The mean value theorem is about a very special arithmetic mean that is calculated using ALL the y ordinate lengths in a given interval for a specified SMOOTH function. Note that the methods of calculus are both NULL and VOID unless one is using a SMOOTH function. The mean value theorem says that the arithmetic mean f'(c) of all the y - ordinates of a function f' can be found from its antecedent function f by the quotient: [f(b)-f(a)]/(b-a). The lengths are considered as magnitudes (not nonsense such as "real number") because many have no number that describe the length of their measure. None of the mainstream theorems (extreme value or Rolle's theorem) are required to prove the mean value theorem. In fact, even the bogus concept of "real number" has no influence on the working of the mean value theorem. The mean value theorem explained using positional derivatives: https://drive.google.com/file/d/0B-mOEooW03iLZG1pNlVIX2RTR0E The positional derivative is a concept I invented to formulate a constructive of the mean value theorem using the flawed mainstream definitions and concepts: https://drive.google.com/file/d/0B-mOEooW03iLSFVKRmx6c0tPWms Constructive proof of the mean value theorem using the rigorous New Calculus: https://drive.google.com/file/d/0B-mOEooW03iLblJNLWJUeGxqV0E Britannica entry: https://www.britannica.com/science/mean-value-theorem It is FALSE that one has to know anything about "real number" or real analysis. You can learn the New Calculus here: https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO My historic geometric theorem of January 2020 (discovered as a result of my New Calculus, but not a part of my New Calculus which is far simpler and rigorous): https://drive.google.com/file/d/1RDulODvgncItTe7qNI1d8KTN5bl0aTXj Fixing the broken mainstream formulation using my geometric identity: https://drive.google.com/file/d/1uIBgJ1ObroIbkt0V2YFQEpPdd8l-xK6y Exactly how stupid are mainstream math professors? https://drive.google.com/file/d/1520NjhgiakcrssQxtbxRCDXus_aHXpI9 The BIG LIE: "Calculus was made rigorous...": https://drive.google.com/file/d/1peL7lzXsmZ4AVsgAwDLvn8QIZbWy8Ebm There are no postulates or axioms in sound mathematics: https://drive.google.com/file/d/1vlU-PJeIk672bFwZyULD1ASTRFF3jXg8 Pi is not a number: https://drive.google.com/file/d/1FFg_9XCkIwTZ9N1jbU4oMYfHHHuFHYf3 Is the "real number line" real? https://drive.google.com/file/d/0B-mOEooW03iLMHVYcE8xcmRZRnc How a genius discovers the concept of number: https://drive.google.com/file/d/1hasWyQCZyRN3RkdvIB6bnGIVV2Rabz8w ... https://www.youtube.com/watch?v=FIx0KrDi2so

Number is the most important concept in mathematics and yet there is no consistent definition of number in the mainstream. Number comes to us from ratio. A prologue to the Elements: https://www.academia.edu/105854738/The_Revised_Elements_Prologue Link to graphic summarizing the development of the number concept (@)7:40): https://drive.google.com/file/d/1Vz-Vg4F8lhZ1dNoKeDcHxk8FRevbgxAn/view Learn about the Holy Grail of Calculus: https://www.academia.edu/105576431/The_Holy_Grail_of_Calculus Download the best book ever written on the number concept: https://www.academia.edu/105399167/The_Ultimate_Book_of_Numbers Get the true Origins and History of Calculus: https://www.academia.edu/106488069/The_Non_fictional_Origins_and_History_of_Calculus Discover closed -form trigonometric functions never realised before: https://www.academia.edu/109334669/Ancient_Greek_trigonometric_formulas_better_than_anything_ever_known Follow me on Academia.edu: https://independent.academia.edu/JohnGabriel30 Donate here: https://gofund.me/af8a5312 or here: https://wise.com/pay/me/johng3933 All my YouTube videos are backed up here: https://odysee.com/@NewCalculus:1 The scum of YouTube are constantly threatening and looking for excuses to shut down my channel. Want to get instant updates for the newest math around? Join our discord server! https://discord.gg/CJ9Ks3WerR Merchandise Store: https://new-calculus.printify.me/products ... https://www.youtube.com/watch?v=x17GK0Id34Y