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Counting Differential Equations, I've had 5 semesters of calculus. (Calc I,II,III,IV, Diff Equ.) Until reading your comment, item 7, I had never heard this si

Counting Differential Equations, I've had 5 semesters of calculus. (Calc I,II,III,IV, Diff Equ.) Until reading your comment, item 7, I had never heard this simple explanation of the meaning of the Fundamental Theorem of calculus ("why areas are the reverse of derivatives"). I was taught the fundamental theorem algebraically, and how to apply it, but none of my professors or textbooks ever explained what it meant.

btilly 7y ago
Thank you for your honesty. I wish it surprised me. But I suspect that your experience is more normal than not.
jacobolus 7y ago
You never had someone in your introductory course show you some problem where you have e.g. a big water tank filling up, and you relate the flow rate with the volume of water in the tank, describing the relation as either sum(flow) = volume or diff(volume) = flow? That’s quite depressing, since this is really the whole point of calculus. * * * Calculus really should be introduced with as much emphasis on physical modeling as possible. Differential equations are at the heart of the past several centuries of science, and understanding the basic ideas involved is crucial for everyone doing any kind of technical work, if not every citizen. I like this version, which uses discrete computer simulation to cut down on some of the obscurantism of a formal heavily algebraic treatment, and lets students jump right into ideas which are significantly delayed in typical university math curricula. http://www.math.smith.edu/~callahan/intromine.html http://www.math.smith.edu/~callahan/intromine.html
gregpetrics 7y ago
That's really sad. You might check out the sequence of activities in the Integral chapter of this book to get another perspective on the FTC. I totally agree. It's the entire point of calculus, and it's what sets the entire field of Differential Equations in motion!
lonelappde 7y ago
I guarantee you it was in your textbook and probably in lecture and you just glossed over and forgot. What does "applying it" even mean if not "finding the intrgral of a dervative or the derivative of an integral"? That's the entire point of the theorem, which is given the most powerful name mathematics pedagogy gives to important ideas, which should motivate you to glance at it more than once. Or do you mean that you never connected "derivative and integral are inverses" with "integrals are areas" and "inverses are opposites"?

btilly 7y ago

Thank you for your honesty. I wish it surprised me. But I suspect that your experience is more normal than not.

jacobolus 7y ago

You never had someone in your introductory course show you some problem where you have e.g. a big water tank filling up, and you relate the flow rate with the volume of water in the tank, describing the relation as either sum(flow) = volume or diff(volume) = flow? That’s quite depressing, since this is really the whole point of calculus. * * * Calculus really should be introduced with as much emphasis on physical modeling as possible. Differential equations are at the heart of the past several centuries of science, and understanding the basic ideas involved is crucial for everyone doing any kind of technical work, if not every citizen. I like this version, which uses discrete computer simulation to cut down on some of the obscurantism of a formal heavily algebraic treatment, and lets students jump right into ideas which are significantly delayed in typical university math curricula. http://www.math.smith.edu/~callahan/intromine.html http://www.math.smith.edu/~callahan/intromine.html

gregpetrics 7y ago

That's really sad. You might check out the sequence of activities in the Integral chapter of this book to get another perspective on the FTC. I totally agree. It's the entire point of calculus, and it's what sets the entire field of Differential Equations in motion!

lonelappde 7y ago

I guarantee you it was in your textbook and probably in lecture and you just glossed over and forgot. What does "applying it" even mean if not "finding the intrgral of a dervative or the derivative of an integral"? That's the entire point of the theorem, which is given the most powerful name mathematics pedagogy gives to important ideas, which should motivate you to glance at it more than once. Or do you mean that you never connected "derivative and integral are inverses" with "integrals are areas" and "inverses are opposites"?