9.7-bo'lim: Polar koordinatali tangentlar
Endi biz ba'zi hisob mavzularini qutb koordinatalari nuqtai nazaridan muhokama qilishimiz kerak.
Biz qutb egri chiziqlariga teginish chiziqlarini topishdan boshlaymiz. Bu holda biz tenglama \(r = f\left( \theta \right)\) shaklida bo'ladi deb faraz qilamiz. Ushbu shakldagi tenglama bilan biz ko'rib chiqqanimizda olingan \(\frac{{dy}}{{dx}}\) hosilasi uchun tenglamadan foydalanishimiz mumkin. parametrik tenglamali tangens chiziqlar. Buning uchun bizdan egri chiziqni ifodalash uchun parametrik tenglamalar to'plamini ishlab chiqishimiz kerak. Buni qilish aslida juda oson.
Oldingidagi ishimizdan biz qutb koordinatalaridan Dekart koordinatalariga o'tish uchun quyidagi konvertatsiya tenglamalari to'plamiga egamiz.
\[x = r\cos \theta \hspace{0.75in}y = r\sin \theta \]Endi biz tenglama \(r = f\left( \theta \right)\) shaklida deb taxmin qilganimizdan foydalanamiz. Buni ushbu tenglamalarga almashtirish egri chiziq uchun quyidagi parametrik tenglamalar to'plamini (parametr sifatida \(\theta\) bilan) beradi.
\[x = f\left( \theta \right)\cos \theta \hspace{0.75in}y = f\left( \theta \right)\sin \theta \]Endi bizga quyidagi hosilalar kerak bo'ladi.
\[\begin{align*}\frac{{dx}}{{d\theta }} & = f'\left( \theta \right)\cos \theta - f\left( \theta \right)\sin \theta & \hspace{0.75in} \frac{{dy}}{{d\theta }} & = f'\left( \theta \right)\sin \theta + f\left( \theta \right)\cos \theta \\ & = \frac{{dr}}{{d\theta }}\cos \theta - r\sin \theta & \hspace{0.75in} & = \frac{{dr}}{{d\theta }}\sin \theta + r\cos \theta \end{align*}\]\(\frac{{dy}}{{dx}}\) hosilasi shundan iboratki,
Polar koordinatali hosila
E'tibor bering, ushbu formulani eslab qolishdan ko'ra, uni qanday qilib olganimizni eslab qolish va parametrik tenglamalar formulasini eslab qolish osonroq bo'ladi.
Keling, bu bilan tezkor misol bilan ishlaylik.
Yechim
Bizga birinchi navbatda quyidagi hosila kerak bo'ladi.
\[\frac{{dr}}{{d\theta }} = 8\cos \theta \]\(\frac{{dy}}{{dx}}\) hosilasi uchun formula quyidagicha bo'ladi:
\[\frac{{dy}}{{dx}} = \frac{{8\cos \theta \sin \theta + \left( {3 + 8\sin \theta } \right)\cos \theta }}{{8{{\cos }^2}\theta - \left( {3 + 8\sin \theta } \right)\sin \theta }} = \frac{{16\cos \theta \sin \theta + 3\cos \theta }}{{8{{\cos }^2}\theta - 3\sin \theta - 8{{\sin }^2}\theta }}\]Tangens chiziqning qiyaligi,
\[m = {\left. {\frac{{dy}}{{dx}}} \right|_{\theta = \frac{\pi }{6}}} = \frac{{4\sqrt 3 + \frac{{3\sqrt 3 }}{2}}}{{4 - \frac{3}{2}}} = \frac{{11\sqrt 3 }}{5}\]Endi \(\theta = \frac{\pi }{6}\) da bizda \(r = 7\) mavjud. Tangens chiziqni olishimiz uchun mos keladigan \(x\)-\(y\) koordinatalarini olishimiz kerak.
\[x = 7\cos \left( {\frac{\pi }{6}} \right) = \frac{{7\sqrt 3 }}{2}\hspace{0.5in}y = 7\sin \left( {\frac{\pi }{6}} \right) = \frac{7}{2}\]Tangens chiziq u holda,
\[y = \frac{7}{2} + \frac{{11\sqrt 3 }}{5}\left( {x - \frac{{7\sqrt 3 }}{2}} \right)\]To'liqlik uchun bu erda egri chiziq va tangensning grafigi keltirilgan.
