Zadanie 2d, Paper 3, maj 2025, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 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
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
The logarithmic profile used for comparison is v(x) = 2ln x, x > 0.
For w(x) = ex/2, , you are given and .
(i) Show that .
(ii) Suggest a geometric reason why v and w have the same maximum curvature.
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Odpowiedź z klucza
(i) The exponential factor is never zero. A stationary point therefore satisfies ex = 2, so x = ln 2. Curvature tends to zero at both infinite ends and its derivative changes from positive to negative at this point, so it is the maximum. Its value is . (ii) The two functions are inverses: their graphs are reflections in y=x. A reflection preserves the amount of geometric bending.
Schemat punktowania
(i) M1: set derivative equal to zero. A1: correct critical equation or x=ln2. M1: substitute into curvature. A1: correct simplified surd value before rationalization. A1: complete derivation of sqrt(3)/9; a correct prior surd simplification from (c) may be referenced. (ii) R1: inverse graphs/reflection preserve curvature. In (i), decimal-only work is capped at M1 A0 M0 A0 A0. For (ii), stating that the functions are inverses, that the graphs are reflections in y=x, or that they have the same shape is sufficient for R1.
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.