DerivativeInterp - 6-masala yechimi | MathModel.uz

3.2-bo'lim : Hosilaning talqini

6. Quyida $f\left( x \right)$ funksiyasining grafigi keltirilgan. Bundan $f'\left( x \right)$ hosilasining grafigini chizish uchun foydalaning.

$The graph of $f\left( x \right)$ on $ - 4 \le x \le 4$. It is made up of three straight line segments. The first runs down from $\left( { - 4,5} \right)$ to a corner at $\left( { - 1,2} \right)$, the second runs up from there to a corner at $\left( {2,4} \right)$, and the third runs back down from there to $\left( {4,0} \right)$.$

Yechim

Funktsiyaning uchta qismi aslida chiziqlar bo'lgani uchun va chiziqqa teginish chizig'i shunchaki chiziqning o'zi bo'ladi, chunki biz egri chiziqning har bir qismida hosilani osongina hisoblashimiz mumkin.

Qismlarning har birida biz har bir qismning qiyaligini hisoblash uchun grafikdagi to'rdan foydalanishimiz mumkin. Grafikning har bir qismdagi qiyaligini bilish, o'z navbatida, har bir qism uchun teginish chizig'ining qiyaligini bildiradi. Bu, o'z navbatida, uchta qismning har biridagi hosila shundan dalolat beradi:

\[\begin{align*} x & < - 1 & \hspace{0.25in}f'\left( x \right) = - 1\\ - 1 < x & < 2 & \hspace{0.25in}f'\left( x \right) = \frac{2}{3}\\ x & > 2 & \hspace{0.25in}f'\left( x \right) = - 2\end{align*}\]

2-qadam

Eslab qoling 4-misol oldingi bo'limdan. Ushbu misolda biz mutlaq qiymat funksiyasining hosilasi $x = 0$ da mavjud emasligini ko'rsatdik. $x = 0$ ning chap tomonidagi chegara (chapdagi chiziqning qiyaligini beradi) va $x = 0$ ning o'ng tomonidagi chegara (o'ngdagi chiziqning qiyaligini beradi) boshqacha edi va shuning uchun umumiy limit mavjud emas edi. Bu, o'z navbatida, hosila o'sha paytda mavjud emasligini ko'rsatadi.

Bu erda bizda bir xil muammo bor. $x = - 1$ da o‘ng va chap qo‘l chegaralari bir xil emasligini va shuning uchun hosila $x = - 1$ da mavjud emasligini tekshirish uchun buni sizga qoldiramiz. Xuddi shunday, hosila $x = 2$ da mavjud emas.

Shunday qilib, ushbu ikki nuqtada grafikda ochiq nuqtalar bo'ladi.

3-qadam

Mana bu funktsiyaning hosilasining eskizi.

$The sketch of the derivative $f'\left( x \right)$. It is made up of three horizontal pieces with breaks between them. For $x < - 1$ the derivative is the constant -1, for $ - 1 < x < 2$ it is a constant of about 0.7, and for $x > 2$ it is the constant -2. Open dots at the ends of each piece show that the derivative does not exist at $x = - 1$ and $x = 2$.$