4.2-bo'lim : Muhim nuqtalar
13. \(V\left( t \right) = {\sin ^2}\left( {3t} \right) + 1\) kritik nuqtalarini aniqlang.
Yechim
Ushbu muammoga javob olish uchun bizga birinchi hosila kerak bo'ladi, shuning uchun uni olamiz.
\[V'\left( t \right) = 6\sin \left( {3t} \right)\cos \left( {3t} \right)\]2-qadam
Eslatib o'tamiz, tanqidiy nuqtalar lotin nolga teng bo'lgan va/yoki mavjud bo'lmagan joydir.
Ushbu lotin hamma joyda mavjud va shuning uchun biz bu haqda tashvishlanishga hojat yo'q. Shuning uchun, biz qilishimiz kerak bo'lgan narsa, hosila nolga teng ekanligini aniqlashdir. Shunday qilib, biz tenglamani yechishimiz kerak,
\[6\sin \left( {3t} \right)\cos \left( {3t} \right) = 0\hspace{0.25in} \to \hspace{0.25in}\sin \left( {3t} \right) = 0\hspace{0.25in}{\mbox{or}}\hspace{0.25in}\cos \left( {3t} \right) = 0\]3-qadam
Shunday qilib, endi biz ushbu ikkita trig tenglamani echishimiz kerak.
Birlik doirasiga tez nazar tashlasak, biz 0 va \(\pi \) da sinus nolga teng ekanligini ko'rishimiz mumkin, shuning uchun \(\sin \left( {3t} \right) = 0\) ning barcha yechimlari shunday bo'ladi:
\[\begin{array}{*{20}{l}}{3t = 0 + 2\pi n}\\{3t = \pi + 2\pi n}\end{array}\hspace{0.5in} \to \hspace{0.5in}\begin{array}{*{20}{l}}{t = {\displaystyle \frac{2}{3}}\pi n}\\{t = {\displaystyle \frac{1}{3}}\pi + {\displaystyle \frac{2}{3}}\pi n}\end{array}\hspace{0.25in}\,n = 0, \pm 1, \pm 2, \pm , \ldots \]Birlik doirasiga yana bir nazar tashlasak, \({\frac{\pi}{2}}\) va \({\textstyle{{3\pi } \over 2}}\) da kosinus nolga teng ekanligini ko'ramiz va shuning uchun \(\cos \left( {3t} \right) = 0\) ning barcha yechimlari shunday bo'ladi:
\[\begin{array}{*{20}{l}}{3t = {\displaystyle \frac{\pi}{2}} + 2\pi n}\\{3t = {\displaystyle \frac{3\pi}{2}} + 2\pi n}\end{array}\hspace{0.5in} \to \hspace{0.5in}\begin{array}{*{20}{c}}{t = {\displaystyle \frac{\pi}{6}} + {\displaystyle \frac{2}{3}}\pi n}\\{t = {\displaystyle \frac{\pi}{2}} + {\displaystyle \frac{2}{3}}\pi n}\end{array}\hspace{0.25in}\,n = 0, \pm 1, \pm 2, \pm , \ldots \]Shuning uchun funktsiyaning kritik nuqtalari:
\[\require{bbox} \bbox[2pt,border:1px solid black]{{t = {\frac{2}{3}}\pi n,\,\,\,t = {\frac{1}{3}}\pi + {\frac{2}{3}}\pi n,\,\,\,\,t = {\frac{\pi}{6}} + {\frac{2}{3}}\pi n,\,\,\,\,\,t = {\frac{\pi}{2}} + {\frac{2}{3}}\pi n\hspace{0.25in}n = 0, \pm 1, \pm 2, \pm , \ldots }}\]Agar trig tenglamalarni qanday yechish kerakligini eslay olmasangiz, orqaga qaytib, eslatmalarning Ko'rib chiqish bo'limida ushbu bo'limlarni ko'rib chiqing.