Zadanie 2d, Paper 3, listopad 2025, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2025, TZ1, Paper 3 (zadania wzorowane), pytanie 2(d). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
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A regional volunteer network had maintained 1000 active memberships before disruption reduced the count to 125. A recovery programme starts at t=0 years. At t=4 the network has250 active members; at t=8 an audit measures 380. Two models use the long-term level 1000: a logistic model x = L/(1 + Ce−kt), with L,C,k>0, and a Gompertz model dx/dt = axln(1000/x), a>0. Fit each model using the observations at t=0 and t=4; reserve the t=8 audit for comparison. In the adjusted model, active memberships also lapse at a continuous proportional rate 0.01 per year.
Polecenie
The adjusted model is dx/dt = .101xln(1000/x) − .01x. (i) Starting at x₀=250 when t=4, use Euler’s method with step 0.4 years to estimate the count at t=8, to the nearest whole member. (ii) Comment on this prediction.
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Use xj+1 = xj + .4[.101xjln(1000/xj) − .01xj], ten steps from t=4 to t=8. The iterates are 263.001573,276.140618,289.392204,302.731704,316.134937,329.578285,343.038799,356.494302,369.923462,383.305869. Thus 383 members. This is above the audit 380 by about 3 members and closer than either unadjusted model.
Schemat punktowania
(i) M1:attempt Euler updating; one minor indexing/notation error may be condoned for this method mark. A1:correct recurrence and step. A1:evidence of a correct iterate after at least two steps. A1:383 whole members after ten steps. (ii) A1:correct comparison with 380 or the earlier models, explicitly or implicitly showing improved agreement. For the first M1, condone at most one of a missing starting x_j term, a subscript error, or a missing/incorrect step factor; this does not earn the recurrence A1.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Show complete reasoning and reproducible numerical work. Give exact results unless a stated approximation is requested.