مشتق و یکنوایی
Derivatives and Monotonicity
مباحث مشتق و یکنوایی توابع.
ریاضی عمومی کنکور ارشد علوم کامپیوتر
کارتها · 11
A real cubic is bijective exactly when its derivative never changes sign, i.e. the discriminant of P' is non-positive.
The symmetric 5-point stencil approximates f'' to order h^4 using coefficients -1, 16, -30, 16, -1 /(12h^2).
Fit data exactly with the unique degree-n polynomial (Lagrange/Newton) or a smooth C^2 cubic spline, and bound the polynomial error by f^(n+1)/(n+1)! times the node product.
Through any n+1 points with distinct nodes there is a unique polynomial of degree ≤ n ; Lagrange and Newton are two ways to build the same polynomial.
On an interval the sign of f' gives whether f rises or falls and the sign of f'' gives whether f' rises or falls; a one-sided monotone-bounded f' has a finite limit that equals the one-sided derivative.
Relative errors of a product don't simply add: δ(ab)=δ(a)+δ(b)+δ(a)δ(b) . The cross term is the catch, and it vanishes exactly when one factor is exact.
The condition number measures how much a linear system amplifies input error, costing you about log10(kappa) reliable digits.
If f is differentiable on [a,b], then f' takes every value between f'(a) and f'(b) — even though f' need not be continuous. So a derivative can never have a jump discontinuity.
To glue two formulas at a breakpoint so the result is n-times continuously differentiable, force the two pieces to agree in value and in every derivative up to order n at that point — that single requirement chains all the lower orders along with it.
Use f' to split the line into monotone pieces, then count exactly one root per piece whose endpoints straddle zero — no factoring required.
Split a rational function into simple poles 1/(x-r) — each differentiates n times to a tidy (-1)^n n!/(x-r)^(n+1), giving the n-th derivative with no product-rule mess.