Asosiy tarkibga o'tish

4.13-bo'lim : Nyuton usuli

1-3-masalalar uchun Nyuton usulidan foydalanib, berilgan funktsiya va \({x_{\,0}}\) ning berilgan qiymati uchun \({x_{\,2}}\) ni aniqlang.

  1. \(f\left( x \right) = 7{x^3} - 8x + 4\), \({x_{\,0}} = - 1\)
  2. \(f\left( x \right) = \cos \left( {3x} \right) - \sin \left( x \right)\), \({x_{\,0}} = 0\)
  3. \(f\left( x \right) = 7 - {{\bf{e}}^{2x - 3}}\), \({x_0} = 5\)

4-8-masalalar uchun Nyuton usulidan foydalanib, berilgan tenglamaning berilgan oraliqda joylashgan oltita kasrgacha aniq ildizini toping.

  1. \(\left[ {1,2} \right]\) da \({x^5} = 6\)
  2. \(\left[ { - 1,1} \right]\) da \(2{x^3} - 9{x^2} + 17x + 20 = 0\)
  3. \(3 - 12x - 4{x^3} - 3{x^4} = 0\), \(\left[ { - 3, - 1} \right]\)
  4. \(\left[ { - 1,1} \right]\) da \({{\bf{e}}^x} = 4\cos \left( x \right)\)
  5. \({x^2} = {{\bf{e}}^{2 - {x^{\,2}}}}\)in \(\left[ {0,2} \right]\)

9-12-masalalar uchun Nyuton usulidan foydalanib, berilgan tenglamaning barcha ildizlarini oltita kasrgacha aniq toping.

  1. \(2{x^3} + 5{x^2} - 10x - 4 = 0\)
  2. \({x^4} + 4{x^3} - 54{x^2} - 92x + 105 = 0\)
  3. \(\displaystyle \frac{3}{2} - {{\bf{e}}^{ - {x^{\,2}}}} = \cos \left( x \right)\)
  4. \(\ln \left( x \right) = 2\cos \left( x \right)\)
  5. Aytaylik, biz \({x^3} - 7{x^2} + 8x - 3 = 0\) ning ildizini topmoqchimiz. Dastlabki nuqta sifatida \({x_{\,0}} = 4\) dan foydalanish mumkinmi? Ushbu misoldan ildizlarni taxmin qilish uchun Nyuton usulidan foydalanish haqida qanday xulosaga kelishingiz mumkin?
  6. Ushbu muammo uchun \(f\left( x \right) = {\cos ^2}\left( x \right) - \sin \left( x \right)\) funksiyasidan foydalaning.
    1. Funksiyani \(\left[ {0,9} \right]\) oraliqda chizing.
    2. Ushbu funktsiyaning ildizlaridan birini oltita kasrgacha topish uchun \({x_{\,0}} = 4\) dan foydalaning. Siz kutgan ildizni oldingizmi?
    3. Ushbu funktsiyaning ildizlaridan birini oltita kasrgacha topish uchun \({x_{\,0}} = 5\) dan foydalaning. Siz kutgan ildizni oldingizmi?
    4. Ushbu funktsiyaning ildizlaridan birini oltita kasrgacha topish uchun \({x_{\,0}} = 6\) dan foydalaning. Siz kutgan ildizni oldingizmi?
    5. Nyuton usuli yordamida tenglamalar ildizlarini topish uchun \({x_{\,0}}\) qiymatlarini tanlash haqida qanday xulosaga kelishingiz mumkin.
  7. \(2{x^5} - 7{x^3} + 3x - 1 = 0\) ildizlaridan birini oltita kasrgacha aniq topish uchun \({x_{\,0}} = 0\) dan foydalaning. Ushbu muammo uchun \({x_{\,0}}\) ning yaxshi qiymatini tanladikmi?