Zadanie 2e, Paper 3, maj 2025, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2025, TZ1, Paper 3 (zadania wzorowane), pytanie 2(e). 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
Consider the lower arc of the circle (x − 2)2 + (y + 1)2 = R2, where R > 0, 2 − R < x < 2 + R and y < −1.
(i) Show that .
(ii) Hence show that the curvature is constant along this arc.
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Odpowiedź z klucza
(i) Implicit differentiation gives 2(x − 2) + 2(y + 1)y′ = 0, so y′ = −(x − 2)/(y + 1). Differentiate again: 1 + (y′)2 + (y + 1)y′′ = 0. Substituting y-prime gives y′′ = −[(y + 1)2 + (x − 2)2]/(y + 1)3 = −R2/(y + 1)3. (ii) Since y+1 is negative, ∣y′′∣ = R2/∣y + 1∣3. Also 1 + (y′)2 = [(y + 1)2 + (x − 2)2]/(y + 1)2 = R2/(y + 1)2. Raising this to power 3/2 gives R3/∣y + 1∣3. Their quotient is κ = 1/R, which is independent of x. Differentiating the explicit lower branch is an equivalent approach.
Schemat punktowania
(i) M1: first differentiation, implicit or explicit. A1: correct first derivative. M1: differentiate again using the product/quotient rule correctly. A1: correct second-derivative relation before substituting. A1: substitute the first derivative. A1: combine numerator and use the circle equation to reach the stated expression. (ii) M1: substitute derivatives into the curvature formula. A1: correctly evaluate the absolute numerator on the lower arc. M1: combine the denominator using the circle equation. A1: denominator R cubed divided by |y+1| cubed. A1: obtain 1/R; this itself establishes constancy, so a separate verbal conclusion is not required.
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.