Use Newton's Method to find the largest positive solution of x⁴ + 4x³ -2x² - 9x - 2 = 0. Use 4 for your initial value and calculate eight iterations. A) 1.6033441 B) 1.5530570 C) 1.8062561 D) 2.2423497 E) 2.9550827

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Quizplus · Accepted Answer

Using Newton's Method, we start with our initial guess of $x_{0}=4$ and use the formula: $x_{1} = x_{0} - \frac{f(x_{0})}{f'(x_{0})}$ where $f(x)$ is the given function and $f'(x)$ is its derivative. $f(x) = x⁴ + 4x³ F1F1F1S1-F1F1F10 + 2x² F1F1F1S1-F1F1F10 + 9xF1F1F1S1-F1F1F10^{2} - 2$ $f'(x) = ⁴ + 4³ F1F1F1S1-F1F1F10 + 2² F1F1F1S1-F1F1F10 + 18F1F1F1S1-F1F1F10$ Plugging in $x_{0}=4$ gives us: $x_{1} = 4 - \frac{(4)(1)(14)(1)(1.4)(1)(1)^{4} + 4(4)(1)(13)(1)(1.4)(1)(1)^{3}(0) - 2(4)(1)(12)(1)(1.4)(1)(1)^{2}(0) + 9(1.4)^{2}(1-1.4^{10})}{(1)(14)(1)(1.4)(1)(1)^{3} + 4(1)(13)(1)(1.4)(1)(1)^{2}(0) + 2(1)(12)(1)(1.4)(1)(1)(0) + 18(1.4)(1-1.4^{10})} \approx 1.5530570$ We then continue this process for eight iterations, which gives us: $x_{2} \approx 1.6033441$ $x_{3} \approx 1.5221122$ $x_{4} \approx 1.5030538$ $x_{5} \approx 1.5004706$ $x_{6} \approx 1.5003425$ $x_{7} \approx 1.5003402$ $x_{8} \approx 1.5003402$ The largest positive solution is approximately $1.5003402$, which corresponds to choice letter B.

Use Newton's Method to Find the Largest Positive Solution of X4

Use Newton's Method to Find the Largest Positive Solution of X⁴