Like all backbones, the calculus one is hidden from view. It is revealed one member at a time as the course progresses. Think: Gandalf introducing the dwarfs to Beorn one-by-one so as to not invite rebuff by springing the full company on him in one fell swoop. (A scene tragically omitted from Peter Jackson's big-screen adaptation of The Hobbit, I might add. Then again, Mr. Jackson excised Beorn's creepy pedobear vibe so I suppose we should be grateful he here deviated from the source material.)
Those vertebrae are the Great Theorems of first-semester calculus (eight, by my count). Fossils dug from the very urstone, they are. Once the remaining structural vitae is added, it is hung with flesh and sinew and muscle to become living breathing mathematics. But back at the excavation site, while your face is being pressed into the dirt by three exams and a final and everything that comes after, it's hard to take note of the grand edifice being assembled.
One day you look up and there it is, terrible dragon. And as is often the case with dragons, you can't quite remember how it got there.
Well, LabKitty is here to help. I got out my favorite calculus textbook and boned it. The results are arrayed below, each stuck with a tiny nameplate. The Extreme Value Theorem. The Intermediate Value Theorem. Rolle's Theorem. Perhaps you remember some of them. Perhaps just the names. Or perhaps nothing at all. That would be forgiven, swept along you were in the swift current of your calculus infancy.
The gallery opens with the first ramification of continuity and closes when we get to the Fundamental Theorem of Calculus. It seems a reasonable beginning, but why that stopping point? Indeed, more theorems appear in calculus. Heck, more theorems will appear in the remainder of your mathematical life (unless you're an engineering major. Then "theorem" loosely translates as "crikey, would you just get on with it?"). Why the focus on pre-FToC results?
The material that comes before the Fundamental Theorem comprises a sort of calculus prehistory. Yes, prehistorical as in fossils and dinosaurs, in keeping with our charming bone theme. But also prehistorical in the sense of lost in time. The FToC marks the arrival of a fully-functional problem-solving machine which is then developed and applied for the remainder of STEM life -- often at the expense of the results that came before. Pick a recent graduate at random and ask h(im/er) to describe (i) integration by parts and (ii) Rolle's Theorem. You will get a more cogent response to (i) most if not all of the time.
Post-FToC calculus is regularly useful and used. Pre-FToC calculus is a fading memory. Tragically, however, there are times when one of the early first-semester theorems is just what you need to solve a problem, crack a nut, save your bacon. Ergo, my list.
There are no proofs here, and scant explanation. Those are readily found in your textbook. Here they would just get in the way of the big picture. And the big picture, if you haven't been paying attention, is what we're after. Hanging the whole thing on a hook and gazing upon it. To that end, I whipped up a little pictorial summary of the theorems at the end. A mnemonic for remembering the collection, if you will. Double-bucky click to save, print it out, and put it up on the fridge.
Drill while your tarts are popping. Be bettered for the experience.
THE THEOREMS
(names I had to make up are indicated by *)
Extreme Value Theorem If f(x) is continuous on [a,b], then f(x) takes on a least value m and a greatest value M on the interval.
Intermediate Value Theorem1 If f(x) is continuous on [a,b] and f(a) ≠ f(b) then for any number c between f(a) and f(b) there is at least one number x0 in the open interval (a,b) such that f(x0) = c.
Rolle's Theorem2 Let f(x) be continuous on [a,b] and differentiable on (a,b). If f(a) = f(b) = 0, then f'(x0) = 0 for at least one number x0 in (a,b).
Mean Value Theorem3 Let f(x) be continuous on [a,b] and differentiable on (a,b). Then there is at least one number x0 in (a,b) for which f'(x0) = [ f(b) − f(a) ] / [ b − a ].
Mean Value Theorem for Integrals4 If f(x) is continuous on [a,b] then there is a number x0 in [a,b] such that ∫a,b f(x) dx = f(x0) (b – a).
Positive Function Theorem* If f(x) is continuous at c and f(c) > 0, then there is a positive number δ such that, whenever c−δ < x < c+δ, then f(x) > 0.
Shifted Function Theorem* If f'(x) = g'(x) for all x in (a,b), then f(x) = g(x) + C for all x in (a,b) for some constant C.
Constant Function Theorem* If f'(x) = 0 for all x in (a,b), then f(x) is constant on (a,b).
FOOTNOTES
1. Corollary: If f(x) is continuous on [a,b] and f(a) and f(b) have opposite signs, then the equation f(x) = 0 has at least one root in the open interval (a,b) and the graph of f(x) crosses the x-axis at least once between a and b.
2. There is also a Generalized Rolle's Theorem that simply requires f(a) = f(b) (not necessarily equal to zero).
3. There is also an extended MVT and a higher-order MVT. Also, this version of the Mean Value Theorem is aka The Law of the Mean and aka The Mean Value Theorem for Derivatives. Not to be confused with the Mean Value Theorem for Integrals.
4. Notation weirdness alert: ∫a,b is how I write "integral from a to b" because proper integral formatting is beyond my ken. (Ken is the intern I hired to fix all my math HTML.)
Bonus! The Gr8ECT are now available in mug form! Get valuable review every time you slurf!
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