calculus in a nutshell:
I will save you some time:
There are 4 principles all calculus of one variable is based on.
I. IVT (Intermediate value theorem) If a function f is continuous on an interval, then the values it takes also form an interval.
Hence if there exist points a,b in the domain interval where f(a) < K < f(b), then f(x) must equal K at some c between a and b.
E.g. the continuous function f(x) = x^3 + x + 1 must equal zero for some x between -1 and 0, since f(-1) = -1, and f(0) = 1.
II. EVT (Extreme value theorem, also called Max Min Value or MMV)
If a function is continuous on a closed bounded interval, then its value also form a closed bounded interval, i.e. particular there is a (finite) smallest and a (finite) largest value.
I.e. if f is continuous on [a,b] then there exist c,d in [a,b] such that for every x in [a,b], f(c) ≤ f(x) ≤ f(d).
E.g. since the volume of the right circular cone of height R+x, inscribed in a sphere of radius R, has volume (π/3)(R^2-x^2)(R+x), is continuous for 0 ≤ x ≤ R, some one of these cones has largest volume.
III. (Rolle’s Theorem) If f is continuous on [a,b] and differentiable on (a,b), and if f(a) = f(b), then f achieves its maximum (or minimum) at a point c between a and b where f’(c) = 0, i.e. at a “critical point” for f.
Cor: If f has two derivatives on [a,b] and f’ takes the same value twice, e.g. if there are two critical points, and if f’’ is zero only a finite number of times, then f has a flex somewhere on (a,b) where f’’=0.
Cor: If f is continuous on [a,b] but has no critical points in the interval (a,b), then f cannot take the same value twice in [a,b], hence cannot change direction, i.e. f is strictly monotone on [a,b].
Cor: If f’’ exists but is never zero on [a,b], then f never changes concavity on [a,b], i.e. f is either concave up or concave down on all of [a,b].
IV. MVT (Mean value theorem) If f is continuous on [a,b] and diffble on (a,b), there is a point c with a<c<b, and f’(c)(b-a) = f(b)-f(a).
Cor: If f’=g’ on [a,b] then f-g is constant on [a,b].
Cor: Since for continuous f, d/dx (integral of f from a to x ) = f(x), then integral of f from a to b = G(b)-G(a), for any G with G’ = f.
