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4.5-bo'lim: Grafik shakli, I qism

1 va 2 masalalar uchun funksiya grafigi berilgan. Funksiyaning ortishi va kamayishi oraliqlarini aniqlang.

  1. The graph of a function on \( - 5 \le x \le 9\). Starting at approximately \(\left( { - 5,4} \right)\) the graph decreases to a valley at approximately \(\left( { - 3, - 1} \right)\), then increases to a peak at approximately \(\left( {1,2} \right)\), then decreases the whole way down to a valley at approximately \(\left( {7, - 5} \right)\) and finally increases to approximately \(\left( {9, - 2} \right)\) at the right end.
    Yechim
  2. The graph of a function on \(0 \le x \le 9.5\). The graph rises steeply from below the x-axis, crosses the x-axis at the origin and continues up to a peak at approximately \(\left( {1,4} \right)\). It then decreases to a valley at approximately \(\left( {4, - 2} \right)\), then increases again, flattening out into a short nearly level stretch at about \(y = 2\) between \(x = 7\) and \(x = 8.5\), and then rises to approximately \(\left( {9.5,3} \right)\) at the right end.
    Yechim

Quyida ning grafigi keltirilgan hosila funktsiyaning. Ushbu grafikdan qaysi intervallarni aniqlang funktsiyasi ortadi va kamayadi.

  1. The graph of the derivative of a function on \( - 8 \le x \le 5\). Starting well below the x-axis at the left the graph rises steeply, crosses the x-axis near \(x = - 7\), reaches a peak at approximately \(\left( { - 6,1.5} \right)\), then decreases and just touches the x-axis at \(x = - 2\) without crossing it. From there it increases to a peak at approximately \(\left( {3,3.5} \right)\) and then falls steeply, crossing the x-axis at \(x = 5\) and continuing down off the bottom of the graph.
    Yechim
  2. Bu muammo ba'zi funktsiyalar bilan bog'liq. Funktsiya haqida biz bilgan narsa shundaki, u hamma joyda mavjud va biz funktsiyaning hosilasi haqida quyida berilgan ma'lumotlarni ham bilamiz. Ushbu funktsiya haqida 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.
    \[\begin{array}{c}f'\left( { - 5} \right) = 0\,\,\,\,\,\,\,\,f'\left( { - 2} \right) = 0\,\,\,\,\,\,\,\,\,\,f'\left( 4 \right) = 0\,\,\,\,\,\,\,\,\,\,\,f'\left( 8 \right) = 0\\ f'\left( x \right) < 0\,\,\,\,\,{\rm{on}}\,\,\,\,\,\left( { - 5, - 2} \right),\,\,\,\left( { - 2,4} \right),\,\,\,\left( {8,\infty } \right)\hspace{0.25in}\hspace{0.25in}f'\left( x \right) > 0\,\,\,\,\,{\rm{on}}\,\,\,\,\,\left( { - \infty , - 5} \right),\,\,\,\left( {4,8} \right)\end{array}\] Yechim

5-12-masalalar 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. \(f\left( x \right) = 2{x^3} - 9{x^2} - 60x\) Yechim
  2. \(h\left( t \right) = 50 + 40{t^3} - 5{t^4} - 4{t^5}\) Yechim
  3. \(y = 2{x^3} - 10{x^2} + 12x - 12\) Yechim
  4. \(p\left( x \right) = \cos \left( {3x} \right) + 2x\) on \(\displaystyle \left[ - \frac{3}{2},\,\,2 \right]\) Yechim
  5. \(\displaystyle R\left( z \right) = 2 - 5z - 14\sin \left( {\frac{z}{2}} \right)\) on \(\left[ { - 10,7} \right]\) Yechim
  6. \(h\left( t \right) = {t^2}\,\sqrt[3]{{t - 7}}\) Yechim
  7. \(f\left( w \right) = w{{\bf{e}}^{2\, - \,\,{\frac{1}{2}}{w^{\,2}}}}\) Yechim
  8. \(g\left( x \right) = x - 2\ln \left( {1 + {x^2}} \right)\) Yechim
  9. Ba'zi \(f\left( x \right)\) funksiyasi uchun \(x = 4\) da nisbiy maksimal mavjudligi ma'lum. Ushbu funktsiya haqida quyidagi savollarning har biriga javob bering.
    1. Bu funksiya hosilasining eng oddiy shakli qanday?
      Eslatma: Haqiqatan ham lotinning ko'plab mumkin bo'lgan shakllari mavjud, shuning uchun bu muammoning qolgan qismini iloji boricha sodda qilish uchun siz o'zingiz o'ylab topish mumkin bo'lgan hosilaning eng oddiy shaklidan foydalanishni xohlaysiz.
    2. (a) dagi javobingizdan foydalanib, funktsiyaning eng umumiy shaklini aniqlang.
    3. \(f\left( 4 \right) = 1\) \(x = 4\) da nisbiy maksimalga ega bo'lgan funktsiyani toping.
      Eslatma: Ushbu qismga javobni aniqlash uchun (b) dan javobingizdan foydalanishingiz kerak.
    Yechim
  10. \(f\left( x \right)\) va \(g\left( x \right)\) ortib borayotgan funktsiyalarni hisobga olsak. Agar biz \(h\left( x \right) = f\left( x \right) + g\left( x \right)\)ni aniqlasak, \(h\left( x \right)\) ortib borayotgan funktsiya ekanligini ko'rsatadi. Yechim
  11. \(f\left( x \right)\) ortib borayotgan funksiya ekanligini hisobga olib, \(h\left( x \right) = {\left[ {f\left( x \right)} \right]^2}\) ni aniqlang. \(h\left( x \right)\) ortib borayotgan funksiya bo'ladimi? Ha bo'lsa, \(h\left( x \right)\) ortib borayotgan funktsiya ekanligini isbotlang. Agar yo'q bo'lsa, \(f\left( x \right)\) funksiyasi uchun zarur bo'lgan boshqa shartlarni aniqlay olasizmi, bu \(h\left( x \right)\) ham oshishiga kafolat beradi? Yechim