Calculus Explained Better Than College Ever Did!
Animated Math
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Become a member: @animatedmathofficial
Get a free ebook: https://animatedmath.gumroad.com/l/se...
Calculus is often taught as two disconnected rulebooks: differentiate this, integrate that, and memorize the formulas. I wanted to understand the one idea that makes both operations belong to the same story.
We begin with a changing flow of water and show exactly where algebra stops being enough. Then we build an instantaneous rate from shrinking nonzero windows, derive the derivative of x squared without dividing by zero, construct an integral from finite sums, and prove the Fundamental Theorem using the thin strip added by a moving endpoint. From there, we explain continuity, F(b) - F(a), +C, signed displacement, probability density, units, initial conditions, and a complete rocket trajectory. No hand-waving and no disconnected rules to memorize.
By the end, you will see why a derivative recovers a local rate, why an integral recovers accumulated change, and why two endpoint values can contain the exact information from an entire interval.
CHAPTERS
0:00 When algebra stops being enough
1:52 Derivatives: shrinking windows to local rates
4:31 Integrals: accumulating local change
7:29 Why derivatives and integrals undo each other
11:24 Proving the Fundamental Theorem
20:57 Signed motion, probability, and the rocket
If I got a detail wrong, tell me in the comments. I read them, and I would rather fix it than defend it.
REFERENCES
ORIGINAL WORKS (FREE)
Isaac Barrow, The Geometrical Lectures of Isaac Barrow: https://openlibrary.org/books/OL66043...
Isaac Newton, The Method of Fluxions and Infinite Series: https://archive.org/details/methodoff...
Gottfried Wilhelm Leibniz, The Early Mathematical Manuscripts of Leibniz: https://quod.lib.umich.edu/u/umhistma...
HISTORY (FREE)
Andrew Leahy, James Gregory and the Pappus-Guldin Theorem, Mathematical Association of America: https://old.maa.org/press/periodicals...
TEXTBOOKS AND COURSES (FREE)
OpenStax, Calculus Volume 1, Section 5.3: The Fundamental Theorem of Calculus: https://openstax.org/books/calculus-v...
MIT OpenCourseWare, 18.01SC Single Variable Calculus: https://ocw.mit.edu/courses/18-01sc-s...
ANIMATION AND COPYRIGHT
All 2D and 3D animations in this video were designed and built from scratch by the Animated Maths team. No stock footage or third-party clips were used. (c) Animated Maths. All rights reserved. Reuse, re-upload, or redistribution without written permission is not allowed.
Covered in this video: calculus explained intuitively, limits, derivatives, instantaneous rate of change, integrals, Riemann sums, the Fundamental Theorem of Calculus, antiderivatives, plus C, definite integrals, signed area, displacement versus distance, probability density, differential equations, acceleration, velocity, position, and initial conditions.
#Calculus #FundamentalTheoremOfCalculus #Derivatives #Integrals #AnimatedMaths
4 Comments
sorry
but truth is truth its always bitter
Calculus Explained Better Than College Ever Did!
Animated Math
9,729 views
Become a member: @animatedmathofficial
Get a free ebook: https://animatedmath.gumroad.com/l/se...
Calculus is often taught as two disconnected rulebooks: differentiate this, integrate that, and memorize the formulas. I wanted to understand the one idea that makes both operations belong to the same story.
We begin with a changing flow of water and show exactly where algebra stops being enough. Then we build an instantaneous rate from shrinking nonzero windows, derive the derivative of x squared without dividing by zero, construct an integral from finite sums, and prove the Fundamental Theorem using the thin strip added by a moving endpoint. From there, we explain continuity, F(b) - F(a), +C, signed displacement, probability density, units, initial conditions, and a complete rocket trajectory. No hand-waving and no disconnected rules to memorize.
By the end, you will see why a derivative recovers a local rate, why an integral recovers accumulated change, and why two endpoint values can contain the exact information from an entire interval.
CHAPTERS
0:00 When algebra stops being enough
1:52 Derivatives: shrinking windows to local rates
4:31 Integrals: accumulating local change
7:29 Why derivatives and integrals undo each other
11:24 Proving the Fundamental Theorem
20:57 Signed motion, probability, and the rocket
If I got a detail wrong, tell me in the comments. I read them, and I would rather fix it than defend it.
REFERENCES
ORIGINAL WORKS (FREE)
Isaac Barrow, The Geometrical Lectures of Isaac Barrow: https://openlibrary.org/books/OL66043...
Isaac Newton, The Method of Fluxions and Infinite Series: https://archive.org/details/methodoff...
Gottfried Wilhelm Leibniz, The Early Mathematical Manuscripts of Leibniz: https://quod.lib.umich.edu/u/umhistma...
HISTORY (FREE)
Andrew Leahy, James Gregory and the Pappus-Guldin Theorem, Mathematical Association of America: https://old.maa.org/press/periodicals...
TEXTBOOKS AND COURSES (FREE)
OpenStax, Calculus Volume 1, Section 5.3: The Fundamental Theorem of Calculus: https://openstax.org/books/calculus-v...
MIT OpenCourseWare, 18.01SC Single Variable Calculus: https://ocw.mit.edu/courses/18-01sc-s...
ANIMATION AND COPYRIGHT
All 2D and 3D animations in this video were designed and built from scratch by the Animated Maths team. No stock footage or third-party clips were used. (c) Animated Maths. All rights reserved. Reuse, re-upload, or redistribution without written permission is not allowed.
Covered in this video: calculus explained intuitively, limits, derivatives, instantaneous rate of change, integrals, Riemann sums, the Fundamental Theorem of Calculus, antiderivatives, plus C, definite integrals, signed area, displacement versus distance, probability density, differential equations, acceleration, velocity, position, and initial conditions.
#Calculus #FundamentalTheoremOfCalculus #Derivatives #Integrals #AnimatedMaths