Asosiy tarkibga o'tish

3.11-bo'lim : Tegishli tariflar

  1. Quyida \(x\) va \(y\) ikkala \(t\) funksiyasi deb faraz qiling. Berilgan \(x = 3\), \(y = 2\) va \(y' = 7\) quyidagi tenglama uchun \(x'\) ni aniqlang. \[{x^3} - {y^4} = {x^2}y - 7\]
  2. Quyida \(x\) va \(y\) ikkala \(t\) funksiyasi deb faraz qiling. Berilgan \(x = {\textstyle{\pi \over 6}}\), \(y = - 4\) va \(x' = 12\) quyidagi tenglama uchun \(y'\) ni aniqlang. \[{x^2}\left( {{y^2} - 16} \right) - 6\cos \left( {2x} \right) = 1 + y\]
  3. Quyida \(x\), \(y\) va \(z\) barcha \(t\) funksiyalari deb faraz qiling. Berilgan \(x = - 1\), \(y = 8\), \(z = 2\), \(x' = - 4\) va \(y' = 7\) quyidagi tenglama uchun \(z'\) ni aniqlang. \[{x^4} + \frac{y}{z} = 2{x^2}{z^2} - 3\]
  4. Quyida \(x\), \(y\) va \(z\) barcha \(t\) funksiyalari deb faraz qilinadi. Berilgan \(x = - 2\), \(y = 3\), \(z = 4\), \(y' = 6\) va \(z' = 0\) quyidagi tenglama uchun \(x'\) ni aniqlang. \[x\,{y^2}{z^2} = {x^3} - {z^4} - 8y\]
  5. Kvadratning tomonlari sekundiga 10 sm tezlikda ortib bormoqda. Maydoni 150 sm bo'lsa, kvadrat bilan o'ralgan maydon qanchalik tez oshadi2.
  6. Teng tomonli uchburchakning tomonlari soatiga 3 dyuym tezlikda kamayib bormoqda. Yon tomonlari 2 fut bo'lsa, uchburchak bilan o'ralgan maydon qanchalik tez kamayadi?
  7. Sferik sharsimon shar shunday to'ldirilmoqdaki, sirt maydoni 20 sm ga oshadi.2/sek radius 2 metr bo'lganda. Radius 2 metr bo'lganda havo shariga qanday tezlikda pompalanadi?
  8. Radiusi 2,5 fut bo'lgan silindrsimon rezervuardan 0,25 fut tezlikda suv to'kiladi.3/sek. Suv balandligi qanchalik tez pasayadi?
  9. Issiq havo shari erda bo'lganida shardan 125 fut masofada joylashgan arqon g'altakchasiga biriktirilgan. Issiq havo shari to'g'ridan-to'g'ri yuqoriga ko'tariladi, arqon uzunligi sekundiga 15 fut tezlikda oshadi. Issiq havo shari ko'tarilganidan keyin 20 soniyadan keyin qanchalik tez ko'tariladi?
    In the lower left corner of this image is a dot labeled “Spool of Rope”. In the upper right corner is a dot labeled “Balloon” and it has an arrow next to it pointing directly up. The two dots are connected with a diagonal line that has the text “This Length Increasing at 15 ft/sec” on it. The sketch also shows that the horizontal distance between the two points is 125 feet.
  10. Tosh yerdan 50 metr balandlikda joylashgan ko‘prikdan to‘g‘ri tushib, 10 m/sek tezlikda quladi. Yana bir kishi xuddi shu ko'prikda 7 metr uzoqlikda. Tosh yerga urilishi bilan tosh va ikkinchi shaxs orasidagi masofa qanday tezlikda oshadi?
    A sketch of the rock and the second person on the bridge. The point at the top left is labeled “Rock Dropped Here” and the point 7 meters to its right is the person. The rock is the point directly below the drop point at the bottom of the sketch with an arrow showing it falling straight down. The line drawn from the person down to the rock is labeled “Find Rate This Distance Changing”.
  11. Bir kishi yo'ldan 550 metr uzoqlikda va dastlab 800 metr uzoqlikda bo'lgan mashina odamga 45 m / sek tezlikda yaqinlashadi. Odam va mashina orasidagi masofa qanday tezlikda o'zgaradi (a) boshlanganidan 5 soniya o'tgach, (b) mashina to'g'ridan-to'g'ri odamning oldida bo'lganda va (c) Mashina odamning yonidan 10 soniya o'tgach.
    In the lower left corner of this image is a dot labeled “Person”. In the upper right corner is a dot labeled “Car” and it has an arrow next to it pointing to the left. The two dots are connected with a diagonal line that has the text “Find Rate This Distance Changing” on it. The sketch also shows that the horizontal distance between the two points is initially 800 meters and that the vertical distance between the two point is 550 meters.
  12. Ikki mashina dastlab bir-biridan 1200 milya masofada joylashgan. Shu bilan birga, A avtomobili sharqqa 35 milya tezlikda harakatlana boshlaydi, B mashinasi esa shimolga 55 milya tezlikda harakatlana boshlaydi (bu dastlabki sozlash uchun quyidagi eskizga qarang). Ikki mashina orasidagi masofa qanday tezlikda o'zgaradi (a) 5 soatlik sayohat, (b) 20 soatlik sayohat va (c) 40 soatlik sayohatmi?
    A sketch of the initial setup for the two cars. Car A’s initial position is the point at the lower left and Car B’s initial position is the point at the lower right, with the 1200 miles between them marked. Car A’s later position is a point to the right of its starting point with an arrow showing it driving east, and Car B’s later position is a point above its starting point with an arrow showing it driving north. The line drawn between the two later positions is labeled “Find Rate This Distance Changing”.
  13. 13. Muammoni takrorlang 12 Yuqoridagi muammodan tashqari, A avtomobili B avtomobili harakatlana boshlaganidan 4 soat o'tgach harakatlana boshlaydi. Qismlar uchun (a), (b) va (c) Bu avtomobil B uchun sayohat vaqtlari deb faraz qiling.
  14. Ikki kishi shahar blokida. Joylashtirish va masofalar uchun quyidagi eskizga qarang. A shaxsi shimoli-sharqiy burchakda, B shaxsi esa janubi-g'arbiy burchakda joylashgan. A shaxsi janubi-sharqiy burchakka 3 fut/sek tezlikda yura boshlaydi. To'rt soniyadan so'ng B shaxsi janubi-sharqiy burchakka 2 fut/sek tezlikda yura boshlaydi. Ularning orasidagi masofa qanday tezlikda o'zgaradi (a) Shaxs A yura boshlaganidan keyin 10 soniya va (b) A shaxsi yarim masofani bosib o'tgandan keyin?
    In the lower left corner of this image is a dot labeled “Person B” and has an arrow next to is pointing to the right. In the upper right corner is a dot labeled “Person A” and it has an arrow next to it pointing down. The two dots are connected with a diagonal line that has the text “Find Rate This Distance Changing” on it. The sketch also shows that the horizontal distance between the two points is initially 500 feet and that the vertical distance between the two point is 300 feet.
  15. Bir kishi uçurtmadan 75 metr uzoqlikda turadi va uçurtmaga ip bog'langan. Uçurtma 2 m/sek tezlikda havoda to'g'ri ko'tarila boshlaydi va bir vaqtning o'zida odam 0,75 m/sek tezlikda uçurtmalarni uchirish nuqtasiga qarab harakatlana boshlaydi. Uzunlik qatori keyin ortib boradimi yoki kamayadi (a) 4 soniya va (b) 20 soniya.
  16. Bir kishi raketa modelida sug'urta yoqadi va raketadan 3 fut / sek tezlikda uzoqlasha boshlaydi. Sug'urta yoqilganidan besh soniya o'tgach, raketa sekundiga 10 fut tezlikda to'g'ridan-to'g'ri havoga uchadi. Odam va raketa orasidagi masofa ortib bormoqda yoki kamaymoqda (a) ishga tushirilgandan keyin 6 soniya va (b) 12 soniyadan keyin?
    A sketch of the person and the model rocket. The person is the point at the lower left with an arrow showing them moving to the left, the initial location of the rocket launch is the point at the lower right, and the rocket is the point above that with an arrow showing it rising straight up. The line drawn between the person and the rocket is labeled “Is This Distance Increasing or Decreasing?”.
  17. Chiroq ustunda va sekundiga 9 dyuym tezlikda erga tushirilmoqda. 6 fut balandlikdagi odam yerda va ustundan 8 fut uzoqlikda. Yorug'lik erdan 15 fut balandlikda bo'lganda, odamlarning soyasi qanday tezlikda oshadi?
    A sketch of the light, the person and the shadow. The light is at the top of a pole on the left with an arrow showing that it is being lowered, the person is a vertical line of height 6 ft standing 8 ft away from the pole, and the light ray from the light 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.
  18. Chiroq poldan 10 metr balandlikdagi devorga o'rnatiladi. Devordan 12 metr narida 45 sm/sek tezlikda ustun to'g'ri ko'tarilmoqda. Ustun balandligi 6 metr bo'lganda, soyaning uchi qanday tezlikda harakatlanadi (a) qutbdan uzoqda va (b) devordan uzoqdami? E'tibor bering, quyidagi eskiz masshtabga mos kelmaydi…
    A sketch of the light, the pole and the shadow. The light is fixed to a wall 10 m above the floor on the left, the pole stands 12 m away from the wall with an arrow showing that it is being raised straight up, and the light ray from the light passes over the top of the pole and hits the floor to the right of the pole. The shadow is the darkened portion of the floor from the pole out to that point.
  19. Chiroq 15 fut balandlikdagi ustunning tepasida joylashgan. 5 fut balandlikdagi odam qutbdan boshlanadi va qutbdan 2,5 fut / sek tezlikda uzoqlashadi. 8 soniya harakat qilgandan so'ng, soyaning uchi qanday tezlikda harakat qiladi (a) odamdan uzoqda va (b) qutbdan uzoqdami?
    A sketch of the light, the person and the shadow. The light is at the top of a 15 ft pole on the left, the person is a vertical line of height 5 ft to the right of the pole with an arrow showing them walking to the right, and the light ray from the top of the pole 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.
  20. Suv idishi konus shaklida (konusning "nuqtasi" pastga qaratilgan deb faraz qiling) va 35 sm tezlikda suv sizib chiqmoqda.3/sek. Tankning asosiy radiusi 1 metr, balandligi esa 2,5 metr. Qachonki, suvning chuqurligi 1,25 metrga teng (a) chuqurlikning o'zgarishi va (b) suv tepasining radiusi o'zgaradimi?
    A sketch of the cone shaped tank as seen from the front, with the point of the cone at the bottom. The radius of the top of the tank is labeled 1 m and the height of the tank is labeled 2.5 m. Water fills the bottom portion of the cone and the surface of the water is a smaller circle part way up the tank.
  21. Uzunligi 20 metr, uchlari kengligi 7 metr va balandligi 10 metr bo'lgan teng yonli uchburchak shaklida. Faraz qilaylik, uchburchakning teng uzunlikdagi ikki tomoni suv idishining tomonlari, uchburchakning boshqa tomoni esa idishning ustki qismi va erga parallel. Tankga 2 m tezlikda suv quyiladi3/min. Suv qancha tezlikda 6 metr chuqurlikda bo'lsa (a) chuqurlikning o'zgarishi va (b) suvning yuqori qismining kengligi o'zgaradimi? E'tibor bering, quyidagi eskiz masshtabga mos kelmaydi…
    A sketch of the trough. It is 20 meters long, the triangular ends are 7 meters across the top and 10 meters deep, and the water in the trough is shaded in and fills the bottom part of the trough up to some depth.
  22. Uzunligi 9 fut bo'lgan suv trubasining uchlari teng qirrali uchburchak shaklida, uning uzunligi 1,5 fut. Tankning yuqori qismini erga parallel deb hisoblang. Agar suv tankdan 2 fut tezlikda pompalansa3/s chuqurligi 0,75 fut bo'lganda suvning chuqurligi qanday tezlikda o'zgaradi?
    A sketch of the trough. It is 9 feet long and the triangular ends are equilateral triangles with sides 1.5 feet long, with the 1.5 foot top side parallel to the ground. The water in the trough is shaded in and fills the bottom part of the trough up to some depth.
  23. Ko'tarilish burchagi (depressiya) - gorizontal chiziq va kuzatuvchining ko'zini gorizontal chiziq ustidagi (pastda) ob'ektga bog'laydigan chiziqdan hosil bo'lgan burchak. Kengligi 25 fut bo'lgan yo'l bilan ajratilgan binolar tomida ikki kishi bor. A shaxsi B shaxsidan 100 fut balandlikda joylashgan va o'z binosining tomidan tosh tushiradi va u sekundiga 3 fut tezlikda tushadi. Quyidagi eskizlar masshtabli emas….

    1. Tog‘ jinsi B shaxsidan 35 fut balandlikda bo‘lganida, B shaxsi tog‘ jinsining qulashini kuzatayotganda, balandlik burchagi qanday tezlikda o‘zgaradi?
      A sketch of the two people and the falling rock. Person A is the point at the top right, on the roof of the taller building, and is labeled as having dropped the rock. Person B is the point at the left, 25 ft away horizontally and 100 ft below Person A. The rock is a point on the vertical line below Person A with an arrow showing it falling and it is still above Person B. The angle at Person B between the horizontal line and the line up to the rock is labeled \(\theta \), Angle of Elevation.
    2. Tog‘ jinsi B shaxsidan 65 fut pastda bo‘lganida, B shaxsi toshning qulashini kuzatayotganda, tushkunlik burchagi qanday tezlikda o‘zgaradi?
      A sketch of the two people and the falling rock. Person A is the point at the top right, on the roof of the taller building, and is labeled as having dropped the rock. Person B is the point at the left, 25 ft away horizontally and 100 ft below Person A. The rock is a point on the vertical line below Person A with an arrow showing it falling and it has now fallen below Person B. The angle at Person B between the horizontal line and the line down to the rock is labeled \(\theta \), Angle of Depression.
  24. Balandlik burchagi - bu gorizontal chiziq va kuzatuvchining ko'zini gorizontal chiziq ustidagi ob'ektga bog'laydigan chiziqdan hosil bo'lgan burchak. Bir kishi binodan 15 metr narida turib, binoning yuzi bo'ylab tashqi liftning harakatlanishini kuzatmoqda. Balandlik burchagi 1 radian bo'lsa, u 0,15 radyan / sek tezlikda o'zgaradi. Bu vaqtda liftning tezligi qanday?
    A sketch of the person and the elevator. The person is the point at the lower left and the elevator is the point at the upper right with an arrow beside it showing that it is moving down the face of the building. A horizontal line 15 m long runs from the person to the base of the building and a second line runs from the person up to the elevator. The angle between them at the person is labeled \(\theta \), Angle of Elevation.
  25. Balandlik burchagi - bu gorizontal chiziq va kuzatuvchining ko'zini gorizontal chiziq ustidagi ob'ektga bog'laydigan chiziqdan hosil bo'lgan burchak. Bir kishi binodan 24 fut uzoqlikda joylashgan va tashqi liftni binoning yuzi bo'ylab harakatlanayotganini kuzatmoqda. Lift 4 fut/sek tezlikda yuqoriga ko'tarilmoqda va odam 0,75 fut/sek tezlikda bino tomon harakatlanmoqda. Lift odam yura boshlagan bir vaqtda erdan harakatlana boshlagan deb faraz qilsak, 10 soniyadan keyin balandlik burchagi ortib boradimi yoki kamayadimi?
    A sketch of the person and the elevator. The person is the point at the lower left with an arrow showing them moving to the right and the elevator is the point at the upper right with an arrow beside it showing that it is moving up the face of the building. A horizontal line 24 ft long runs from the person to the base of the building and a second line runs from the person up to the elevator. The angle between them at the person is labeled \(\theta \), Angle of Elevation.