AA AI Jungwon
AA AI Jungwon
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Bernoulli Numbers & Tangent: Journey through IB P3 and Textbook
00:00 Introduction
01:28 Importance of Learning with Books
02:33 Appearance of Hyperbolic Function in books
04:02 Complex Numbers and Hyperbolic Trigonometry
14:15 Power Sum and Bernoulli Numbers
18:52 **Animated Proof**
28:02 Generating Function For Bernoulli Numbers
40:05 General Expression of tangent for Maclaurin Series
In this video, we dive deep into advanced mathematical concepts, exploring how IB students can make the most of their syllabus by combining various topics to derive something truly beautiful. We’ll explore:
• IB exam questions and how they can unfold into a potential IA topic.
• The fascinating world of Bernoulli numbers and their relation to the general formula of the tangent.
• The Maclaurin series of the tangent and how it ties into the broader world of hyperbolic functions and complex numbers in trigonometry.
• The power sum and its connection to Bernoulli numbers.
• A generating function that appears in textbooks, allowing you to derive Bernoulli numbers as coefficients of the Maclaurin series of that generating function.
• How these different pieces come together to reveal the general expression of the tangent using the hyperbolic tangent Maclaurin series and trigonometric identities.
This is an amazing opportunity for IB students to combine key elements of their syllabus and enjoy the beauty of mathematics at a deeper level. Don’t miss out on this mathematical adventure!
이 비디오에서는 고급 수학 개념을 깊이 탐구하며 IB 수학 과정에서 다양한 주제를 결합하여 아름다운 수학적 발견을 이루는 방법을 알아봅니다. 다룰 내용은 다음과 같습니다:
• IB 시험 문제가 잠재적인 IA 주제로 어떻게 전개될 수 있는지.
• 베르누이 수와 그와 관련된 탄젠트의 일반 공식.
• 맥클로린 급수와 탄젠트의 관계, 그리고 쌍곡선 함수와 복소수의 삼각함수와의 밀접한 연관성.
• 멱합과 베르누이 수와의 관계.
• 생성 함수를 통해 맥클로린 급수의 계수로 베르누이 수를 도출하는 방법.
• 쌍곡선 탄젠트 맥클로린 급수와 삼각함수 항등식을 이용해 탄젠트의 일반 표현식을 도출하는 방법.
IB 학생들에게 수학 과정을 심도 있게 결합하여 수학의 아름다움을 경험할 수 있는 훌륭한 기회입니다. 놓치지 마세요!
#IBMath #BernoulliNumbers #MaclaurinSeries #TangentFunction #ComplexNumbers #Trigonometry #HyperbolicFunctions #PowerSum #GeneratingFunction #MathIA #IBStudents #MathSyllabus #IBExam #수학 #베르누이수 #맥클로린급수 #삼각함수 #복소수 #쌍곡선함수 #IB수학 #수학IA #수학과정 #탄젠트
Переглядів: 45

Відео

Second-Order Differential Equations using IB Mathematics: P3 and Textbook
Переглядів 21Місяць тому
“Mastering Second-Order Homogeneous Differential Equations: Techniques & Real-Life Applications” 00:00 Introduction 00:21 Importance of Learning with Books 03:00 Solving Second Order Differential Equations: Using Maclaurin Series 12:55 Solving Second Order Differential Equations: Using Auxiliary Equations 20:53 Solving Second Order Differential Equations: Using Laplace Transformation 35:15 Conc...
IB Complex Numbers: Exploring Maclaurin & Fourier Series with Exam Problems
Переглядів 63Місяць тому
00:00 Introduction 00:51 2011 Nov Question: Complex and Geometric 06:00 Trigonometric Sum from 2016 Nov 10:20 Hyperbolic Function 17:33 Fourier Series 23:30 Maclaurin and Fourier From Complex Number 30:24 2009 May Question: Complex and Logarithm 40:50 Sawtooth Function (Fourier Series) 43:33 Arctan(x) (Maclaurin Series) 48:44 Conclusion Dive into the fascinating world of complex numbers and dis...
Uncovering the Beauty of IB Paper 3: From Riemann Sums to the Gamma Function!
Переглядів 1,5 тис.Місяць тому
00:00 Introduction 00:22 P3 Problems from 2022 & 2023 02:37 Riemann Sum: Surface Area I 13:26 Riemann Sum: Surface Area II 31:08 Time Functional 38:04 Gamma Function 44:41 Gamma(1/2) 49:01 Riemann Sum: Double Integral I 58:34 Fubini's Theorem 01:01:20 Riemann Sum: Double Integral II 01:09:22 Gaussian Integral 01:13:05 Stirling's Formula 01:30:46 Conclusion #ibmath #ibmathematics #analysis_and_a...
방원지니: Surface Area Formula
Переглядів 382 місяці тому
00:00 Surface Area Formula 01:22 Attempt with rectangle 06:20 Attempt with better rectangle 14:14 Substitution for x and y 18:00 Example with Sphere 22:05 5th Application with Time 27:54 Conclusion
AA HL: Integral and Complex Number
Переглядів 2,4 тис.4 місяці тому
00:00 Introduction 00:13 Question: Integration 04:42 Visualisation of the problem: n=2 06:44 Visualisation of the problem: n=4 09:44 Visualisation of the problem: n=3 13:28 Roots of Unity with n=5 16:55 Fundamental Theorem of Algebra 18:23 Solution for n=5 #ibmath #ibmathematics #analysis_and_approaches #applications_and_interpretation #mathematicsvideo #revision #complexnumbers #cotes #rootsof...
Untangling Trigonometry and M24 Limit Problem
Переглядів 2,7 тис.4 місяці тому
00:00 Introduction 00:53 Pythagorean Identity and unit circle 02:29 Double angle formula and unit circle 05:44 Compound angle formula and unit circle 11:11 Exam Question (M11) 15:02 Squeeze Theorem 15:51 Exam Question (N23) 17:24 First Principles on Sine 22:03 Fundamental Limits of Trigonometry 25:36 Exam Question (M21) 26:46 Exam Question (M24) #ibmath #ibmathematics #analysis_and_approaches #...
l'hopital's Rule and Fundamental Limit of Trigonometry
Переглядів 9205 місяців тому
l'hopital's Rule and Fundamental Limit of Trigonometry
Legendary Problem: Arctan and Ramanujan series
Переглядів 2 тис.5 місяців тому
Legendary Problem: Arctan and Ramanujan series
IB school life: IA and Crit T
Переглядів 9395 місяців тому
IB school life: IA and Crit T
Last Minute Revision for AA P3
Переглядів 2,3 тис.6 місяців тому
00:00 Introduction 01:12 Use of Calculator 06:06 Learning Phase 13:05 Proving Phase 22:10 What will come out? 25:38 P3 GOAT from Wuhan 26:53 Basel Problem 28:13 1. Basel by DMT and FTA 46:25 2. Basel by Series 57:16 3. Basel by Maclaurin Series 01:08:24 4. Basel by Reduction Formula 01:37:39 Conclusion #ibmath #ibmathematics #analysis_and_approaches #complexnumbers #mathematics #gauss
AI: Matrix Trilogy
Переглядів 2,2 тис.6 місяців тому
#ibmath #ibmathematics #analysis_and_approaches #complexnumbers #mathematics #gauss #matrix #applications_and_interpretation
Chicken or Egg: Exponential and Complex Numbers
Переглядів 1,2 тис.7 місяців тому
#ibmath #ibmathematics #analysis_and_approaches #complexnumbers #mathematics #gauss
N23 AA HL: Question 12 and Heptadecagon Constructability
Переглядів 3,4 тис.7 місяців тому
00:00 Introduction 00:18 Geometry 02:55 Complex Number exam questions 04:29 AA HL N23 Q12 05:59 Exam questions 17-gon Constructability 19:38 Chebyshev Polynomials 29:54 M16 Complex Number 41:00 Quadratics #1 45:25 Quadratics #2 47:49 Quadratics #3 55:22 Construction in Animation 58:04 Conclusion #ibmath #ibmathematics #analysis_and_approaches #complexnumbers #mathematics #gauss
(e)mportance of being (e)
Переглядів 1,3 тис.9 місяців тому
Video Chapters 00:00 Introduction 00:37 super star e 01:37 Aim 02:05 Topic 1: Convergence 17:03 Topic 5: Derivative of e 21:25 Euler’s Identity 23:04 Maclaurin Series 28:42 Maclaurin Theorem 31:45 Topic 1, 3 and 5: Euler’s Identity Proof 32:23 Topic 4: Binomial to Poisson 42:31 Topic 4: Poisson to Normal 1:00:07 Topic 1: Irrationality of e 1:14:04 Conclusion #ibmath #ibmathematics #analysis_and...
국제학교 수학생활: 영어의 중요성 (Defence Against The Dark Arts)
Переглядів 869 місяців тому
국제학교 수학생활: 영어의 중요성 (Defence Against The Dark Arts)
IA: Sell Me This Pen
Переглядів 91610 місяців тому
IA: Sell Me This Pen
IB Fokedex: Half Angle Formula (Visual Demonstration)
Переглядів 5410 місяців тому
IB Fokedex: Half Angle Formula (Visual Demonstration)
IB Fokedex: Double Angle Formula (Visual Demonstration)
Переглядів 7910 місяців тому
IB Fokedex: Double Angle Formula (Visual Demonstration)
IB Fokedex: Sum of Arctangents (Visual Demonstration)
Переглядів 3110 місяців тому
IB Fokedex: Sum of Arctangents (Visual Demonstration)
IB Fokedex: Integration by Parts (Visual Demonstration)
Переглядів 12310 місяців тому
IB Fokedex: Integration by Parts (Visual Demonstration)
IB Question to solve AMC 12A Q25 (Complex Numbers)
Переглядів 10710 місяців тому
IB Question to solve AMC 12A Q25 (Complex Numbers)
Paper 3 vs IA: Gamma Function from May 2023
Переглядів 1,2 тис.Рік тому
Paper 3 vs IA: Gamma Function from May 2023
2023 May AI HL Exam Review: USA
Переглядів 81Рік тому
2023 May AI HL Exam Review: USA
2023 May AA SL Exam Review: Doppelgänger
Переглядів 56Рік тому
2023 May AA SL Exam Review: Doppelgänger
2023 May AA HL Review (Last Minute Revision Works)
Переглядів 202Рік тому
2023 May AA HL Review (Last Minute Revision Works)
Paper 3 Style: Hyperbolic Function and Integration
Переглядів 938Рік тому
Paper 3 Style: Hyperbolic Function and Integration
Integration by Parts and Reduction Formula (Teacher's Love)
Переглядів 118Рік тому
Integration by Parts and Reduction Formula (Teacher's Love)
Maths IA: Tom Riddle Diary and Severus Snape
Переглядів 120Рік тому
Maths IA: Tom Riddle Diary and Severus Snape
IA Class 2: How to Write for IA
Переглядів 248Рік тому
IA Class 2: How to Write for IA

КОМЕНТАРІ

  • @adrenalinejunki3e
    @adrenalinejunki3e 8 днів тому

    Thank you so much, Mr. Jungwon!! Just subscribed! Could you also do the same but for the november 2024 exam session?

    • @AAAIJungwon
      @AAAIJungwon 7 днів тому

      Thank you too. What topic do you want to go over 👨🏻‍🏫 ?

    • @adrenalinejunki3e
      @adrenalinejunki3e 7 днів тому

      ​@@AAAIJungwon Preferably both Calculus and Functions ( Algebra and Statistics too, if it's possible ) Thankss! :))

  • @blacksky492
    @blacksky492 3 місяці тому

    I GOT A 7!!!!! WITH 78 LETS GOOO TEACHERRR

    • @AAAIJungwon
      @AAAIJungwon 3 місяці тому

      Congratulations, my friend. Your remarkable enthusiasm for learning has earned you a well-deserved score of 7. Be proud of this achievement and enjoy your summer 👨🏻‍🏫

  • @yiniagarcia508
    @yiniagarcia508 3 місяці тому

    ❤ The best!!!

  • @HanaHashemi-jy6sd
    @HanaHashemi-jy6sd 4 місяці тому

    does this video include the hl topics as well?

    • @AAAIJungwon
      @AAAIJungwon 4 місяці тому

      Yes it covers all the topic in the syllabus and I’m going through them with examples from the actual exam 👨🏻‍🏫

    • @HanaHashemi-jy6sd
      @HanaHashemi-jy6sd 4 місяці тому

      @@AAAIJungwon that's great! will you be making one for the May 2025 exam?

  • @baqi1
    @baqi1 4 місяці тому

    Hello sir where are these questions from?

    • @AAAIJungwon
      @AAAIJungwon 4 місяці тому

      They are mainly from IB exams 👨🏻‍🏫

  • @blacksky492
    @blacksky492 5 місяців тому

    Did you see paper 3, what was your thoughts?

    • @AAAIJungwon
      @AAAIJungwon 5 місяців тому

      Did you have question with 1. Differential Equations 2. Quadratics and Distribution?

    • @blacksky492
      @blacksky492 5 місяців тому

      @@AAAIJungwon yeah some derivative stuff and also quadratics with dice

  • @Zawgs
    @Zawgs 5 місяців тому

    If this shows up in tomorrow’s exam I am beyond cooked

    • @AAAIJungwon
      @AAAIJungwon 5 місяців тому

      Only try to gain the deduction skills my friend 👨🏻‍🏫. Remember, always try to see if you already know the answer by glancing through the questions 👨🏻‍🏫

    • @flooberr
      @flooberr 5 місяців тому

      real

  • @sadie9205
    @sadie9205 5 місяців тому

    bruh this came up after I took p2 lol 💀💀💀💀💀💀

  • @mandygoat8493
    @mandygoat8493 5 місяців тому

    oh i love you

  • @blacksky492
    @blacksky492 5 місяців тому

    MR JUNGWON, APPARENTLY THE PAPER GOT LEAKED, WHAT WILL IB DOOO. IM SO SCARED THE BOUNDERIES WILL BE BAD. I HOPE THEY MAKE US DO ANOTHER PAPER 1

    • @AAAIJungwon
      @AAAIJungwon 5 місяців тому

      It has always been leaked. I graduated a school in Germany back in 2008 and even then it was a thing. I thought they've fixed it with the timezone A, B and C implementation? I don't think they would do jack-sh*t. Leakage happens in almost every exam in the world 👨🏻‍🏫

    • @blacksky492
      @blacksky492 5 місяців тому

      @@AAAIJungwon that’s really sad that it happens. But I aced paper 2, thanks teacher 💪 (only didn’t have time to do a question cause my calculator refused to define the function and confused myself on some other question so decided to skip after doing some work)

    • @AAAIJungwon
      @AAAIJungwon 5 місяців тому

      @@blacksky492 Well done 👨🏻‍🏫 Please continue to revise for P3 👨🏻‍🏫

  • @blacksky492
    @blacksky492 5 місяців тому

    The final countdown has arrived.

  • @UnbrokenMango21
    @UnbrokenMango21 5 місяців тому

    Hey! Loved the vid! Could you fix the PPT link please!

  • @tamimalkawari271
    @tamimalkawari271 5 місяців тому

    i still didn't start studying is this video enough for me to get a 5? is this for sl or hl too?

    • @AAAIJungwon
      @AAAIJungwon 5 місяців тому

      It's for both 👨🏻‍🏫

  • @whoami5590
    @whoami5590 6 місяців тому

    chat is this indeed the goat?

  • @NoIax
    @NoIax 6 місяців тому

    I'm a SL student, barely managing to scrape by with a 3 before the exams, is this like a more difficulty appropriate revision guide or video, or is this the best one. Just thinking there is a possibility that this is far beyond my understanding

  • @blacksky492
    @blacksky492 6 місяців тому

    7:48 why does Pythagoras identity come in?

    • @AAAIJungwon
      @AAAIJungwon 6 місяців тому

      modulus z implies we gotta find the length, and e^(ix)=cis(x), so we find the modulus by putting into pythagoras theorem.

    • @blacksky492
      @blacksky492 6 місяців тому

      @@AAAIJungwonthank you I see it

  • @blacksky492
    @blacksky492 6 місяців тому

    1:35:53 why does it need to be positive when sin is negative even though it should be positive

    • @AAAIJungwon
      @AAAIJungwon 6 місяців тому

      Because you are doing something INSIDE cosine, so we have to watch out the quadrant for cosine. When we do x-pi/2 it's in the 4th quadrant, holding cosine to be positive. So even if it converts to sine by complementary angle, the expression should remain as positive. For example, what do you think about sin(-13pi/2+x)?

    • @blacksky492
      @blacksky492 6 місяців тому

      Ah ok I get it, using the same bounderies, since sin is - in 4 quadrant then it is the same as negative cosine x, since 13pi/2 is same as pi/2

    • @blacksky492
      @blacksky492 6 місяців тому

      @@AAAIJungwon by the way which is your fav course between ai and aa, and which is more challenging in your opinion?

    • @AAAIJungwon
      @AAAIJungwon 6 місяців тому

      yep that's right. odd pi/2 = always complementary, so please convert. THEN decide whether +/- depending on the quadrant you have shifted. does not matter acute or obtuse.@@blacksky492

    • @AAAIJungwon
      @AAAIJungwon 6 місяців тому

      My heart says everyone SHOULD take AI in this era. What students will be learning in college will be so much more relatable with AI syllabus. AA is too old fashioned and they no longer have to put too much efforts working on hard AA style problems. BUT this does not mean every school and teacher has been trained for teaching AI well. I think it would take at least 7 years to circulate AI syllabus well for global standard, so at the moment conventional AA path would be better off for general students. BUT again, AI all the way if students consider for their knowledge to be in zeitgeist.@@blacksky492

  • @clauviol1403
    @clauviol1403 6 місяців тому

  • @probaye
    @probaye 6 місяців тому

    i love this please make more!

  • @blacksky492
    @blacksky492 6 місяців тому

    YESSSS I LOVE YOU

  • @blacksky492
    @blacksky492 6 місяців тому

    Yo this video is golden

    • @AAAIJungwon
      @AAAIJungwon 6 місяців тому

      Thank you 👨🏻‍🏫

  • @yumoli-mi5yi
    @yumoli-mi5yi 6 місяців тому

    You are the greatest of all time!! Thank you for posting this quality information, the best I've seen.

    • @AAAIJungwon
      @AAAIJungwon 6 місяців тому

      Thank you liking. Please share with your friends 👨🏻‍🏫

  • @weronikabudzynska
    @weronikabudzynska 7 місяців тому

    Is it also for higher level? What do you think is the best way to split your video to prepare well for the May 2024 exams?

    • @AAAIJungwon
      @AAAIJungwon 7 місяців тому

      Yes it is for higher level. Going topic by topic is good 👨🏻‍🏫

  • @Abhi-yh5mo
    @Abhi-yh5mo Рік тому

    hi! any tips for aa p3?

    • @AAAIJungwon
      @AAAIJungwon Рік тому

      drive.google.com/open?id=11b-kKT2P2jrQQLuwu0af7mdnRsLAtl9x&usp=drive_copy I was making the video but it was taking so much time. I organized some ideas that could possibly come up for P3

    • @Abhi-yh5mo
      @Abhi-yh5mo Рік тому

      @@AAAIJungwon thanks!

    • @AAAIJungwon
      @AAAIJungwon Рік тому

      @@Abhi-yh5mo how was p3 bud?

    • @Abhi-yh5mo
      @Abhi-yh5mo Рік тому

      @@AAAIJungwon Definitely was tough but it was very doable.The first question went quite well but the second one was kind of hard

  • @davidwong7234
    @davidwong7234 Рік тому

    bro your content is superb!

    • @AAAIJungwon
      @AAAIJungwon Рік тому

      Thank you. I made a AI HL revision video just now: ua-cam.com/video/6ELsltdbAW0/v-deo.html in case you have some friends revising

    • @AAAIJungwon
      @AAAIJungwon Рік тому

      Hello, I've made a video on the use of calculator. Hope it helps for your exam tomorrow. ua-cam.com/video/PvV_K5F8mOU/v-deo.html