Zadanie 2b, Paper 3, maj 2025, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2025, TZ1, Paper 3 (zadania wzorowane), pytanie 2(b). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
To compare bends in planar profiles, use the curvature for a twice-differentiable function f. Write for its greatest curvature. Investigate straight lines, parabolas, logarithmic and exponential profiles, and a circular arc.
Polecenie
For f(x) = a(x − b)2 + c, where a ≠ 0, you are given and .
(i) Solve κ′(x) = 0 to find where the maximum occurs; no maximum test is required.
(ii) Express in terms of a.
(iii) State and interpret it geometrically.
(iv) Let p(x) = −3(x − 4)2 + 7 and . State whether the maximum curvature of p is greater than, less than or equal to that of q, with a reason.
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Odpowiedź z klucza
(i) The numerator vanishes only at x = b. (ii) Substitution gives . (iii) The limit is zero: far along the parabola, its curvature becomes arbitrarily small and it locally bends less sharply. This does not assert that its distance from a straight-line asymptote tends to zero. (iv) The maximum for p is 6, whereas that for q is 3, so p has the greater maximum curvature.
Schemat punktowania
(i) A1: x=b; a=0 is excluded. (ii) M1: substitute the maximizing argument. A1: 2|a|. (iii) A1: zero limit. R1: valid interpretation in terms of vanishing local bending. (iv) A1: greater for p. R1: compare 6 with 3 using the absolute leading coefficients. In (iv), do not award A1 R0: a bare selection without justification earns neither mark. Follow through an omitted absolute value from (ii): a consistently justified comparison -6<3 may receive the two follow-through marks.
Komentarz. Paper 3: calculator use is permitted, but reasoning must be shown. Exact equivalent forms and appropriate numerical answers to three significant figures are accepted unless stated otherwise.