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3.2-bo'lim : Hosilaning talqini

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

  1. \(a = 1\)
  2. \(a = 4\)
The graph of \(f\left( x \right)\) on \(0 \le x \le 5\). The graph starts at approximately \(\left( {0, - 5} \right)\) and increases the whole way, slowly at first and then much more steeply. It crosses the x-axis at about \(x = 3.3\) and reaches approximately \(\left( {5,7} \right)\) at the right end.

a \(a = 1\)
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 \(x = 1\). The point on the graph at \(x = 1\) is marked with a dot and the tangent line drawn through it rises gently 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 1 ga ko'tarilamiz va shuning uchun biz buni taxmin qilishimiz mumkin,

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


b \(a = 4\)
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 \(x = 4\). The point on the graph at \(x = 4\) is marked with a dot and the tangent line drawn through it rises steeply 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 5 ga ko'taramiz va shuning uchun biz buni taxmin qilishimiz mumkin,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f'\left( 4 \right) = 5}}\]