Taylor remainder
When you approximate a function by a Taylor polynomial, you are deliberately discarding everything beyond a certain degree. The Taylor remainder is the honest accounting of exactly what you threw away — the gap between the true function and its polynomial approximation. Mastering analysis means never approximating without bounding this gap.
Definition: with P_n the degree-n Taylor polynomial of f at a, the remainder is R_n(x) = f(x) - P_n(x). Its size is governed by the next derivative. The Lagrange form states R_n(x) = (f^(n+1)(xi)/(n+1)!)(x - a)^(n+1) for some xi strictly between a and x (assuming f^(n+1) exists on the interval). The integral form R_n(x) = (1/n!) integral_a^x f^(n+1)(t)(x - t)^n dt requires f^(n+1) continuous but needs no unknown xi.
The practical payoff is a numerical error bound: if |f^(n+1)| <= M on the interval between a and x, then |R_n(x)| <= M |x - a|^(n+1) / (n+1)!. This is how one proves, for example, that the Taylor series of e^x, sin, and cos converge to those functions everywhere — by showing the remainder tends to 0. A standing caution: a small remainder for one fixed n at one point says nothing global; convergence of the series requires R_n(x) -> 0 as n -> infinity, which can fail even for smooth functions.
Approximating cos(0.5) by 1 - x^2/2 (n = 2 at a = 0): since |f'''| = |sin| <= 1, the bound is |R_2| <= 1 * 0.5^3 / 6 = 0.0208. Indeed 1 - 0.125 = 0.875 versus cos(0.5) = 0.8776, an error of 0.0026, comfortably inside the bound.
The remainder bound is a guarantee, usually generous; the true error sits well under it.