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En appliquant la m\'ethode de Newton, avec une approximation de d\'epart \'egale \`a : x0=2x_0 = 2, pr\'esenter les 55 premi\`eres \'etapes de calcul it\'eratif d'un z\'ero (d'une racine) de la fonction est est : f(x)=3x5−2x4+4x2−7f(x) = 3x^5 - 2x^4 + 4x^2 - 7 ; donner les 55 r\'esultats avec au moins 55 d\'ecimales exactes ; commencer par \'ecrire la d\'eriv\'ee de cette fonction et la formule de Newton.

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Q. En appliquant la m\'ethode de Newton, avec une approximation de d\'epart \'egale \`a : x0=2x_0 = 2, pr\'esenter les 55 premi\`eres \'etapes de calcul it\'eratif d'un z\'ero (d'une racine) de la fonction est est : f(x)=3x5−2x4+4x2−7f(x) = 3x^5 - 2x^4 + 4x^2 - 7 ; donner les 55 r\'esultats avec au moins 55 d\'ecimales exactes ; commencer par \'ecrire la d\'eriv\'ee de cette fonction et la formule de Newton.
  1. Derivative of f(x)f(x): Step 11: Write the derivative of the function f(x)f(x).f(x)=3x5−2x4+4x2−7f(x) = 3x^5 - 2x^4 + 4x^2 - 7f′(x)=15x4−8x3+8xf'(x) = 15x^4 - 8x^3 + 8x
  2. Newton's Formula: Step 22: Apply Newton's formula to find x1x_1. Newton's formula: xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} x0=2x_0 = 2 x1=2−3(2)5−2(2)4+4(2)2−715(2)4−8(2)3+8(2)x_1 = 2 - \frac{3(2)^5 - 2(2)^4 + 4(2)^2 - 7}{15(2)^4 - 8(2)^3 + 8(2)} x1=2−96−32+16−7240−64+16x_1 = 2 - \frac{96 - 32 + 16 - 7}{240 - 64 + 16} x1=2−73192x_1 = 2 - \frac{73}{192} x1=1.61979x_1 = 1.61979
  3. Calculate x2x_2: Step 33: Calculate x2x_2 using Newton's formula.\newlinex2=1.61979−3(1.61979)5−2(1.61979)4+4(1.61979)2−715(1.61979)4−8(1.61979)3+8(1.61979)x_2 = 1.61979 - \frac{3(1.61979)^5 - 2(1.61979)^4 + 4(1.61979)^2 - 7}{15(1.61979)^4 - 8(1.61979)^3 + 8(1.61979)}\newlinex2=1.61979−18.546−8.423+10.471−759.217−16.963+12.958x_2 = 1.61979 - \frac{18.546 - 8.423 + 10.471 - 7}{59.217 - 16.963 + 12.958}\newlinex2=1.61979−13.59455.212x_2 = 1.61979 - \frac{13.594}{55.212}\newlinex2=1.37623x_2 = 1.37623
  4. Find x3x_3: Step 44: Find x3x_3.
    x3=1.37623−3(1.37623)5−2(1.37623)4+4(1.37623)2−715(1.37623)4−8(1.37623)3+8(1.37623)x_3 = 1.37623 - \frac{3(1.37623)^5 - 2(1.37623)^4 + 4(1.37623)^2 - 7}{15(1.37623)^4 - 8(1.37623)^3 + 8(1.37623)}
    x3=1.37623−6.825−2.599+7.548−728.845−6.150+11.009x_3 = 1.37623 - \frac{6.825 - 2.599 + 7.548 - 7}{28.845 - 6.150 + 11.009}
    x3=1.37623−4.77433.704x_3 = 1.37623 - \frac{4.774}{33.704}
    x3=1.24035x_3 = 1.24035
  5. Compute x4x_4: Step 55: Compute x4x_4.x4=1.24035−3(1.24035)5−2(1.24035)4+4(1.24035)2−715(1.24035)4−8(1.24035)3+8(1.24035)x_4 = 1.24035 - \frac{3(1.24035)^5 - 2(1.24035)^4 + 4(1.24035)^2 - 7}{15(1.24035)^4 - 8(1.24035)^3 + 8(1.24035)}x4=1.24035−2.862−1.209+6.135−714.433−3.150+9.922x_4 = 1.24035 - \frac{2.862 - 1.209 + 6.135 - 7}{14.433 - 3.150 + 9.922}x4=1.24035−0.78821.205x_4 = 1.24035 - \frac{0.788}{21.205}x4=1.20298x_4 = 1.20298

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