9.2-bo'lim : Parametrik tenglamalar bilan tangentlar
4. Berilgan nuqtada quyidagi parametrik tenglamalar to‘plamiga teguvchi chiziq(lar)ning tenglamasini toping.
\[x = {t^2} - 2t - 11\hspace{0.25in}y = t{\left( {t - 4} \right)^3} - 3{t^2}{\left( {t - 4} \right)^2} + 7\mbox{ at } \left( { - 3,7} \right)\]Yechim
Bizga parametrik tenglamalar to'plami uchun birinchi hosila kerak bo'ladi. Bizga quyidagi hosilalar kerak bo'ladi,
\[\begin{align*}\frac{{dx}}{{dt}} & = 2t - 2\\ & \\ \frac{{dy}}{{dt}} & = {\left( {t - 4} \right)^3} + 3t{\left( {t - 4} \right)^2} - 6t{\left( {t - 4} \right)^2} - 6{t^2}\left( {t - 4} \right) = {\left( {t - 4} \right)^3} - 3t{\left( {t - 4} \right)^2} - 6{t^2}\left( {t - 4} \right)\end{align*}\]Birinchi hosila shundan iboratki,
\[\frac{{dy}}{{dx}} = \frac{{\,\,\frac{{dy}}{{dt}}\,\,}}{{\,\,\frac{{dx}}{{dt}}\,\,}} = \frac{{{{\left( {t - 4} \right)}^3} - 3t{{\left( {t - 4} \right)}^2} - 6{t^2}\left( {t - 4} \right)}}{{2t - 2}}\]2-qadam
Yaxshi, biz yuqorida topilgan hosila \(t\) nuqtai nazaridan va biz \(\left( { - 3,7} \right)\) parametrik egri chizig'ini qo'yadigan \(t\) qiymatini (qiymatlarini) aniqlashimiz kerak.
Buni \(x\) va \(y\) koordinatalarini ma'lum parametrik tenglamalarga tenglashtirib, ikkala tenglamani qanoatlantiradigan \(t\) qiymat(lar)ini aniqlash orqali amalga oshirish juda oson.
Shunday qilib,
\[\begin{align*} - 3 & = {t^2} - 2t - 11\\ 0 & = {t^2} - 2t - 8\\ 0 & = \left( {t - 4} \right)\left( {t + 2} \right)\hspace{0.5in}\hspace{0.25in}\,\,\, \to \hspace{0.5in}\hspace{0.5in}t = - 2,t = 4\end{align*}\] \[\begin{align*}7 & = t{\left( {t - 4} \right)^3} - 3{t^2}{\left( {t - 4} \right)^2} + 7\\ 0 & = {\left( {t - 4} \right)^2}\left[ {t\left( {t - 4} \right) - 3{t^2}} \right]\\ 0 & = {\left( {t - 4} \right)^2}\left[ { - 4t - 2{t^2}} \right]\\ 0 & = - 2t{\left( {t - 4} \right)^2}\left[ {2 + t} \right]\hspace{0.5in}\hspace{0.25in}\,\,\,\,\,\, \to \hspace{0.5in}\hspace{0.25in}\,\,\,t = - 2,t = 0,t = 4\end{align*}\]Ushbu ro'yxatda parametrik egri chiziq \(t = - 2\) va \(t = 4\) uchun \(\left( { - 3,7} \right)\) da bo'lishini ko'rishimiz mumkin.
3-qadam
Oldingi bosqichdan biz nuqtada ikkita tangens chiziqqa ega bo'lishini ko'rishimiz mumkin. Bu erda har bir teginish chizig'i uchun qiyaliklar mavjud.
\(t = - 2\) da tangens chizig'ining qiyaligi,
\[m = {\left. {\frac{{dy}}{{dx}}} \right|_{t = - 2}} = - 24\]va \(t = 4\) da tangens chiziqning qiyaligi,
\[m = {\left. {\frac{{dy}}{{dx}}} \right|_{t = 4}} = 0\]4-qadam
\(t = - 2\) uchun teginish chizig'i shunday bo'ladi:
\[y = 7 - 24\left( {x + 3} \right)\hspace{0.25in} \to \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{y = - 24x - 65}}\]\(t = 4\) uchun teginish chizig'i shunday bo'ladi:
\[y = 7 - \left( 0 \right)\left( {x + 3} \right)\hspace{0.25in} \to \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{y = 7}}\]Ikkinchi tangens chiziqdan hayajonlanmang! Bu faqat ikkinchi tangens chiziq gorizontal chiziq ekanligini aytmoqda.