Asosiy tarkibga o'tish

3.2-bo'lim : Hosilaning talqini

1. \(f\left( x \right)\) funksiya grafigidan foydalanib, \(f'\left( a \right)\) qiymatini baholang.

  1. \(a = - 2\)
  2. \(a = 3\)
The graph of \(f\left( x \right)\) on \( - 3 \le x \le 4\). It is a parabola shaped curve that comes down from approximately \(\left( { - 3,6} \right)\), crosses the x-axis at about \(x = - 1.8\), reaches its lowest point at approximately \(\left( { - 0.2, - 2.5} \right)\) and then increases, crossing the x-axis at \(x = 2\) and reaching approximately \(\left( {4,3.5} \right)\) at the right end.

a \(a = - 2\)
Yechim

Hosilaning talqinlaridan biri uning ma'lum bir nuqtadagi funktsiyaga teginish chizig'ining qiyaligi ekanligini hisobga olsak, avval grafikdagi nuqtada teginish chizig'ini chizamiz.

The graph of \(f\left( x \right)\) with a tangent line sketched in at \(x = - 2\). The point on the graph at \(x = - 2\) is marked with a dot and the tangent line drawn through it falls steeply from the upper left to the lower right, showing that the function is decreasing and the derivative is negative there.
2-qadam

Bu erda funktsiya aniq kamayib bormoqda va shuning uchun biz bilamizki, bu nuqtada hosila salbiy bo'ladi. Endi, tangens chizig'ining bu eskizidan shunday ko'rinadiki, agar biz 1 dan oshib o'tsak, biz 4 ga tushamiz va shuning uchun biz buni taxmin qilishimiz mumkin,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f'\left( { - 2} \right) = - 4}}\]


b \(a = 3\)
Yechim

Hosilaning talqinlaridan biri bu funksiyaga ma'lum bir nuqtadagi teginish chizig'ining qiyaligi ekanligini hisobga olsak. Keling, avval nuqtada teginish chizig'ini chizamiz.

The graph of \(f\left( x \right)\) with a tangent line sketched in at \(x = 3\). The point on the graph at \(x = 3\) is marked with a dot and the tangent line drawn through it rises from the lower left to the upper right, showing that the function is increasing and the derivative is positive there.
2-qadam

Bu erda funktsiya aniq ortib bormoqda va shuning uchun biz bilamizki, bu nuqtada hosila ijobiy bo'ladi. Endi, tangens chizig'ining bu eskizidan shunday ko'rinadiki, agar biz 1 dan oshib o'tsak, biz 2 ga ko'taramiz va shuning uchun biz buni taxmin qilishimiz mumkin,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f'\left( 3 \right) = 2}}\]