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Huh? I learned differentiation with delta-epsilon proofs in the 11th grade. It was pretty standard at my public high school. I'm sure I didn't understand it

Huh? I learned differentiation with delta-epsilon proofs in the 11th grade. It was pretty standard at my public high school. I'm sure I didn't understand it very well at the time, but it was good to have seen it when I took real analysis.

Izkata 8y ago
Had to look it up to be sure, since I didn't recognize the name, but it looks like this refers to using limits to "invent" differentiation? If so, yeah, we did it that way too, in 11th or 12th grade. (Had the same teacher both years, can't remember when exactly Calc was) I remember it involving a lot of drawings of graphs so we could understand what each term referred to, and can't really imagine an easier way to learn it.
- sevensor 8y ago
  Yeah, exactly. I honestly don't know how else you'd teach the derivative either, but I guess it's done.
- giornogiovanna 8y ago
  The limit definition of differentiation is definitely intuitive, but epsilon-delta refers to the most common definition of what it means to take a limit. It's also simple, because when you say it in words it makes perfect sense, but it's not at all obvious why someone would define limits in such a way, or what the alternatives are.
- nextos 8y ago
  Epsilon-delta is the formalization introduced by Cauchy, Bolzano and Weirstrass during the 19th Century. It is not the way calculus was invented by Newton and Leibniz. They thought in terms of infinitesimals. However, this approach is relatively hard to formalize. It was only done by Robinson in the 1960s. To see why Cauchy et al had to work on epsilon-delta, take a look at [1], an excellent book. [1] https://www.macalester.edu/aratra/ https://www.macalester.edu/aratra/
fatnoah 8y ago
Same here. That's how I learned it as well. That said, I didn't really "get it" until I was well into the class. I still remember that "aha moment" very vividly, even though it was in 1992 or so. I was sitting at a table in the school library working on a problem about a pool being emptied while it was raining and my brain said "Holy crap! This is just about how fast things are changing!". With all of the proofs, graphs, etc. things were never put into such a simple statement. The end result of months of flailing followed by the "aha" was a grade of C for the class AND being the only one to get a 5 on the AP test. EDIT: As a fun corollary, I really hated Calculus class in high school. Ironically, I ended up getting a Masters in EE, so I ended up spending the next several years doing Calculus in > 50% of my courses. EDIT EDIT: Also fun to note was that Calculus was only offered to Seniors at the time and required 4 years of math beforehand: Algebra, Geometry, Advance Math, and Trigonometry. That meant you had to double-up on math in either Sophomore or Junior years, which basically limited class size to a small handful of people.

Izkata 8y ago

Had to look it up to be sure, since I didn't recognize the name, but it looks like this refers to using limits to "invent" differentiation? If so, yeah, we did it that way too, in 11th or 12th grade. (Had the same teacher both years, can't remember when exactly Calc was) I remember it involving a lot of drawings of graphs so we could understand what each term referred to, and can't really imagine an easier way to learn it.

sevensor 8y ago
Yeah, exactly. I honestly don't know how else you'd teach the derivative either, but I guess it's done.
giornogiovanna 8y ago
The limit definition of differentiation is definitely intuitive, but epsilon-delta refers to the most common definition of what it means to take a limit. It's also simple, because when you say it in words it makes perfect sense, but it's not at all obvious why someone would define limits in such a way, or what the alternatives are.
nextos 8y ago
Epsilon-delta is the formalization introduced by Cauchy, Bolzano and Weirstrass during the 19th Century. It is not the way calculus was invented by Newton and Leibniz. They thought in terms of infinitesimals. However, this approach is relatively hard to formalize. It was only done by Robinson in the 1960s. To see why Cauchy et al had to work on epsilon-delta, take a look at [1], an excellent book. [1] https://www.macalester.edu/aratra/ https://www.macalester.edu/aratra/

fatnoah 8y ago

Same here. That's how I learned it as well. That said, I didn't really "get it" until I was well into the class. I still remember that "aha moment" very vividly, even though it was in 1992 or so. I was sitting at a table in the school library working on a problem about a pool being emptied while it was raining and my brain said "Holy crap! This is just about how fast things are changing!". With all of the proofs, graphs, etc. things were never put into such a simple statement. The end result of months of flailing followed by the "aha" was a grade of C for the class AND being the only one to get a 5 on the AP test. EDIT: As a fun corollary, I really hated Calculus class in high school. Ironically, I ended up getting a Masters in EE, so I ended up spending the next several years doing Calculus in > 50% of my courses. EDIT EDIT: Also fun to note was that Calculus was only offered to Seniors at the time and required 4 years of math beforehand: Algebra, Geometry, Advance Math, and Trigonometry. That meant you had to double-up on math in either Sophomore or Junior years, which basically limited class size to a small handful of people.