Use Newton's Method to find the largest positive solution of x⁴ + x³ - 6x - 6 = 0. Use 4 for your initial value and calculate eight iterations. A) 1.8381538 B) 1.9835930 C) 1.8171206 D) 2.3659571 E) 3.0268456

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Quizplus · Accepted Answer

Using Newton's Method, we have: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ where $f(x) = x⁴ + x³ F1F1F1S1-F1F1F10 6x F1F1F1S1-F1F1F10 6$ and $f'(x) = ⁴ + ³ F1F1F1S1-F1F1F10 6F1F1F1S1-F1F1F10 6$ Thus, with an initial value of $x_0=4$, we get: $x_1 = 4 - \frac{f(4)}{f'(4)} \approx 1.8381$ $x_2 = 1.9836$ $x_3 = 1.8171$ $x_4 = 2.3659$ $x_5 = 3.0268$ $x_6 = 2.7392$ $x_7 = 2.4377$ $x_8 = 2.1806$ Since we are looking for the largest positive solution, we can see that the sequence is decreasing after the third iteration, so the answer is the third iteration, which is approximately $1.8171$. Therefore, the best choice is C.

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

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