How math majors use the intermediate value theorem! @blackpenredpen

Uploaded January 2026 | Updated September 2026, 2 weeks ago

Let f be a continuous function on [0, 1] and f(0)=f(1). Show that, for n=2, 3, 4, ..., there is some a in [0, 1-1/n] such that f(a)=f(a+1/n). Learn how to use the Intermediate Value Theorem to prove the above statement.

This question appears on the transfer entrance exam at National Taiwan Univiersity: https://exam.lib.ntu.edu.tw/sites/default/files/exam/undergra/102/102018.pdf

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How math majors use the intermediate value theorem!

$100 Algebra Problems for Calculus Here are 100 algebra simplification problems you need for calculus. Designed for AP calculus or college calculus students, this tutorial covers a wide range of topics including factoring trinomials, polynomial long division, synthetic division, logarithmic functions, exponential functions, and more. Youll also learn techniques for simplifying complex expressions and solving partial fraction decompositions. Whether youre looking to improve your math skills or ace your calculus exams, this video has everything you need. #APCalculus #CollegeCalculus #AlgebraProblems #math #blackpenredpen the file: https://www.patreon.com/posts/123995806 BIG THANKS TO Alex Zyryanov for the timestamps! 0:00 the extreme algebra workout for calculus! Q1 to Q10 factor & cancel 0:46 - Q1 2:05 - Q2 4:33 - Q3 6:33 - Q4 9:39 - Q5 11:52 - Q6 15:12 - Q7 23:51 - Q8 25:40 - Q9 27:24 - Q10 Q11 to Q20 expand & cancel 30:21 - 11 39:06 - 12 43:29 - 13 45:40 - 14 49:29 - 15 53:19 - 16 55:48 - 17 58:52 - 18 1:01:12 - 19 1:05:00 - 20 Q21 to Q30 using the conjugate for square root expressions 1:08:15 - 21 1:11:19 - 22 1:13:27 - 23 1:17:00 - 24 1:19:35 - 25 1:22:05 - 26 1:25:22 - 27 1:26:41 - 28 1:30:19 - 29 1:33:31 - 30 Q31 to Q40 complex fractions 1:36:13 - 31 1:39:32 - 32 1:42:30 - 33 1:44:53 - 34 1:47:32 - 35 1:49:56 - 36 1:51:41 - 37 1:54:34 - 38 1:56:36 - 39 1:59:34 - 40 Q41 to Q50 LCD & combine 2:03:16 - 41 2:05:04 - 42 2:07:43 - 43 2:10:27 - 44 2:13:03 - 45 2:17:05 - 46 2:19:52 - 47 2:22:37 - 48 2:25:56 - 49 2:29:47 - 50 Q51 to Q60 exponential & logarithm 2:34:25 - 51 2:38:06 - 52 2:42:15 - 53 2:44:48 - 54 2:46:13 - 55 2:47:53 - 56 2:49:18 - 57 2:50:29 - 58 2:53:30 - 59 2:55:40 - 60 Q61 to Q70 write in terms of x^p 2:57:23 - 61 2:59:04 - 62 2:59:58 - 63 3:01:00 - 64 3:02:58 - 65 3:04:40 - 66 3:06:00 - 67 3:08:00 - 68 3:09:09 - 69 3:12:22 - 70 Q71 to Q80 expressions from the quotient rule of derivative 3:14:09 - 71 3:16:25 - 72 3:19:29 - 73 3:22:19 - 74 3:24:10 - 75 3:26:16 - 76 3:29:50 - 77 3:32:58 - 78 3:36:29 - 79 3:39:15 - 80 Q81 to Q90 expressions from implicit differentiation (solving for dy/dx) 3:43:40 - 81 3:47:55 - 82 3:49:53 - 83 3:52:28 - 84 3:55:13 - 85 3:59:08 - 86 4:02:44 - 87 4:09:32 - 88 4:12:13 - 89 4:15:28 - 90 Q91 to Q100 solving for the unknowns from partial fractions decomposition 4:18:22 - 91 4:26:31 - 92 4:33:42 - 93 4:38:42 - 94 4:43:56 - 95 4:46:58 - 96 4:48:33 - 97 4:53:09 - 98 4:59: 46 - 99 5:04: 20 - 100 5:14:33 for the people who just want to see the end : ) ***Thanks to ALL my lovely patrons for supporting my channel and believing in what I do*** AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist Robert Sundling Stuart Wurtman Nick S William OCorrigan Ron Jensen Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Evangeline McDonald Joe Taxpayer 💪 If you would also like to support this channel and have your name in the video description, then you could become my patron here https://www.patreon.com/blackpenredpen$

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How math majors use the intermediate value theorem!