen in any instructor elsewhere,,,, thank yoooooooou Leonard,, I love man
@fartleka71007 жыл бұрын
Hi Leonard. I have been watching your videos for the past few days and you are amazing. I failed my calc exam because I hate studying the material that I don't even fully understand (so I didn't study), but watching your videos has changed my perspective completely. It now makes sense. I'm determined to actually study now that I have these resources. Thank you for making this material available for us struggling students for free (although I'd love it if you were my actual calculus teacher!!!). I actually PAY ATTENTION and ENJOY watching your videos because you explain things in a way that retains its complexity and terminology... but in a simple, step-by-step way. I have already watched 3 of your videos totaling to ~8 hours. I can't even stay fully engaged for a 1.5 hour lecture at my university! That is where the teaching difference is
@ProfessorLeonard7 жыл бұрын
I'm so glad you find them helpful!! I'm also very pleased that you can stay engaged and are learning. Keep it up!
@MasonMcKinney-jx7jp5 ай бұрын
These videos are a miracle from heaven, allowing me to self-study for AP Cal BC in just 3 months as a homeschooler. Anyway, I feel bad by pointing out a silly mistake because of how perfect Prof Leonard always is. But I'll do it for everyone watching in 2024 and beyond. At 1:10:00-1:11:15 he used a subtraction cos identity incorrectly. Leonard accidentally put a minus in between the identity's solution as opposed to a plus. Because cos(a-b) results in cosacosb+sinasinb. Hope this helps!
@echo-channel7710 жыл бұрын
cos(A - B) should be [ cosA cosB + sinA sinB ] but the calc still works out in the demo as 1, i.e. 1-0 => 1+0. Awesome videos!
@winstonava16 жыл бұрын
I KNEW IT
@venkatamarutiramtarigoppul2078 Жыл бұрын
1:03:05 this shows how considerate he is towards his students . Thank you proffesor leonard fo explaining so good
@adamharoon60213 жыл бұрын
My Calc 2 course is covering this topic now. Thank you so much for these wonderful videos Professor Leonard! I never thought I would come to enjoy mathematics.
@vivienne_lavida8 жыл бұрын
Best calc professor ever!! Thank you so much for your handwork and videos!
@OVER9000ORLY8 жыл бұрын
Dude How much money do you want to teach linear algebra
@charmendro5 жыл бұрын
I Hope he does before i take it i the spring semester
@imanabu58624 жыл бұрын
@@charmendro same here 😂😂😂
@ahmedkarem27184 жыл бұрын
Not all the money in the world
@timjjt98344 жыл бұрын
calc 3 my boi
@melanyanguilar56203 жыл бұрын
he actually announced he was gonna film linear algebra videos but idk when:( Covid probably got it delayed. Also, yes I know this is 5 years late!
@UnforsakenXII8 жыл бұрын
Well... I head off to my first calculus class next month and you've already taught me up to calculus II. Thanks a lot, Leonard. Your videos have truly made me love math again. Sincerely, some 18 year old who once despised any form of math.
@humayra87032 жыл бұрын
feeling blessed because I've found the best teacher
@aizadabadiyeva7828 жыл бұрын
Professor Leonard, thank you so much for making these videos! They're very helpful. One thing to ask, do you plan to do videos for Calculus III and Linear Algebra? Thank you in advance.
@thebassofmontecristo9 жыл бұрын
The "buttioid". The best thing I got from this video! :) Just kidding! Fantastic work!
@satyamverma19105 жыл бұрын
Yes lol
@loic22058 жыл бұрын
This chapter is so much easier than the previous chapter... dumb series. I have my exam tomorrow and im really confident :) Thank you so much :D
@OVER9000ORLY8 жыл бұрын
You are a godsend my teacher blasted through this in a day and I was so lost Thank You so much! I'm so happy I found your channel
@tutahosman83804 жыл бұрын
Pro.Leonard, I sincerly have to say you have a a gift of teaching math!!
@Matchbox20100010 жыл бұрын
I really enjoy watching these lectures on calculus that you made, I have studied Convex Analysis, Advanced Calculus II a Real Analysis Approach and Multidimensional Analysis. I love math, thanks for making these lecture videos their great.
@marjavanderwind42514 жыл бұрын
Hi @Professor Leonard I think I see a mistake at 1:10:06 because the cosine addition formula for cos(x - y) is with a plus-sign in the middle instead of a minus sign. In this case it doesn't matter because of the sin(pi) becoming zero, but it might be relevant for you guys.
@marjavanderwind42514 жыл бұрын
Oh wait this thing was said by others before.
@montsamaisamamuka22938 жыл бұрын
Incredible staff, this video just saved my semester.
@dildobaggins27593 жыл бұрын
lol buttioid cracked me up...
@ShivamYadav-ys3bp2 жыл бұрын
Thank you sir thanks🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏
@alexprocel917 жыл бұрын
Show up hands if you feel ok with this one!
@spongebobtentacles24585 жыл бұрын
i find it disrespectful how his students start packing up (loudly) before he even plans to end the class
@KhadijaAlshemeili4 жыл бұрын
ikr:)
@sla-hf1op3 жыл бұрын
these guys don't deserve this maniacly handsome professor i should be sit there i would never stand up til my ass stick the chair
@sla-hf1op3 жыл бұрын
thus he'd have to hold me to save me
@iamasian3296 жыл бұрын
Hi professor Leonard, just wanted to say your videos are truly amazing and helpful on many different levels. I used to just watch math videos on khanacademy which are great, but I tend to watch your videos when I feel I don't understand what Sal on khanacademy teaches. I'm not exactly sure if you're able to make more videos since you mentioned in your channel trailer that you might do something else, but I was wondering if you're able to make a video series on introductory Physics that involves calculus? Once again, thanks a lot for your hard work and dedication!
@IlhamAzlanSyah8 жыл бұрын
1:09:14 In the cosine identities, the sign of cos becomes negative, doesn't it? But it doesn't matter because sin 2 pi equals to zero.
@edwardhaughney96658 жыл бұрын
I like the soft spoken manner of these lectures. Surprising from one wearing a superman shirt- but quite effective. Can see you playing the superman role in Hollywood. thanks
@godsonUnderscoreY5 жыл бұрын
Dude, you're a Savior!!!! Thank you so much!!
@ahmadelbliwi56266 жыл бұрын
I dont speak english fluently but i really have understood all what you said thanks Follower from jordan
@georgesadler7830 Жыл бұрын
Professor Leonard, thank you for an excellent Introduction to Polar Coordinates and Polar Equations in Calculus Two. This material is a solid review of Precalculus and is very helpful in Multivariable Calculus. Graphing Polar Equations is not difficult; however, I found the examples a bit confusing. Problem solving will increase my knowledge and deep understanding of the material. This is an error free video/lecture on KZbin.
@johnholme7832 жыл бұрын
Yes I understand, it’s the chain rule. my old maths teacher used to come into class stinking of booze! You could see that he was just going through the motions, unlike professor Leonard who has a real passion for the subject! Those students are very lucky to have a professor like him. If you can’t pass in Leonard’s class you’ve got no aptitude at all for maths! ,
@rmdir Жыл бұрын
That teacher was me, and I needed the booze after dealing with you stupid kids. But I have to say, Leonard is a better teacher than I ever was.
@saeedsaleh6873 Жыл бұрын
Hi Professor Leonard, I'm a big fan of you, I've just noticed a little mistake with the subtraction identity for cosine in 1:09:00 , it should be 2[Cos(2pi)Cos(2theta) + Sin(2pi)Sin(2theta)], luckily that mistake didn't change or effect the answer.
@davidlunacruz11504 жыл бұрын
The greatest professor of all time
@animeshmajumdar663 жыл бұрын
At 1:09:36 i think there should be a '+' sign in the identity and not ' -' 🙂🙂🙂
@dandyremix23607 жыл бұрын
I can't thank you enough. Great work.
@fraze12288 жыл бұрын
Hi Prof Leonard. Love your lecture. I think that the symmetry about the pole should be f(theta) = - f(theta) or f(pi + theta) = f(theta) instead. Just to check. Thanks a lot.
@ShivamYadav-ys3bp2 жыл бұрын
Thank you sir thanks🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🌹🌹🌹🌹🌹🌹🌹🙏🙏🙏🙏🙏🙏🌹🌹🌹🌹🌹❤❤❤❤❤🌹🌹🙏🙏🙏🌹❤❤❤❤❤❤❤
@jacksonr2603 ай бұрын
I love when he says, "I'm gonna go fast and not explain it all" then proceeds to give a fairly thorough explanation 🤣 much preferred to the other way around!
@Kybeline7 жыл бұрын
Thanks that you care about us, who watch your videos, too :)
@deepakbeniwal48464 жыл бұрын
Thanks a lot SIR ...your teaching method is very excellent ...."""""""".LOVE """FROM ""'"""""INDIA"""""'''"""
@sagarsinghbk3 жыл бұрын
Thank you, professor Love from Nepal
@westonmulenga64218 жыл бұрын
i wish i knew abt your Tutorials sometimz back,but i think you helping me here,you have thumb Sir
@adamsam31868 жыл бұрын
hope you'll teach liner algebra, or differential equation. I couldn't find a teacher who teach math same as you thank you for making these videos
@avery4528 Жыл бұрын
Teacher workshops should be required every summer by the state just like for accounting and real estate. "These kids are so stupid today" No, you've lost your ability to relate.
@hindalamri90267 жыл бұрын
شرحححههه رهيبب 👍🏻
@tybargky35544 жыл бұрын
Professor, I thought the subtraction Identity for cos(A-B)= cosAcosB+sinAsineB? correct me if i'm wrong. Thank you
@mohammedalameri86508 жыл бұрын
best instructor ever
@rupaktiwari51952 жыл бұрын
Thank You Professor Legend!
@Emma-hg1vi3 ай бұрын
From now on my jesus is professor leonard I would fail calculus without you but now I get the highest point in class
@WaterDrinkersUnited10 жыл бұрын
Clark Kent + George McFly = Professor Leonard
@adnanmohamed65173 жыл бұрын
@Professor Leonard I think there is a mistake in the video. Look the board @48:50 (or in the minute 48). Point 3 of symmetry says: If f(theta + pi) = - f(theta) then symmetric about the origin. But I think the negative sign should not be there in the R.H.S. The correct equation in my opinion is: f(theta + pi) = - f(theta) (without the negative)
@fromfilmtocode4 жыл бұрын
Best video on the matter I found as a newbie.
@ucbuixuan99405 жыл бұрын
At 1:08:31, can you just think of 2π as a full circle and get rid of it? Hence: 2cos(2π-2θ) = 2cos(-2θ) = 2cos(2θ)
@jaxonrholden4 жыл бұрын
In metrology when converting between coordinate systems, I use theta = 2*arctan[y/(x+r)] which automatically gives the angle in the correct quadrant without adding pi.
@vk62974 жыл бұрын
plz explain @professor leonard the symmetry rule is not working for r=theta sin(theta)
@malevip7 жыл бұрын
@ProfessorLeonard, You have a mistake at 1:10:34 cos(A-B) = cosA cosB + sinA sinB not minus..
@Racoalexandra3 жыл бұрын
yesss
@abdullahmohammad56137 жыл бұрын
An easy way to approach the right graph of the polar segment when it's negative is to add/subtract 180 Deg for the given angle :)
@user-mx6qr2id7e7 жыл бұрын
What a great job 👍🏻
@edwardhaughney96658 жыл бұрын
Pretty interesting/cool presentation. If you think you can suceed in math then you are well on your way. These videos assist and your manner of instruction imparts a "can do attitude." thanks
@shutnawi6635 Жыл бұрын
thank you so much you have been very helpful
@kqnrqdtqqtttel17783 жыл бұрын
1:08:32 Just note that cos has a period of 2pi, so adding 2pi to the angle -2theta and taking the cos gives the same result as simply taking the cos of -2theta, or the cos of 2theta.
@BaySelvi3 жыл бұрын
Identity function is written wrong at(1:09:46) It must be written as 2[cos(2π)cos(2θ)+sin(2π)sin(2θ)] However it gives the same result because sin(2π) is 0 and it makes the sine part of the identity 0 in both cases whether its sign is minus(-) or not
@mathprof77766 жыл бұрын
Thank you professor Leonard.
@constantinknab4 ай бұрын
PERFECT VID BEFORE THE TEST!!!
@KiNgS4EvAful7 жыл бұрын
you always save my ass not all hero's wear capes!!!!
@amiryamini30758 жыл бұрын
Hey Professor Leonard! Your videos are amazing!! Will you be doing Linear Algebra?
@quant-prep28433 жыл бұрын
who is that laughing girl ?? I really need to meet that girl
@FuriousJones8 жыл бұрын
I sneezed at 3:04 and thought Professor Leonard was saying 'bless you' to me.
@Peter_19864 жыл бұрын
I once texted on my phone and then Professor Leonard suddenly looked at me and exclaimed "stop texting!", lol. I think he did it in one of the videos about Geometric Series in this playlist.
@seabasschukwu6988Ай бұрын
@@Peter_1986 brother u r everywhere
@sameterdogan18205 жыл бұрын
worth needs men like you, proffessor...
@kinkehpanda9 жыл бұрын
The trigonometric identity at 1:10:10 is slightly incorrect. It should be [cos2pi cos2thetha + sin2pi sin2thetha]. It didn't affect the outcome this time because the right term was zero. NVM it was mentioned in the comments already.
@greense657 жыл бұрын
But you added the timestamp, which was missing from the earlier comment identifying this small error. Thanks.
@satyamverma19105 жыл бұрын
Thanks mate!! I was wondering the same
@yasminasabra88754 жыл бұрын
yes true
@jeramibellofatto8177 жыл бұрын
2Cos2(2pi - 2theta) = 2[Cos(2pi)Cos(2theta) + Sin(2pi)Sin(2theta)] In this case the second term cancels out but would matter for other problems. 1:10:17
@littleprincess2502 жыл бұрын
Sir are u form isi ???
@kristinewaite75399 жыл бұрын
I am very thankful for all your help. Do you do Calculus 3, or differential equations, linear algebra ?
@robertflores55207 жыл бұрын
Post your workout routine
@Salamanca-joro4 ай бұрын
Tip for you guys , If you get theta as pi over something or in pi and you want to convert it to degrees use this : Multiply it by 180/pi For example this example 26:08 To find x we use this formula X=rcostheta Since theta was pi over 4 convert it to degrees Pi/4 times 180/pi Is 45 So x=-sqrt2 times cos 45 Which equals -1 And you can do plug 45 for sin as well to find y and you will get same answer -1 Try to do it with the other examples and you will see how good this method is
@sidsingh93877 жыл бұрын
When you increased the "size" of the origin so the poorly drawn curves pass through the point - I do that every single time too! :>
@sandeepgrover3957 жыл бұрын
very good lecture. Thanks a ton sir! :)
@Inspired_Cat5 ай бұрын
Just to inform one minor error (1:08:46)... The result was still correct since it was zero but the subtracting identity of cos is positive at the center sign. Not negative.
@hopefulee84028 жыл бұрын
Is this a High school class or a college class you have when you film these calc 2 videos?
@kyledavies38718 жыл бұрын
College, but it seems like high school
@p0intblAnkwaziT3 жыл бұрын
@@kyledavies3871 why does it seem like high school?
@M5xThully9 жыл бұрын
Thanks for making these videos.
@tshepomohlomi96347 жыл бұрын
THANKS , Professor Leonard,! we're waiting for you at university of kwa zulu natal in South Africa, please come!!!!!
@faiqfarhan23099 жыл бұрын
May I Know Where Can I Download The Polar Coordinate Graph Just Like The One On This Video?
@amre32379 жыл бұрын
+Faiq Farhan Just google polar coordinate graph paper fam
@ChrisChoi1235 жыл бұрын
the last thing you evaluated, the slope at pi/6, could be simplified to -1.
@anonymoususer43562 жыл бұрын
thank you
@anthonysantiago60177 жыл бұрын
hey professor, your sum and difference identity was actually incorrect but you got the right answer. its cos cos sin sin then change sign for cos.
@MrCoolrohitb977 жыл бұрын
Sqrt(2)=1.414 but you said 1.7 at 1:16:20 anyway love your videos, they are really helpful. Your efforts are appreciated. 😃
@eng18974 жыл бұрын
thanks so so much ❤❤❤❤❤❤❤
@shahabasov7627 Жыл бұрын
35.00 why not in clockwise ..ı dont understand
@CristianRodriguez-ct6df8 жыл бұрын
This motivates me to workout my biceps
@Festus20224 жыл бұрын
I think Leonard made a mistake at 1:55:20 when he "factors out a negative". I think the numerator should be -cos2theta MINUS cosTHETA. This changes the result.
@Festus20224 жыл бұрын
Correction--> Dr. Leonard was correct. I am wrong!!
@justinli199010275 жыл бұрын
seriously awesome teacher
@wildchild11615 жыл бұрын
@ Professor Leonard @1:08:59 your identity is wrong , Cos(a-b) = Cos(a)Cos(b)+Sin(a)Sin(b) . It doesn't affect the out come though because your sin value ends up being 0
@ProfessorLeonard5 жыл бұрын
Yep, I seem to remember watching this back a while ago and thinking the same thing. Good catch!