Asosiy tarkibga o'tish

3.11-bo'lim : Tegishli tariflar

  1. Quyida \(x\) va \(y\) ikkalasi ham \(t\) funksiyasi deb faraz qiling. Berilgan \(x = - 2\), \(y = 1\) va \(x' = - 4\) quyidagi tenglama uchun \(y'\) ni aniqlang. \[6{y^2} + {x^2} = 2 - {x^3}{{\bf{e}}^{4 - 4y}}\] Yechim
  2. Quyida \(x\), \(y\) va \(z\) barcha \(t\) funksiyalari deb faraz qiling. Berilgan \(x = 4\), \(y = - 2\), \(z = 1\), \(x' = 9\) va \(y' = - 3\) quyidagi tenglama uchun \(z'\) ni aniqlang. \[x\left( {1 - y} \right) + 5{z^3} = {y^2}{z^2} + {x^2} - 3\] Yechim
  3. Muayyan to'rtburchaklar uchun bir tomonning uzunligi har doim boshqa tomonning uzunligidan uch baravar ko'pdir.

    1. Agar qisqa tomon 2 dyuym/minut tezlikda kamayayotgan bo'lsa, uzunroq tomon qanday tezlikda kamayadi?
    2. Qisqa tomoni 6 dyuym uzunlikda va 2 dyuym/minut tezlikda kamayganda, yopiq maydon qanday tezlikda kamayadi?
    Yechim
  4. Yupqa muz qatlami aylana shaklida. Agar muz shunday erigan bo'lsa, varaqning maydoni 0,5 m ga qisqaradi.2/sek varaqning maydoni 12 m bo'lsa, radius qanday tezlikda kamayadi2? Yechim
  5. Bir kishi to'g'ridan-to'g'ri havoga sekundiga 15 fut tezlikda uchirilgan raketa modelidan 350 fut uzoqlikda turibdi. Odam va raketa orasidagi masofa qanday tezlikda oshib bormoqda (a) Ko'tarilgandan keyin 20 soniya? (b) Ko'tarilgandan keyin 1 daqiqa? Yechim
  6. Samolyot havoda 750 metr balandlikda, erga parallel ravishda 100 m/s tezlikda uchadi va dastlab radar stantsiyasidan 2,5 kilometr uzoqlikda joylashgan. Samolyot va radar stantsiyasi orasidagi masofa qanday tezlikda o'zgaradi (a) dastlab va (b) 30 soniyadan keyin u radar stantsiyasidan o'tib ketganmi?
    A sketch of the plane and the radar station. The radar station is the point at the lower left and the plane is the point at the upper right, flying to the left as shown by an arrow. The plane is 750 meters above the ground and is 2.5 kilometers horizontally from the radar station. The line drawn from the radar station up to the plane is labeled “Find Rate This Distance Changing”.
    Yechim
  7. Liftda ikki kishi bor. Shu bilan birga, bir kishi liftdan 2 fut/sek tezlikda uzoqlasha boshlaydi, ikkinchisi esa 7 fut/sek tezlikda liftga ko'tarila boshlaydi. Ikki kishi orasidagi masofa 15 soniyadan keyin qanday tezlikda o'zgaradi?
    A sketch of the person and the elevator. The initial position of both people is the point at the lower right. The person walking away is the point at the lower left with an arrow showing them moving left, and the elevator is the point at the upper right with an arrow showing it moving up. The line drawn between the person and the elevator is labeled “Find Rate This Distance Changing”.
    Yechim
  8. Velosipedda ikki kishi bir joyda. Velosipedchilardan biri to'g'ridan-to'g'ri shimolga 8 m/sek tezlikda minishni boshlaydi. Birinchi velosipedchi shimolga minishni boshlaganidan besh soniya o'tgach, ikkinchisi to'g'ridan-to'g'ri sharqqa 5 m/sek tezlikda yura boshlaydi. Ikkinchi odam minishni boshlaganidan keyin 20 soniyadan keyin ikki chavandoz orasidagi masofa qanday tezlikda oshadi?
    A sketch of the two bike riders. The initial position of both people is the point at the lower left. The “North” biker is the point at the upper left with an arrow showing them riding up and the “East” biker is the point at the lower right with an arrow showing them riding to the right. The line drawn between the two riders is labeled “Find Rate This Distance Changing”.
    Yechim
  9. Erdan 5 metr balandlikdagi devorga chiroq o'rnatilgan. 2 metr balandlikdagi odam dastlab devordan 10 metr masofada joylashgan va devorga 0,5 m / sek tezlikda harakat qilmoqda. 4 soniya harakatlanishdan keyin soyaning uchi harakatlanadi (a) shaxsga qarab yoki undan uzoqda va (b) devorga qarab yoki undan uzoqdami?
    A sketch of the lamp, the person and the shadow. The lamp is mounted at the top of a wall of height 5 on the left, the person is a vertical line of height 2 to the right of the wall with an arrow showing them walking to the left towards the wall, and the light ray from the lamp passes over the person’s head and hits the ground to the right of the person. The shadow is the darkened portion of the ground from the person out to that point.
    Yechim
  10. Konus shaklidagi suv idishi 12 m tezlikda suv bilan to'ldirilmoqda3/sek. Tankning tayanch radiusi 26 metr, balandligi esa 8 metr. Suv tepasining radiusi 10 metr bo'lganda, idishdagi suvning chuqurligi qanday tezlikda o'zgaradi? E'tibor bering, quyidagi rasm to'liq hajmga mos kelmaydi….
    A sketch of the cone shaped tank as seen from the front, with the point of the cone at the bottom. The tank is wide and shallow, with the radius of the top of the tank labeled 26 meters and the height of the tank labeled 8 meters. Water fills the bottom portion of the cone and the surface of the water is a smaller circle part way up the tank.
    Yechim
  11. Balandlik burchagi - bu gorizontal chiziq va kuzatuvchining ko'zini gorizontal chiziq ustidagi ob'ektga bog'laydigan chiziqdan hosil bo'lgan burchak. Bir kishi issiq havo sharining uchish nuqtasidan 500 fut masofada joylashgan. Issiq havo shari sekundiga 15 fut tezlikda qaytib tusha boshlaydi. Issiq havo shari erdan 200 fut balandlikda bo'lganda, \(\theta \) balandlik burchagi qanday tezlikda o'zgaradi. Agar ko'rishda qiynalayotgan bo'lsangiz, balandlik burchagini tasavvur qilish uchun quyidagi (ehtimol yomon) eskizga qarang.
    A sketch of the person and the hot air balloon. The person is the point at the lower left and the balloon is the point at the upper right with an arrow beside it showing that it is coming back down. A horizontal line runs from the person out to a point directly below the balloon and a second line runs from the person up to the balloon. The angle between these two lines at the person is labeled \(\theta \), Angle of Elevation.
    Yechim