3.4-bo'lim : Mahsulot va miqdor qoidasi
9. \(x = 9\) da \(f\left( x \right) = \left( {1 + 12\sqrt x } \right)\left( {4 - {x^2}} \right)\) ga teguvchi chiziq tenglamasini toping.
Yechim
Biz bilamizki, funktsiyaning hosilasi bizga teginish chizig'ining qiyaligini beradi, shuning uchun bizga funktsiyaning hosilasi kerak bo'ladi. Biz hosilani olish uchun mahsulot qoidasidan foydalanamiz.
\[f'\left( x \right) = \left( {6{x^{ - \,\,\frac{1}{2}}}} \right)\left( {4 - {x^2}} \right) + \left( {1 + 12\sqrt x } \right)\left( { - 2x} \right) = \left( {\frac{6}{{\sqrt x }}} \right)\left( {4 - {x^2}} \right) - 2x\left( {1 + 12\sqrt x } \right)\]2-qadam
Esda tutingki, biz hosilani “soddalashtirish” bilan ovora emasmiz (kasr ko‘rsatkichini ildizga qaytarishdan tashqari), chunki buning uchun bizga haqiqatan ham tezkor baholash kerak bo‘ladi.
Aytgancha, bu muammo uchun bizga kerak bo'lgan baholashlar.
\[f\left( 9 \right) = \left( {37} \right)\left( { - 77} \right) = - 2849\hspace{0.5in}f'\left( 9 \right) = \left( 2 \right)\left( { - 77} \right) - 18\left( {37} \right) = - 820\]3-qadam
Endi biz qilishimiz kerak bo'lgan narsa teginish chizig'ining tenglamasini yozishdir.
\[y = f\left( 9 \right) + f'\left( 9 \right)\left( {x - 9} \right) = - 2849 - 820\left( {x - 9} \right)\hspace{0.25in} \to \hspace{0.25in}\,\require{bbox} \bbox[2pt,border:1px solid black]{{y = - 820x + 4531}}\]