Mavzular

Mavzular / Hosila va integral

Optimallashtirish masalalari

🟢 — masalani tezlashtiradi · 🟡 — ilg'or daraja · 🔵 — chuqur o'rganish (olimpiada darajasi)

Amaliy masalalar (optimallashtirish)

  • Usul:
    1. Kattalikni bitta o'zgaruvchi orqali ifodalang (berilgan shartdan ikkinchi o'zgaruvchini yo'qoting).
    2. Hosilani nolga tenglang: dVdx=0\dfrac{dV}{dx} = 0.
    3. d2Vdx2\dfrac{d^2V}{dx^2} ishorasi orqali maksimum yoki minimum ekanini ko'rsating.

Ko'p ishlatiladigan geometrik formulalar

ShaklHajm / yuza
SharV=43πr3V = \frac{4}{3}\pi r^3,   S=4πr2\;S = 4\pi r^2
SilindrV=πr2hV = \pi r^2 h,   Syon=2πrh\;S_{\text{yon}} = 2\pi r h
KonusV=13πr2hV = \frac{1}{3}\pi r^2 h,   Syon=πrl\;S_{\text{yon}} = \pi r l
KubV=x3V = x^3,   S=6x2\;S = 6x^2

🟢 Klassik masalalar

MasalaJavob
Perimetri PP bo'lgan to'g'ri to'rtburchakning eng katta yuzikvadrat, S=(P4)2S = \left(\dfrac{P}{4}\right)^2
Tomoni aa bo'lgan kvadrat listdan burchaklarini xx dan kesib qopqoqsiz quti: V=x(a−2x)2V = x(a - 2x)^2x=a6x = \dfrac{a}{6} da maksimum
Hajmi berilgan yopiq silindrning eng kichik sirtih=2rh = 2r
Yig'indisi ss bo'lgan ikki sonning eng katta ko'paytmasiikkalasi s2\dfrac{s}{2}, ko'paytma s24\dfrac{s^2}{4}