Zadanie 1d, Paper 3, maj 2026, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2026, TZ1, Paper 3 (zadania wzorowane), pytanie 1(d). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
A numerical model samples the positive odd inputs 3, 5, 7, … . For a non-negative integer p and a positive integer n, define . This investigation compares geometric bounds with exact finite-difference calculations.
Polecenie
Rectangle k has width 1, spans k − 1 ≤ x ≤ k, and has height (2k + 1)2. A quadratic curve y=f(x) passes through their top-right corners. (i) State f(x). (ii) Compare areas to show . (iii) Use a similar area comparison to show .
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Odpowiedź z klucza
(i) f(x) = (2x + 1)2. (ii) On [0,n] f is strictly increasing. The rectangles lie above f except at their right edges, and have total area T₂(n). Therefore . (iii) The translated curve (2x + 3)2 passes through the top-left corners and lies above each rectangle except at that corner. Thus . The inequalities are strict since n>0 and the regions differ on intervals of positive length. Equivalently integrate f from 1 to n+1 for the upper bound.
Schemat punktowania
(i) A1: (2x+1)². (ii) R1: justify rectangle area strictly exceeds the curve area for general n. A1: identify total rectangle area T₂(n). M1: use the appropriate definite integral. A1: correctly evaluate the lower integral. (iii) R1: justify an upper area comparison for all n. M1: identify the translated curve or shifted integration interval. A1: correct definite integral. A1: correct evaluation. Numerical examples alone do not prove either bound.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Give exact values or three significant figures unless stated otherwise. Proofs and technology results require supporting working.