Zadanie 2i, Paper 3, listopad 2025, TZ3, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2025, TZ3, Paper 3 (zadania wzorowane), pytanie 2(i). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
Darmowe konto pozwoli wrócić do niego później.
Zadanie
Wspólne informacje do wiązki zadań
Let and . Define . You may use this theorem: for an alternating series whose term magnitudes decrease to zero, the absolute error after finitely many terms is at most the magnitude of the next term.
Polecenie
Find the smallest N for which . Use the alternating-series bound or directly compare the actual errors.
Korepetytor
Nie wiesz, jak ruszyć? Korepetytor zna to zadanie i poprowadzi Cię krok po kroku, bez gotowca.
3 rozmowy · bez karty · Zaloguj się
Sprawdź się
Jak poszło?
Pokaż rozwiązanieUkryj rozwiązanie
Odpowiedź z klucza
The next arctan term afterN terms has magnitudex^(2N+1)/(2N+1). Multiply by 2x and integrate: E_N=2a^(2N+3)/[(2N+1)(2N+3)]. At N=6 the bound is 1.859345531×10⁻⁸, above the tolerance; at N=7 it is 2.439513156×10⁻⁹, below it. The bounds strictly decrease, so N=7 is the smallest guaranteed count. Alternatively actual integral errors at N=6 and 7 are 1.6444171311×10⁻⁸ and 2.1492839975×10⁻⁹, giving the same smallest N=7. Their magnitudes decrease: the error is ∫₀ᵃ(a²−t²)t^(2N)/(1+t²)dt.
Schemat punktowania
M1: use the integral of the next weighted term. A1: correct bound formula or a correct boundary value. M1: solve its inequality or evaluate adjacent integerN values. A1: correct consecutive bounds identifying the crossing 6→7. A1:N=7. Equivalent actual-error route: M1 calculate exactI minus the N-term weighted polynomial integral; A1 one correct boundary error; M1 compare consecutiveN; A1 second correct boundary error; A1 smallest N=7. Accept equivalent indices for the omitted term or its degree, with correct conversion to number of retained terms. Condone omitted integral limits for the initial method mark.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Use radians. Retain unrounded intermediate values; provide the requested precision and full proof reasoning.