Nope. An alternative definition of the derivative that is computationally easier in some cases is the limit as x goes to a of the fraction (f(x) - f(a))/(x - a). The only issue is that strictly speaking he should perform the cancellation *before* he thinks about m approaching n -- if m approaches n before the cancellation he's cancelling out 0/0. Most students get boggled enough by one definition of the derivative that introducing two really throws them for a loop. But they are absolutely the same as your son has noted. If you can afford it I'd really recommend getting Spivak's calc book for your son as well -- I saw it on bookdepository which iirc has free shipping worldwide. I think it would be 100% up his alley.

I googled "calc 1 final exam" and found a bunch with solutions: http://cims.nyu.edu/~kiryl/Calculus/Tests/Old%20Final%20Exams.html http://www.math.pitt.edu/~athanas/MATH-0220-CALCULUS-I/ http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/final-exam/ I'm sure you could find more if you need them.

FTR, calculus at an Anton-level really isn't that difficult for a student with a solid algebra/geometry/trigonometry foundation who already thinks mathematically. The big issue is with students who are weak in prerequisites or do not think well mathematically and try to learn by rote. When I was about your son's age (and much less well-prepared -- there was NO aops! alas!) I went through calc 1 in 3 weeks and was perfectly well-prepared for calc 2 at the state university in the fall. I mention this not as a boast (please don't think that) but simply as reassurance.

I really doubt doing the whole problem set from Anton is going to be necessary. I think unless he's struggling on something (one topic that's notorious for requiring immense amounts of practice is integration by substitution) I'd choose problems very judiciously. For example -- I really, really doubt he's going to require 20 practice problems on differentiating polynomials. I think his school has assignments posted for calc 1, with solutions, as well as an old mid-term -- those may be helpful.

I think he'd be fine studying theoretical single-variable calc AND aops precalc if he's got the time to do it after his brief run through trig. If not, quite honestly, given how much he'd prefer to do it theoretically, I'd postpone the aops precalc and study calc theoretically. Given his mathematical maturity, if he has solid algebra and trig skills he'll be fine without it.

Yeah, he sounds like he'd be more interested in Spivak or Apostol than that calculus text. I'd recommend getting one of those for fun. Book Depository has Spivak for a reasonable price and Apostol for an unreasonable price. Apostol is what MIT uses for calculus with theory, Spivak is what Ohio State uses for the same course (or did last time I looked) and both are excellent books for someone who's very mathematically mature but hasn't been exposed to calculus. I really don't think he needs to follow Anton, but more that he needs to be able to do the problem sets in Anton, if that makes sense?

Yes, I'd just classify whatever happened before the new 9th grade year as "pre-high-school courses". If the student needed an extra year in math and science, but had done good work in history and english, you could use the new senior year for a fun elective related to their interests in those areas. If they were missing what's usually a core course -- say they'd done American History in 9th grade (now 8th) and not again, one good solution could be to use the senior year for something like AP US -- another one could be to just do an advanced USH topics course without the pressure of the AP. MIT OCW has a bunch of special topics in USH.

Rifles for Watie is an amazing book and it's the book that turned my 9yo brother who was a non-reader into an avid reader. I recommend it very highly.

Is this the DS who did MUS Geometry last year? Did he do MUS Algebra before that? You might find that MUS really didn't adequately prepare him for a different algebra 2. It might be better either to go back to MUS and start over at the beginning of algebra 2, or to re-do algebra 1 with a different curriculum, testing through the easier bits. Algebra 1 is a perfectly reasonable high school class and there is no reason for a struggling sophomore to be doing algebra 2 (which puts him in calculus in 12th grade). It'll look fine if you organize the transcript by subject rather than grade.

There's a band called 1023 MB. They haven't had any gigs yet.

If you can't find "recent documents", make another file, type one word, save it, and pay attention to the path. It should save in the same place.

As far as dry, there are a couple good ways to keep your turkey's bosom moist and tender. 1) Roast it upside down -- flip it over about 45 minutes before you serve to crisp up the skin. 2) Slap a slab of some fatty pork (bacon is traditional but one year we just did a pork steak) over it before roasting.

I prefer not to use them myself, but I would put it on a guest bed. It is easier for a guest to remove an unwanted top sheet (or sleep on top of it) than to ask for one if s/he really did mind.

A farmer counted 196 cows in his field, but when he rounded them up he had 200. eta: mistyped the joke the first time :/

What about Jann's geometry class?

I think we're talking about two different things. I'm talking about the gifted programs for elementary school which tend to be based on IQ tests and not subject-specific tests. You're talking about specific cutoffs for individual courses in middle/high school, based on tests relevant to that specific course. I absolutely agree with using those.

Too big is not a problem. The day after Thanksgiving, chop all the meat off and put it in ziplocks -- 1 lb per ziplock -- and put them in the freezer. It's great for emergency meals -- you just make anything that needs meat (stirfry, gravy to go over rice, whatever) and then dump it in.

If he *wants* to be held back I'd go ahead and do it. But ... just in case ... I would keep very good records of his (new) 8th grade year, and try and have him do courses that *could* be high school -- e.g. algebra, physical science, world geography. That way if he changes his mind, you can come up with a legitimate high school transcript and graduate him anyway at 18.

I'm not talking about pointing things out in a 'let's solve this problem!' manner, but just for discussion and "I wonder if". Try "Math for Smarty Pants" and "The I Hate Mathematics! Book" -- both available very cheaply, both are primarily words with very little computation. Don't just hand them to your kids -- read a few pages a day until you see something interesting, and then bring it up in conversation. I'll be astonished if you can't find at least a few interesting things in there.

I am aware of how the cutoffs are formed, and yet there is very, very little difference between two children two points apart, but one is labeled gifted and one is not -- simply because one is two points further away from average. In actuality a difference this small is not statistically significant and it is quite possible that re-testing would lead to a different result -- and this really prejudices enrollment in gifted programs in favor of the parents who can pay for re-testing and re-re-testing.

Not for Algebra; there are significant differences for Geometry.

Exactly. And that's one big problem I have with "gifted" programs -- they tend to pick some arbitrary cutoff, as if you are either gifted or not, and then either you do the "gifted" program or the "normal" program. It's as ridiculous as having an "athletically talented" test for PE, and then having kids either do the "athlete" program or the "normal" program based on where they test. This is not to say that I don't believe we should have individualized learning. We should. We should take advantage of the wonderful advances in technology to enable even more customization and individualization than we ever could before.

I rather like Wells' The Time Machine: http://www.gutenberg.org/ebooks/35 Campbell's The Black Star Passes is also an old favorite: http://www.gutenberg.org/files/20707/20707-h/20707-h.htm BTW here's a list of public domain sci-fi on gutenberg, including multiple issues of Astounding Stories: http://www.gutenberg.org/wiki/Science_Fiction_%28Bookshelf%29 And while we're at it, for more recent stuff that's freely available (but not public domain, strictly speaking): https://www.baenebooks.com/c-1-free-library.aspx On Basilisk Station and The Honor of the Queen (the first two Honor Harrington books) are in the Baen Free Library and I *love* this series.

I agree with this. Some children make some remarkable leaps in maturity during middle school/early high school. I'm going to address one of your pros explicitly: "Being Top of the Class in math (college), rather than hanging on b/c we pushed through disregarding mechanical details...I think this may be a life-changer for him, tbh." -- regardless of his grade level, you should not be pushing on disregarding mechanical details, even if it means he only has algebra and geometry at graduation (though in that case I would probably do an extra year of high school), he will still be better off than if he had more math on his transcript but did not understand it.

I remember when I was middle-school-aged, crying to my mother "But Mom, I just want to be *normal*".