Asosiy tarkibga o'tish

4.5-bo'lim : Grafik shakli, I qism

1 – 4 masalalar uchun funksiya grafigi berilgan. Funksiyaning ortishi va kamayishi oraliqlarini aniqlang.

  1. This graph has no y scale. It starts at x=-6 in the second quadrant that is the highest point on the graph. The graph has a valley at x=-4 in the second quadrant, followed by a peak at x=-2 in the second quadrant, then a valley at x=5 in the 4th quadrant and ending at a point in fourth quadrant at x=7.
  2. This graph has no y scale. It starts at x=-4 in the second quadrant that is the followed by a peak at x=-3 in the second quadrant. The graph then has a valley at x=1 in the first quadrant, followed by a peak at x=3 in the first quadrant, then a valley at x=6 in the 4th quadrant and ending at a point in slightly above the x-axis in the first quadrant at x=7.
  3. This graph has no y scale. It starts at x=-7 in the second quadrant that is the followed by a peak at x=-6 in the second quadrant. The graph then goes through a point at x=-2 perfectly flat and then has a valley at x=1 in the fourth quadrant and ending at a point in the first quadrant at x=2.
  4. This graph has no y scale. It starts at approximately x=-2.5 on the x-axis and is the followed by a peak at x=-2 in the second quadrant. The graph then goes through the origin perfectly flat and then has a valley at x=2 in the fourth quadrant. It then goes through a point at x=4 perfectly flat and ends at a point in the first quadrant at x=5 that is the highest point on the graph.

5-7-masalalar uchun grafigi hosila funktsiya berilgan. Ushbu grafikdan qaysi intervallarni aniqlang funktsiyasi ortadi va kamayadi.

  1. This graph has no y scale. It starts in the fourth quadrant at x=-9 and goes through the x-axis at x=-8 reaching a peak at approximately x=5. It then goes through the axis at x=-1 and reaches a valley at approximately x=1. Finally it goes back through the x-axis at x=3 and ends at a point in the first quadrant at x=4.
  2. This graph has no y scale. It starts in the second quadrant at x=-6 and goes down and just touches the x-axis at x=-5 and then is followed by a peak at approximately x=-3.8. It then goes through the axis at x=-3 and reaches a valley at approximately x=-1. Finally, it goes through the origin and ends at a point in the first quadrant at approximately x=0.5.
  3. This graph has no y scale. It starts in the fourth quadrant at approximately x=-2.5 and goes up and just touches the x-axis at x=-2 and then is followed by a valley at approximately x=-0.8. It then goes back up to just touch the x-axis at x=1 followed by another valley at approximately x=2.2. It goes back up to just touch the x-axis again at x=3 and then ends at a point in the fourth quadrant at approximately x=3.5.

8 – 10-masalalar uchun funktsiyaning hosilasi haqida ma’lum ma’lumotlar berilgan. Ushbu ma'lumotlardan quyidagi savollarning har biriga javob bering.

  1. Funktsiyaning kritik nuqtalarini aniqlang.
  2. Funksiyaning ortishi va kamayishi oraliqlarini aniqlang.
  3. Kritik nuqtalarni nisbiy maksimal, nisbiy minimum yoki hech biri sifatida tasniflang.

  1. \[\begin{array}{c}f'\left( 1 \right) = 0\hspace{0.5in}f'\left( 3 \right) = 0\hspace{0.5in}f'\left( 8 \right) = 0\\ f'\left( x \right) < 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( { - \infty ,1} \right),\,\,\,\left( {3,8} \right)\hspace{0.25in}f'\left( x \right) > 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( {1,3} \right),\,\,\,\left( {8,\infty } \right)\end{array}\]
  2. \[\begin{array}{c}g'\left( { - 2} \right) = 0\hspace{0.5in}\,g'\left( 0 \right) = 0\hspace{0.5in}g'\left( 3 \right) = 0\hspace{0.5in}\,g'\left( 6 \right) = 0\\ g'\left( x \right) < 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( {0,3} \right),\,\,\,\left( {6,\infty } \right)\hspace{0.5in}g'\left( x \right) > 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( { - \infty , - 2} \right),\,\,\,\left( { - 2,0} \right),\,\,\,\left( {3,6} \right)\end{array}\]
  3. \[\begin{array}{c}h'\left( { - 1} \right) = 0\hspace{0.5in}\,h'\left( 2 \right) = 0\hspace{0.5in}h'\left( 5 \right) = 0\\ h'\left( x \right) < 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( { - \infty , - 1} \right),\,\,\,\left( { - 1,2} \right)\hspace{0.5in}h'\left( x \right) > 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( {2,5} \right),\,\,\,\left( {5,\infty } \right)\end{array}\]

11-28 muammolar uchun quyidagi savollarga javob bering.

  1. Funktsiyaning kritik nuqtalarini aniqlang.
  2. Funksiyaning ortishi va kamayishi oraliqlarini aniqlang.
  3. Kritik nuqtalarni nisbiy maksimal, nisbiy minimum yoki hech biri sifatida tasniflang.

  1. \(f\left( t \right) = {t^3} - 15{t^2} + 63t + 3\)
  2. \(g\left( x \right) = 20 + 8{x^2} + 4{x^3} - {x^4}\)
  3. \(Q\left( w \right) = 8{w^3} - 18{w^2} - 24w - 10\)
  4. \(\displaystyle f\left( x \right) = {x^5} + \frac{5}{4}{x^4} - 20{x^3} - 7\)
  5. \(P\left( x \right) = 5 - 4x - 9{x^2} - 3{x^3}\)
  6. \(R\left( z \right) = {z^5} + {z^4} - 6{z^3} + 5\)
  7. \(h\left( z \right) = 1 - 12{z^2} - 9{z^3} - 2{z^4}\)
  8. \(Q\left( t \right) = 7 - t + \sin \left( {4t} \right)\)on \(\displaystyle \left[ - \frac{3}{2},\frac{3}{2} \right]\)
  9. \(\displaystyle f\left( z \right) = 6z - 20\cos \left( \frac{z}{2} \right)\), \(\left[ {0,22} \right]\)
  10. \(\displaystyle g\left( x \right) = 24\cos \left( \frac{x}{3} \right) + 8x + 2\), \(\left[ { - 30,25} \right]\)
  11. \(h\left( w \right) = 9w - 5\sin \left( {2w} \right)\), \(\left[ { - 5,0} \right]\)
  12. \(h\left( x \right) = \sqrt[5]{x}\,\,\left( {x + 7} \right)\)
  13. \(W\left( z \right) = \left( {10 - {w^2}} \right){\left( {w + 2} \right)^{\frac{2}{3}}}\)
  14. \(f\left( t \right) = \left( {{t^2} - 8} \right)\,\,\,\,\sqrt[3]{{{t^2} - 4}}\)
  15. \(f\left( x \right) = {{\bf{e}}^{\frac{1}{3}{x^{\,3}} - {x^{\,2}} - 3x}}\)
  16. \(h\left( z \right) = \left( {{z^2} - 8} \right){{\bf{e}}^{3\, - z}}\)
  17. \(A\left( t \right) = \ln \left( {{t^2} + 5t + 8} \right)\)
  18. \(g\left( x \right) = x - 3 + \ln \left( {1 + x + {x^2}} \right)\)
  19. Quyidagi savollarning har biriga javob bering.
    1. Aynan bitta nisbiy ekstremalga ega bo‘lgan ko‘phadning minimal darajasi qancha?
    2. Aniq ikkita nisbiy ekstremitaga ega boʻlgan koʻphadning minimal darajasi qancha?
    3. To'liq uchta nisbiy ekstremitaga ega bo'lgan ko'phadning minimal darajasi qancha?
    4. Aynan \(n\) nisbiy ekstremasi bo'lgan ko'phadning minimal darajasi qancha?
  20. Ba'zi \(f\left( x \right)\) funksiyasi uchun \(x = - 4\) da nisbiy minimum mavjudligi ma'lum. Ushbu funktsiya haqida quyidagi savollarning har biriga javob bering.
    1. Bu funktsiyaning hosilasi eng oddiy shakl qaysi?
      Eslatma: Haqiqatan ham lotinning ko'plab mumkin bo'lgan shakllari mavjud, shuning uchun bu muammoning qolgan qismini iloji boricha soddalashtirish uchun hosilaning eng oddiy shaklidan foydalanishni xohlaysiz.
    2. (a) dagi javobingizdan foydalanib, funktsiyaning o'zi olishi mumkin bo'lgan eng umumiy shaklni aniqlang.
    3. \(f\left( { - 4} \right) = 6\) \(x = - 4\) da nisbiy minimumga ega bo'lgan funktsiyani toping.
      Eslatma: Bu erda ko'plab javoblar mavjud, shuning uchun ulardan birini bering.
  21. Ba'zi \(f\left( x \right)\) funksiyasi uchun \(x = - 1\) da nisbiy maksimal mavjudligi ma'lum. Ushbu funktsiya haqida quyidagi savollarning har biriga javob bering.
    1. Bu funktsiyaning hosilasi eng oddiy shakl qaysi?
      Eslatma: Haqiqatan ham lotinning ko'plab mumkin bo'lgan shakllari mavjud, shuning uchun bu muammoning qolgan qismini iloji boricha soddalashtirish uchun hosilaning eng oddiy shaklidan foydalanishni xohlaysiz.
    2. (a) dagi javobingizdan foydalanib, funktsiyaning o'zi olishi mumkin bo'lgan eng umumiy shaklni aniqlang.
    3. \(f\left( { - 1} \right) = 3\) \(x = - 1\) da nisbiy maksimalga ega bo'lgan funktsiyani toping.
      Eslatma: Bu erda ko'plab javoblar mavjud, shuning uchun ulardan birini bering.
  22. Ba'zi \(f\left( x \right)\) funksiyasi uchun \(x = 3\) da nisbiy minimal yoki nisbiy maksimal bo'lmagan kritik nuqta mavjudligi ma'lum. Ushbu funktsiya haqida quyidagi savollarning har biriga javob bering.
    1. Bu funktsiyaning hosilasi eng oddiy shakl qaysi?
      Eslatma: Haqiqatan ham lotinning ko'plab mumkin bo'lgan shakllari mavjud, shuning uchun bu muammoning qolgan qismini iloji boricha soddalashtirish uchun hosilaning eng oddiy shaklidan foydalanishni xohlaysiz.
    2. (a) dagi javobingizdan foydalanib, funktsiyaning o'zi olishi mumkin bo'lgan eng umumiy shaklni aniqlang.
    3. \(f\left( 3 \right) = 2\) nisbiy minimal yoki nisbiy maksimal bo'lmagan \(x = 3\) da kritik nuqtaga ega bo'lgan funktsiyani toping.
      Eslatma: Bu erda ko'plab javoblar mavjud, shuning uchun ulardan birini bering.
  23. \(f\left( x \right)\) ba'zi funksiyalari uchun \(x = 1\)da nisbiy maksimal va \(x = 4\)da nisbiy minimum mavjudligi ma'lum. Ushbu funktsiya haqida quyidagi savollarning har biriga javob bering.
    1. Bu funktsiyaning hosilasi eng oddiy shakl qaysi?
      Eslatma: Haqiqatan ham lotinning ko'plab mumkin bo'lgan shakllari mavjud, shuning uchun bu muammoning qolgan qismini iloji boricha soddalashtirish uchun hosilaning eng oddiy shaklidan foydalanishni xohlaysiz.
    2. (a) dagi javobingizdan foydalanib, funktsiyaning o'zi olishi mumkin bo'lgan eng umumiy shaklni aniqlang.
    3. \(f\left( 1 \right) = 6\) va \(f\left( 4 \right) = - 2\) \(x = 1\) da nisbiy maksimalga va \(x = 4\) da nisbiy minimumga ega bo‘lgan funksiyani topishini hisobga olsak.
      Eslatma: Bu erda ko'plab javoblar mavjud, shuning uchun ulardan birini bering.
  24. \(f\left( x \right)\) va \(g\left( x \right)\) ortib borayotgan funktsiyalarni hisobga olsak, \(h\left( x \right) = f\left( x \right) - g\left( x \right)\) har doim ortib borayotgan funktsiya bo'ladimi? Agar shunday bo'lsa, \(h\left( x \right)\) ortib borayotgan funktsiya bo'lishini isbotlang. Agar yo'q bo'lsa, \(f\left( x \right)\) va \(g\left( x \right)\) o'suvchi funktsiyalarni toping, shunda \(h\left( x \right)\) kamayuvchi funktsiya bo'ladi va \(h\left( x \right)\) ortib boruvchi funksiya bo'lishi uchun boshqa ortib borayotgan funktsiyalar to'plamini toping.
  25. \(f\left( x \right)\) ortib borayotgan funksiya ekanligini hisobga olsak. \(h\left( x \right) = f\left( x \right) - g\left( x \right)\) ortib borayotgan funktsiya bo'lishi uchun biz \(g\left( x \right)\) ga qo'yishimiz mumkin bo'lgan bir nechta shartlar mavjud. Ushbu mumkin bo'lgan shartlardan iloji boricha ko'proq narsani aniqlang.
  26. \(f\left( x \right)\) funksiyasi uchun \(f\left( x \right)\) da №15 da berilgan amaliyotdan farqli shartlar to‘plamini aniqlang, ular uchun \(h\left( x \right) = {\left[ {f\left( x \right)} \right]^2}\) ortib boruvchi funksiya bo‘ladi.
  27. \(f\left( x \right)\) funksiyasi uchun \(f\left( x \right)\) da bitta shartni aniqlang, u uchun \(h\left( x \right) = {\left[ {f\left( x \right)} \right]^3}\) ortib borayotgan funktsiya bo'ladi.
  28. \(f\left( x \right)\) va \(g\left( x \right)\) ijobiy funksiyalar ekanligini hisobga olsak. Ular bo'yicha \(h\left( x \right) = f\left( x \right)g\left( x \right)\) ortib borayotgan funksiya bo'ladigan shartlar to'plamini aniqlang. E'tibor bering, bu erda bir nechta mumkin bo'lgan shartlar to'plami mavjud, ammo "eng oddiy" shartlarni aniqlashga harakat qiling.
  29. \(\displaystyle h\left( x \right) = \frac{{f\left( x \right)}}{{g\left( x \right)}}\) uchun #38 ni takrorlang.
  30. \(f\left( x \right)\) va \(g\left( x \right)\) ortib borayotgan funktsiyalarni hisobga olsak, \(h\left( x \right) = f\left( {g\left( x \right)} \right)\) ham ortib borayotgan funksiya bo'lishini isbotlaydi.