Utter garbage like set theory and topology may be called mathematics but neither of these has anything to do with mathematics. Become a follower on Academia: https://independent.academia.edu/JohnGabriel30 The Holy Grail of Calculus solves the slope and area problem for the first time in human history: https://www.academia.edu/105576431/The_Holy_Grail_of_Calculus Feeling generous? Donate here: https://gofund.me/af8a5312 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=Zk6hadhdGss

For all the goodies, go here: https://drive.google.com/drive/folders/1Nd8PfOnCrpbI_M32rrM8WJTcJX9yIvLL Takeaways from PART 5: 1. In order for the nonsensical and circular limit definition to work, the prize apes of mainstream math academia (your math lecturers and teachers) need for there to be a hole at the point of tangency (they like holes because their brains are full of holes). This is so they can peddle their drivel about a function having a limit but not being defined at the point of tangency (THIS IS A FALSEHOOD THAT HAS BEEN PEDDLED AND PUBLISHED OVER AND OVER IN THE BULLSHIT KNOWN AS MAINSTREAM CALCULUS TEXTBOOKS. IF THERE IS A LIMIT, A FUNCTION IS ALSO DEFINED AT THE POINT OF TANGENCY. MULTIPLICATION BY 1 IS THEIR FRAUDULENT METHOD OF PUNCHING A HOLE IN ANY FUNCTION TO BUTTRESS THEIR BULLSHIT INEQUALITY DEFINITION USING EPSILONS AND DELTAS). 2. There is no finite difference of the form (f(x+h)-f(x))/h that can ever represent a tangent line slope, meaning you can't find the derivative in a valid systematic way using your flawed mainstream formulation of calculus. 3. My historic geometric identity explains why mainstream results work and it has nothing to do with the drivel preached by those morons called calculus professors and teachers! Calculus works because of the Mean Value Theorem which IS the fundamental theorem. A proof that your mainstream math educators are primates is the fact that they have several forms of the mean value theorem: www.academia.edu/81300370/Mainstream_mathematics_academics_are_arrogant_and_incorrigible_ignoramuses_The_mean_value_theorem_IS_the_fundamental_theorem_of_calculus Without a smooth function, the methods of calculus are both NULL and VOID. In PART 6, we will conclude by making an example of a moronic mainstream math professor who doesn't understand the very theory he peddles in his classes and on behalf of all his idiot colleagues in the mainstream, someone called Yosen Lin. Prepare for my next video here: https://www.academia.edu/102705943/Why_your_calculus_lecturer_or_mathematics_teacher_is_most_likely_a_buffoon Want to see how I expose Newton, Leibniz and Madhava as idiots? Go here: https://www.academia.edu/109334669/Ancient_Greek_trigonometric_formulas_better_than_anything_ever_known Entire article here: https://www.academia.edu/79881709/Six_simple_reasons_why_the_mainstream_derivative_definition_of_calculus_is_flawed Download the free eBook on the Single Variable New Calculus: https://www.academia.edu/41616655/An_Introduction_to_the_Single_Variable_New_Calculus Learn about the Holy Grail of Calculus: https://www.academia.edu/105576431/The_Holy_Grail_of_Calculus Learn about the Gabriel Polynomial (GP) which is a very advanced concept not possible using mainstream derivative or integral definitions. The GP is always a closed form polynomial and has no Error term: https://www.academia.edu/45229087/The_Gabriel_Polynomial 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://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 ... https://www.youtube.com/watch?v=FPP50gqyFZY

Functions are not "rates of change" and there is no such thing as an "instantaneous rate of change" or "rate of change" in mathematics, only a finite difference where "change" is irrelevant. What exactly is a rate? https://www.academia.edu/89274499/What_exactly_is_a_rate How we got numbers: https://www.academia.edu/67862197/How_we_got_numbers Download the free eBook on the Single Variable New Calculus: https://www.academia.edu/41616655/An_Introduction_to_the_Single_Variable_New_Calculus Learn about the Holy Grail of Calculus: https://www.academia.edu/105576431/The_Holy_Grail_of_Calculus Learn about the Gabriel Polynomial (GP) which is a very advanced concept not possible using mainstream derivative or integral definitions. The GP is always a closed form polynomial and has no Error term: https://www.academia.edu/45229087/The_Gabriel_Polynomial 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://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=tVDBOw8YPqc

Proposition 14. ᾿Εάν πρώτον πρός δεύτερον τόν αυτόν έχη λόγον καί τρίτον πρός τέταρτον, το δε πρώτον του τρίτου μείζον ή, καί το δεύτερον του τετάρτου μείζον έσται, καν ίσον, ίσον, καν έλαττον, έλαττον. Heath: If a first (magnitude) has the same ratio to a second that a third (has) to a fourth, and the first (magnitude) is greater than the third, then the second will also be greater than the fourth. And if (the first magnitude is) equal (to the third then the second will also be) equal (to the fourth). And if (the first magnitude is) less (than the third then the second will also be) less (than the fourth). Heath claims in a remark: In modern notation, this proposition reads that if α : β :: γ : δ then α T γ as β T δ where T means less than, equal to or greater than. Proposition 14 spells out the relationship between magnitudes which I have already demonstrated in the previous propositions. If α:β=γ:δ then α=γ and β=δ. If α is greater than γ then β is greater than δ. If α is smaller than γ then β is smaller than δ. Proposition 15. Τα μέρη τοίς ώσαυτως πολλαπλασίοις τόν αυτόν έχει λόγον ληφθέντα κατάλληλα. Heath: Parts have the same ratio as similar multiples, taken in corresponding order. Heath claims in a remark: In modern notation, this proposition reads that α:β :: mα:mβ. This has been stated and used in proposition 12. _ : _ _ _ (m=1) :: _ _ : _ _ _ _ _ _ (m=2) :: _ _ _ : _ _ _ _ _ _ _ _ _ (m=3) Proposition 16. Heath: If four magnitudes are proportional then they will also be proportional alternately. Another way to state this is to say: If two ratios p:q and r:a are proportional then p:r :: q:a and q:p :: a:r and r:p :: a:q. This has been stated and used in proposition 12. Proposition 17. ᾿Εάν συγκείμενα μεγέθη ανάλογον ή, καί διαιρεθέντα ανάλογον έσται. Heath: If composed magnitudes are proportional then they will also be proportional (when) separated. Heath claims in a remark: In modern notation, this proposition reads that if α+β : β :: γ+δ : δ then α : β :: γ : δ. If one wanted to state the above correctly in modern notation according to Heath’s interpretation, then α/β+k :: γ/δ + k where k is any number. But is this proposition just telling us the following? 3/4 + 2 : 9/12 + 2 = 11/4 : 33/12 No! Rather, (2+1):4 :: (6+3):12 is what the proposition is saying, implying that 2:4 :: 6:12 and 1:4 :: 3:12. Generally, (a+b):c :: (d+e):f implies that a:c :: d:f and b:c :: e:f. ... https://www.youtube.com/watch?v=wP9gKK_DlGo