Taylor's theorem
A tangent line is a straight-line stand-in for a curve, accurate only very near the point of contact. Taylor's theorem upgrades this: instead of a line, use a polynomial that matches not just the value and slope but also the curvature and higher bends. The more derivatives you match, the longer the polynomial shadows the true function — and, crucially, the theorem tells you exactly how big the leftover error can be.
Statement: if f is n times differentiable at a (with suitable continuity of the lower derivatives near a), then f(x) = P_n(x) + R_n(x), where P_n(x) = sum_{k=0}^{n} (f^(k)(a)/k!) (x - a)^k is the degree-n Taylor polynomial at a, and R_n(x) is the remainder. The polynomial is the unique degree-n polynomial agreeing with f in value and first n derivatives at a; the remainder R_n(x) -> 0 faster than (x - a)^n as x -> a.
The theorem is not the claim that f equals its Taylor polynomial — it generally does not, and the entire content lies in controlling R_n. The remainder has explicit forms (Lagrange, Cauchy, integral), the Lagrange form reading R_n(x) = (f^(n+1)(xi)/(n+1)!)(x - a)^(n+1) for some xi between a and x. Honest use of Taylor's theorem always carries this error term; dropping it turns a theorem into wishful thinking.
For f(x) = e^x at a = 0, P_2(x) = 1 + x + x^2/2. At x = 0.1 this gives 1.105, against e^0.1 = 1.10517..., an error under 0.0002 — matching the (x^3)/6 size predicted by the remainder.
A short polynomial already approximates e^x well near 0, and the remainder pins down how well.
When a = 0 the Taylor polynomial is called the Maclaurin polynomial. The Lagrange remainder is proved by applying the Cauchy mean value theorem (or repeated Rolle) to f against the polynomial.