3 ms·

It's been a while, but I seem to remember that this is actually how I was thought calculus...

• analog31 12y ago
  That was my impression too... >>> You can calculate the derivative, or rate of change, of a function f by doing >>> (f(x+ε) - f(x)) / ε >>> and then at the end throwing out terms involving ε. ... is exactly how the derivative was defined in both high school and college calc, except that the last bit was formalized by taking a limit. Now, the concept of a limit was presented rather informally in high school, then with progressively more rigor in college calculus, and ultimately in real analysis. This approach got most of the STEM students through calculus in a finite time, but meant that only the math majors got to do calculus at a deeper level. And grad school, well, I went into physics instead. Every math course skips the nasty bits that get re-visited at a higher level later. Maybe that's just how math is. My understanding of infinities and infinitesimals is that they are "not numbers," but are essentially a short hand notation for the processes involved in taking limits. And limits are based on sets. Granted, there may be other ways to understand the same stuff.
  • mcguire 12y ago
    I didn't take calculus in high school (I'm not even sure it was offered, actually. But, football!), so in my first semester of college I got hit with limits right out of the gate. I didn't know there was an alternative until much later. "My understanding of infinities and infinitesimals is that they are 'not numbers,'" Nope, they're perfectly good numbers. Check out the origins of "nonstandard analysis" in the mid-20th century for the one stupid trick that makes mathematicians jealous. Or something. Now, irrationals and complexes? Them's not numbers; them's eeeeevil.