Zadanie 1a
Sketch f_1, stating the coordinates of its local maximum.
Matematyka AA IB HL — maj 2023, TZ1, Paper 3 (zadania wzorowane)
Zadania wzorowane na pytaniach z tego arkusza: te same umiejętności, format i punktacja, inne dane i kontekst. Treści arkusza IB nie publikujemy.
W bazie 16 zadań z tego arkusza, razem 55 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
For b>0, show that the area under f_1 from 0 to b is (eb² − b2 − 1)/(2eb²).
Zadanie 1c
(i) Use l’Hôpital’s rule to evaluate ; its conditions may be assumed satisfied. (ii) Hence state A_1.
Zadanie 1d
It is given that A_2=1 and A_3=3. Use calculator integration with a suitable finite upper bound to estimate (i) A_4; (ii) A_5.
Zadanie 1e
Conjecture A_n in terms of n.
Zadanie 1f
Prove the conjecture by mathematical induction. You may assume for every fixed m≥0.
Zadanie 2a
For two parameters and budget 18, express their product as a function f of x_1 alone.
Zadanie 2b
(i) State the maximizing x_1. (ii) Hence show that the maximum product is 8.
Zadanie 2c
The proposed formula for two parameters is M_2(S)=(S−2)/2. Verify it at S=18.
Zadanie 2d
(i) Show that the arithmetic and geometric means of z_j=x_j² are equal when all parameters x_j are equal. (ii) Prove Mn(S) = (S/n − 1)n/2 for every admissible n.
Zadanie 2e
Find (i) M_3(18); (ii) M_4(18); (iii) M_5(18).
Zadanie 2f
State P(18) and the integer n at which it occurs.
Zadanie 2g
Determine P(30) and its maximizing integer n.
Zadanie 2h
Show that the maximum of g occurs at x=λS for a constant λ independent of S, and determine λ numerically.
Zadanie 2i
Verify that g(x)=M_x(S) for every admissible integer x.
Zadanie 2j
Use the continuous maximum to find the largest attainable product for budget 120. Express it as a×10^k, with 1≤a<10 and positive integer k.