3.4-bo'lim : Funktsiyaning ta'rifi
13. \(f\left( x \right)\) funksiyasi uchun ayirma qismi quyidagicha aniqlanadi:
\[\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\]\(f\left( x \right) = 2{x^2} - x\) funksiyasi uchun ayirma qismini hisoblang.
Yechim
Biz bu muammoni qismlarga bo'lib hal qilamiz. Avval \(f\left( {x + h} \right)\) ni hisoblaymiz.
\[f\left( {x + h} \right) = 2{\left( {x + h} \right)^2} - \left( {x + h} \right) = 2\left( {{x^2} + 2xh + {h^2}} \right) - \left( {x + h} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{2{x^2} + 4xh + 2{h^2} - x - h}}\]2-qadam
Endi biz \(f\left( {x + h} \right) - f\left( x \right)\) ni hisoblaymiz va biroz soddalashtiramiz.
\[\begin{align*}f\left( {x + h} \right) - f\left( x \right) & = 2{x^2} + 4xh + 2{h^2} - x - h - \left( {2{x^2} - x} \right)\\ & = 2{x^2} + 4xh + 2{h^2} - x - h - 2{x^2} + x = \require{bbox} \bbox[2pt,border:1px solid black]{{4xh + 2{h^2} - h}}\end{align*}\]\(f\left( x \right)\) ni ayirishda qavsdan ehtiyot bo'ling. Biz funktsiyani ayirishimiz kerak va shuning uchun funktsiyani ayirishimiz uchun hamma narsa atrofida qavs kerak.
3-qadam
Endi biz to'liq farq ko'rsatkichini hisoblash orqali muammoni tugatishimiz mumkin.
\[\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h} = \frac{{4xh + 2{h^2} - h}}{h} = \frac{{h\left( {4x + 2h - 1} \right)}}{h} = \require{bbox} \bbox[2pt,border:1px solid black]{{4x + 2h - 1}}\]