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4.6-bo'lim: Grafik shakli, II qism

1-1 bo'lim: Grafik shakli, I qism
  1. Funktsiyaning grafigi quyida keltirilgan. Funktsiya yuqoriga va pastga botiq bo'lgan intervallarni aniqlang.
    The graph of a function on \( - 3 \le x \le 8\). Starting below the x-axis at the left the graph rises to a small peak at approximately \(\left( { - 2,1} \right)\), decreases to the origin, then increases to a peak at approximately \(\left( {4,5} \right)\). It then falls steeply, crossing the x-axis near \(x = 6\), down to a valley at approximately \(\left( {7, - 3} \right)\) and then turns back up to approximately \(\left( {8, - 2} \right)\) at the right end.
    Yechim
  2. Quyida grafigi keltirilgan 2nd hosila funktsiyaning. Ushbu grafikdan qaysi intervallarni aniqlang funktsiyasi yuqoriga va pastga botiq bo'ladi.
    The graph of the second derivative of a function on \( - 5 \le x \le 4.5\). Coming down steeply from the upper left the graph crosses the x-axis at \(x = - 4\), dips to a shallow valley just below the axis and crosses back up at \(x = - 2\). It then rises to a peak at approximately \(\left( {0,2} \right)\), decreases and touches the x-axis at \(x = 3\) without crossing it, and then increases steeply to the upper right.
    Yechim

3-8 muammolar uchun quyidagi savollarga javob bering.

  1. Funktsiya uchun mumkin bo'lgan burilish nuqtalari ro'yxatini aniqlang.
  2. Funktsiya yuqoriga va pastga botiq bo'lgan intervallarni aniqlang.
  3. Funksiyaning burilish nuqtalarini aniqlang.

  1. \(f\left( x \right) = 12 + 6{x^2} - {x^3}\) Yechim
  2. \(g\left( z \right) = {z^4} - 12{z^3} + 84z + 4\) Yechim
  3. \(h\left( t \right) = {t^4} + 12{t^3} + 6{t^2} - 36t + 2\) Yechim
  4. \(h\left( w \right) = 8 - 5w + 2{w^2} - \cos \left( {3w} \right)\) on \(\left[ { - 1,2} \right]\) Yechim
  5. \(R\left( z \right) = z{\left( {z + 4} \right)^{\,{\frac{2}{3}}}}\) Yechim
  6. \(h\left( x \right) = {{\bf{e}}^{4 - {x^{\,2}}}}\) Yechim

9-14-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.
  4. Funktsiya yuqoriga va pastga botiq bo'lgan intervallarni aniqlang.
  5. Funksiyaning burilish nuqtalarini aniqlang.
  6. Bosqichlardagi ma'lumotlardan foydalaning (a) – (e) funksiyaning grafigini chizish uchun.

  1. \(g\left( t \right) = {t^5} - 5{t^4} + 8\) Yechim
  2. \(f\left( x \right) = 5 - 8{x^3} - {x^4}\) Yechim
  3. \(h\left( z \right) = {z^4} - 2{z^3} - 12{z^2}\) Yechim
  4. \(Q\left( t \right) = 3t - 8\sin \left( {{\displaystyle \frac{t}{2}}} \right)\) on \(\left[ { - 7,4} \right]\) Yechim
  5. \(f\left( x \right) = {x^{\,{\frac{4}{3}}}}\left( {x - 2} \right)\) Yechim
  6. \(P\left( w \right) = w{{\bf{e}}^{4w}}\) Yechim
  7. To'liq bitta burilish nuqtasi bo'lgan ko'phadning minimal darajasini aniqlang. Yechim
  8. Faraz qilaylik, biz \(f\left( x \right)\) kritik nuqtalari \(x = - 1\), \(x = 2\) va \(x = 6\) bo‘lgan polinom ekanligini bilamiz. Agar biz 2-chi hosila \(f''\left( x \right) = - 3{x^2} + 14x - 4\) ekanligini ham bilsak. Iloji bo'lsa, kritik nuqtalarning har birini nisbiy minimumlar, nisbiy maksimallar sifatida tasniflang. Agar tanqidiy nuqtalarni tasniflashning iloji bo'lmasa, nima uchun ularni tasniflash mumkin emasligini aniq tushuntiring. Yechim