3.5-bo'lim : Trig funksiyalarining hosilalari
13. \(\left[ { - 2,7} \right]\) diapazonida \(f\left( x \right) = 4\cos \left( x \right) - x\) funksiyasi ortib boradi va kamayadi.
Yechim
Bizga birinchi navbatda hosila kerak bo'ladi, chunki hosila bizga funktsiyaning o'zgarish tezligini berishini bilamiz. Mana lotin.
\[\require{bbox} \bbox[2pt,border:1px solid black]{{f'\left( x \right) = - 4\sin \left( x \right) - 1}}\]2-qadam
Keyinchalik, funktsiya qayerda o'zgarmasligini bilishimiz kerak va shuning uchun biz qilishimiz kerak bo'lgan narsa lotinni nolga tenglashtirish va hal qilishdir.
\[ - 4\sin \left( x \right) - 1 = 0\hspace{0.5in} \Rightarrow \hspace{0.5in}\sin \left( x \right) = - \frac{1}{4}\]Tez kalkulyator hisobi bizga shuni aytadi:
\[x = {\sin ^{ - 1}}\left( { - \frac{1}{4}} \right) = - 0.2527\]Trig tenglamalarni echish bo'yicha "Ko'rib chiqish" bo'limidagi ishimizni eslab, biz bilamizki, bu yechimga mos keladigan musbat burchak: \(x = 2\pi - 0.2527 = 6.0305\). Ikkalasidan ham foydalanish mumkin, lekin biz ijobiy burchakdan foydalanamiz.
Bundan tashqari, birlik doirasini tezkor tekshirishdan biz \(x = \pi + 0.2527 = 3.3943\) ikkinchi yechim bo'lishini ko'rishimiz mumkin.
Bularning barchasini birlashtirib, hosila nolga teng bo'lishini ko'ramiz,
\[x = 6.0305 + 2\pi n\,\,\,\,\,\,\,{\mbox{and}}\hspace{0.25in}x = 3.3943 + 2\pi n\hspace{0.25in}\,n = 0, \pm 1, \pm 2, \pm 3, \ldots \]Va nihoyat, biz ishlayotgan \(\left[ { - 2,7} \right]\) oralig'ida qaysi yechimlar to'g'ri kelishini aniqlash uchun ba'zi \(n\)-larni ulashimiz kerak.
Shunday qilib, \(\left[ { - 2,7} \right]\) oralig'ida funktsiya quyidagi uchta nuqtada o'zgarishni to'xtatadi.
\[x = - 0.2527,\,\,\,\,\,3.3943,\,\,\,\,6.0305\]3-qadam
Ushbu muammoga javob olish uchun biz bilishimiz kerak bo'lgan narsa, hosila ijobiy (va shuning uchun funktsiya ortib bormoqda) yoki salbiy (va shuning uchun funktsiya kamayishi) qaerda ekanligini bilishimiz kerak. Hosil uzluksiz bo'lgani uchun biz bilamizki, uning belgisini o'zgartirishi mumkin bo'lgan yagona joy hosila nolga teng bo'lgan joydir. Shunday qilib, biz ushbu bo'limda qilganimizdek, tezkor raqamlar qatori bizga turli intervallar uchun hosilaning belgisini beradi.
Mana bu muammo uchun raqam chizig'i.
![A number line for the derivative of \(f\left( x \right) = 4\cos \left( x \right) - x\) over the interval \(\left[ { - 2,7} \right]\), with dashed vertical lines at \(x = - 0.2527\), \(x = 3.3943\) and \(x = 6.0305\) dividing it into four regions. In the first region \(f'\left( { - 1} \right) = 2.37\) so \(f'\left( x \right) > 0\), in the second \(f'\left( 0 \right) = - 1\) so \(f'\left( x \right) < 0\), in the third \(f'\left( 4 \right) = 2.03\) so \(f'\left( x \right) > 0\), and in the fourth \(f'\left( 7 \right) = - 3.63\) so \(f'\left( x \right) < 0\).](/Solutions/CalcI/DiffTrigFcns//Solutions/CalcI/DiffTrigFcns/DiffTrigFcns_Files/image001.webp)
Bundan biz quyidagi o'sish/kamayish ma'lumotlarini olamiz.
\[\require{bbox} \bbox[2pt,border:1px solid black]{\begin{align*}{\mbox{Increasing :}}&\,\, - 2 \le x < - 0.2527,\,\,\,\,3.3943 < x < 6.0305\\ {\mbox{Decreasing :}}&\,\, - 0.2527 < x < 3.3943,\,\,\,\,6.0305 < x \le 7\end{align*}}\]E'tibor bering, biz faqat \(\left[ { - 2,7} \right]\) oralig'ida nima sodir bo'layotganini ko'rib chiqdik, chunki biz ushbu intervaldan tashqarida funktsiyaning ortishi/kamayishi haqida hech narsa deya olmaymiz.