Penjelasan dengan langkah-langkah:
batas galat = 0,0001
f(x) = x³ - 5x + 1
turunan: f'(x) = 3x² - 5
misalkan x0 = 3
Menentukan nilai x1:
[tex] \rm x_{0} = 3 \to \: f(x_{0}) = f(3) = {3}^{3} - 5(3) + 1 = 13[/tex]
[tex] \rm x_{0} = 3 \to \: f'(x_{0}) = f'(3) = {3(3)}^{2} - 5 = 22[/tex]
k = 0
[tex]x_{k + 1} = x_{k} - \frac{f(x_{k})}{f'(x_{k})} [/tex]
[tex]x_{0 + 1} = x_{0} - \frac{f(x_{0})}{f'(x_{0})}[/tex]
[tex]x_{1} = 3 - \frac{13}{22}[/tex]
[tex]x_{1} = 2.40909090909[/tex]
[tex]galat: x_{k} = |\frac{x_{k} - x_{k - 1}}{x_{k}}| [/tex]
[tex]galat: x_{1} = | \frac{x_{1} - x_{0}}{x_{1}}| [/tex]
[tex]galat: x_{1} = | \frac{2.40909090909 - 3}{2.40909090909} | [/tex]
[tex]galat: x_{1} = 0.24528301[/tex]
Nilai galat lebih dari 0,0001. Maka, perhitungan dilanjutkan.
nilai x2:
[tex] \rm x_{1} = 2.40909090909 \to \: f(x_{1}) = f(2.40909090909) = {2.40909090909}^{3} - 5(2.40909090909) + 1 = 2.93623215[/tex]
[tex] \rm x_{1} = 2.40909090909 \to \: f'(x_{1}) = f'(2.40909090909) = {3(2.40909090909)}^{2} - 5 = 12.41115702[/tex]
k = 1
[tex]x_{1 + 1} = x_{1} - \frac{f(x_{1})}{f'(x_{1})}[/tex]
[tex]x_{2} = 2.40909090909 - \frac{2.93623215}{12.41115702}[/tex]
[tex]x_{2} = 2.17251085[/tex]
[tex]galat: x_{2} = | \frac{x_{2} - x_{1}}{x_{2}}| [/tex]
[tex]galat: x_{2} = | \frac{2.17251085 - 2.40909090909}{2.17251085} | [/tex]
[tex]galat: x_{2} = 0.10889706[/tex]
nilai x3:
[tex] \rm x_{2} = 2.17251085 \to \: f(x_{2}) = f(2.17251085) = {2.17251085}^{3} - 5(2.17251085) + 1 = 0.39126983[/tex]
[tex] \rm x_{2} = 2.17251085 \to \: f'(x_{2}) = f'(2.17251085) = {3(2.17251085)}^{2} - 5 = -0.2801966[/tex]
k = 2
[tex]x_{2 + 1} = x_{2} - \frac{f(x_{2})}{f'(x_{2})}[/tex]
[tex]x_{3} = 2.17251085 - \frac{0.39126983}{-0.2801966}[/tex]
[tex]x_{3} = 3.56892261[/tex]
dst...
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Penjelasan dengan langkah-langkah:
batas galat = 0,0001
f(x) = x³ - 5x + 1
turunan: f'(x) = 3x² - 5
misalkan x0 = 3
Menentukan nilai x1:
[tex] \rm x_{0} = 3 \to \: f(x_{0}) = f(3) = {3}^{3} - 5(3) + 1 = 13[/tex]
[tex] \rm x_{0} = 3 \to \: f'(x_{0}) = f'(3) = {3(3)}^{2} - 5 = 22[/tex]
k = 0
[tex]x_{k + 1} = x_{k} - \frac{f(x_{k})}{f'(x_{k})} [/tex]
[tex]x_{0 + 1} = x_{0} - \frac{f(x_{0})}{f'(x_{0})}[/tex]
[tex]x_{1} = 3 - \frac{13}{22}[/tex]
[tex]x_{1} = 2.40909090909[/tex]
[tex]galat: x_{k} = |\frac{x_{k} - x_{k - 1}}{x_{k}}| [/tex]
[tex]galat: x_{1} = | \frac{x_{1} - x_{0}}{x_{1}}| [/tex]
[tex]galat: x_{1} = | \frac{2.40909090909 - 3}{2.40909090909} | [/tex]
[tex]galat: x_{1} = 0.24528301[/tex]
Nilai galat lebih dari 0,0001. Maka, perhitungan dilanjutkan.
nilai x2:
[tex] \rm x_{1} = 2.40909090909 \to \: f(x_{1}) = f(2.40909090909) = {2.40909090909}^{3} - 5(2.40909090909) + 1 = 2.93623215[/tex]
[tex] \rm x_{1} = 2.40909090909 \to \: f'(x_{1}) = f'(2.40909090909) = {3(2.40909090909)}^{2} - 5 = 12.41115702[/tex]
k = 1
[tex]x_{k + 1} = x_{k} - \frac{f(x_{k})}{f'(x_{k})} [/tex]
[tex]x_{1 + 1} = x_{1} - \frac{f(x_{1})}{f'(x_{1})}[/tex]
[tex]x_{2} = 2.40909090909 - \frac{2.93623215}{12.41115702}[/tex]
[tex]x_{2} = 2.17251085[/tex]
[tex]galat: x_{k} = |\frac{x_{k} - x_{k - 1}}{x_{k}}| [/tex]
[tex]galat: x_{2} = | \frac{x_{2} - x_{1}}{x_{2}}| [/tex]
[tex]galat: x_{2} = | \frac{2.17251085 - 2.40909090909}{2.17251085} | [/tex]
[tex]galat: x_{2} = 0.10889706[/tex]
Nilai galat lebih dari 0,0001. Maka, perhitungan dilanjutkan.
nilai x3:
[tex] \rm x_{2} = 2.17251085 \to \: f(x_{2}) = f(2.17251085) = {2.17251085}^{3} - 5(2.17251085) + 1 = 0.39126983[/tex]
[tex] \rm x_{2} = 2.17251085 \to \: f'(x_{2}) = f'(2.17251085) = {3(2.17251085)}^{2} - 5 = -0.2801966[/tex]
k = 2
[tex]x_{k + 1} = x_{k} - \frac{f(x_{k})}{f'(x_{k})} [/tex]
[tex]x_{2 + 1} = x_{2} - \frac{f(x_{2})}{f'(x_{2})}[/tex]
[tex]x_{3} = 2.17251085 - \frac{0.39126983}{-0.2801966}[/tex]
[tex]x_{3} = 3.56892261[/tex]
dst...