A Shifted Polynomial Glued Together
Difficulty: Hard · Topics: Calculus, Derivatives, Integration
Bir-biriga ulangan siljitilgan ko‘phad
The polynomial f(x) satisfies
lim x→ 1(f(x)-1)/(x-1)=0, lim x→∞fracf(x)-1x^4=1.
For each natural number n, let g(x)=f(x-n+1)+n-1 on n-1≤ x<n. When g(x) is differentiable on the set of all real numbers, find ∫ 0^15g(x) dx.
f(x) ko‘phad quyidagilarni qanoatlantiradi:
lim x→ 1(f(x)-1)/(x-1)=0, lim x→∞fracf(x)-1x^4=1.
Har bir natural n uchun n-1≤ x<n da g(x)=f(x-n+1)+n-1 bo‘lsin. g(x) barcha haqiqiy sonlar to‘plamida differensiallanuvchi bo‘lganda ∫ 0^15g(x) dx ni toping.