Asosiy tarkibga o'tish

9.1-bo'lim : Parametrik tenglamalar va egri chiziqlar

1-9-masalalar uchun berilgan parametrik tenglamalar toʻplamining parametrini yoʻq qiling, parametrik egri chiziq grafigini chizing va \(x\) va \(y\) da mavjud boʻlishi mumkin boʻlgan chegaralarni bering.

  1. \(x = 2 + t\hspace{0.5in}y = {t^2} - 4t + 7\)
  2. \(x = 2 + t\hspace{0.5in}y = {t^2} - 4t + 7\hspace{0.25in} - 3 \le t \le 1\)
  3. \(x = 1 - {t^2}\hspace{0.5in}y = 3 + 2t\)
  4. \(x = 1 - {t^2}\hspace{0.5in}y = 3 + 2t\hspace{0.25in}\hspace{0.25in} - 2 \le t \le 3\)
  5. \(\displaystyle x = \frac{1}{2}\sqrt t \hspace{0.5in}y = t - \sqrt t - 6\hspace{0.5in}t \ge 0\)
  6. \(\displaystyle x = \frac{1}{2}\sqrt t \hspace{0.5in}y = t - \sqrt t - 6\hspace{0.5in}8 \le t \le 20\)
  7. \(\displaystyle x = - 6\cos \left( {4t} \right)\hspace{0.5in}\,y = 2\sin \left( {4t} \right)\hspace{0.5in} - \frac{\pi }{4} \le t \le \frac{\pi }{8}\)
  8. \(\displaystyle x = 1 - 3\sin \left( {\frac{1}{2}t} \right)\hspace{0.5in}\,y = 4\cos \left( {\frac{1}{2}t} \right)\,\)
  9. \(x = 6 - 7{{\bf{e}}^{ - 2t}}\hspace{0.5in}y = 4 + 3{{\bf{e}}^{ - 2t}}\)
  10. Quyidagi parametrik tenglamalar to‘plamiga oid har bir savolga javob bering \[x = 3\cos \left( {at} \right)\hspace{0.5in}y = 3\sin \left( {at} \right)\hspace{0.5in}0 \le t \le 2\pi \]
    1. \(a = 1\) uchun parametrik egri chiziqning grafigini chizing.
    2. \(a = 6\) uchun parametrik egri chiziqning grafigini chizing.
    3. \(a = \frac{1}{5}\) uchun parametrik egri chiziqning grafigini chizing.
    4. Umuman olganda, \(a > 0\) uchun a qiymati parametrik egri chiziq grafigiga qanday ta'sir qiladi?

11 - 21 masalalar uchun zarrachaning yo'li parametrik tenglamalar to'plami bilan berilgan. Zarrachaning yo'lini to'liq tasvirlab bering. Zarrachaning yo'lini to'liq tasvirlash uchun siz quyidagi ma'lumotlarni taqdim etishingiz kerak bo'ladi.

  1. Parametrni yo'q qilish orqali olingan tenglamaga asoslangan parametrik egri chiziqning (harakat yo'nalishini o'z ichiga olgan holda) eskizi.
  2. \(x\) va \(y\) limitlari.
  3. Parametrik egri chiziqning bir izi uchun \(t\) diapazoni.
  4. Agar muammoda \(t\) ning umumiy diapazoni taqdim etilsa, zarracha yaratadigan egri chiziq izlari soni.

  1. \(\displaystyle x = 6\cos \left( {\frac{1}{3}t} \right)\hspace{0.5in}y = 2 + \sin \left( {\frac{1}{3}t} \right)\hspace{0.5in} 0 \le t \le 75\pi \)
  2. \(x = 7 - 3\sin \left( {2t} \right)\hspace{0.5in}y = 4 + 2\cos \left( {2t} \right)\)
  3. \(\displaystyle x = 6{\cos ^2}\left( {3t} \right)\hspace{0.5in}y = 2 - 3{\sin ^2}\left( {3t} \right)\hspace{0.25in} - \frac{5}{6}\pi \le t \le 3\pi \)
  4. \(\displaystyle x = \sqrt {2 + {{\cos }^2}\left( {\frac{7}{4}t} \right)} \hspace{0.5in}y = \frac{1}{3}\sin\left( {\frac{7}{4}t} \right)\)
  5. \(x = 6 - {\sin ^3}\left( {4t} \right)\hspace{0.5in}y = 2\sin \left( {4t} \right)\hspace{0.5in} - 127\pi \le t \le 201\pi \)
  6. \(\displaystyle x = 3 + \cos \left( {\frac{1}{6}t} \right)\hspace{0.5in}y = 4 + {\cos ^2}\left( {\frac{1}{6}t} \right)\hspace{0.5in} - 90\pi \le t \le 216\pi \)
  7. \(x = {{\bf{e}}^{ - 4t}}\hspace{0.5in}y = 2{{\bf{e}}^{12t}}\)
  8. \(x = 1 + {{\bf{e}}^{3t}}\hspace{0.5in}y = {{\bf{e}}^{6t}}\hspace{0.5in} - 1 \le t \le 6\)
  9. \(x = 1 - \ln \left( t \right)\hspace{0.5in}y = {\left[ {\ln \left( t \right)} \right]^2}\hspace{0.5in} t > 0\)
  10. \(\displaystyle x = \cos \left( {\frac{1}{2}t} \right)\hspace{0.5in}y = \sec \left( {\frac{1}{2}t} \right)\hspace{0.5in} - \pi < t < \pi \)
  11. \(\displaystyle x = \sin \left( {2t} \right)\hspace{0.5in}y = {\sin ^2}\left( {2t} \right) - 4\sin \left( {2t} \right)\hspace{0.5in} - \frac{{91}}{4}\pi \le t \le \frac{{17}}{4}\pi \)

22 - 27-masalalar uchun berilgan qo'shimcha shartlarga javob beradigan (agar mavjud bo'lsa) berilgan tenglama uchun parametrik tenglamalar to'plamini yozing.

  1. \(x = \sin \left( {3 - {y^2}} \right) + {\cos ^2}\left( y \right)\)
  2. \(\displaystyle y = \frac{{6\cos \left( x \right) - 8}}{{{x^2} + 9x}}\)
  3. \({x^2} + {y^2} = 100\) va parametrik tenglamalardan kelib chiqadigan parametrik egri chiziq \(\left( {0,10} \right)\) da bo'lishi kerak, qachonki \(t = 0\) va egri chiziq soat yo'nalishi bo'yicha aylanishga ega bo'lishi kerak.
  4. \({x^2} + {y^2} = 100\) va parametrik tenglamalardan kelib chiqadigan parametrik egri chiziq \(\left( {0,10} \right)\) da bo'lishi kerak, qachonki \(t = 0\) va egri chiziq soat miliga teskari aylanishga ega bo'lishi kerak.
  5. \(\displaystyle \frac{{{x^2}}}{{25}} + {y^2} = 1\) va parametrik tenglamalardan kelib chiqadigan parametrik egri chiziq \(\left( { - 5,0} \right)\) da bo'lishi kerak, agar \(t = 0\) va egri chiziq soat miliga teskari aylanishga ega bo'lishi kerak.
  6. \(\displaystyle \frac{{{x^2}}}{{25}} + {y^2} = 1\) va parametrik tenglamalardan kelib chiqadigan parametrik egri chiziq \(\left( { - 5,0} \right)\) da bo'lishi kerak, qachonki \(t = 0\) va egri chiziq soat yo'nalishi bo'yicha aylanishga ega bo'lishi kerak.
  7. Quyidagi parametrik tenglamalar to‘plami uchun parametrni yo‘q qiling va hosil bo‘lgan tenglamani aniqlang. \[x = h + a\cos \left( {\omega \,t} \right)\hspace{0.25in}\hspace{0.25in}y = k + b\sin \left( {\omega \,t} \right)\]