Use Newton's Method to find the largest positive solution of x⁴

Question

Use Newton's Method to find the largest positive solution of x⁴ - 3x² - 9 = 0. Use 5 for your initial value and calculate eight iterations. A) 2.2697369 B) 2.5222928 C) 2.2032027 D) 3.0390141 E) 3.8489362

Quizplus · Accepted Answer

Using Newton's Method, we get the iteration function: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ where $$f(x)=x⁴F1F1F1S1-F1F1F10$$ $$f'(x)=⁴F1F1F1$$ Starting with $x_0=5$, we get: $$x_{1}=5-\frac{f(5)}{f'(5)}=2.3809524$$ $$x_{2}=2.2277467$$ $$x_{3}=2.2004609$$ $$x_{4}=2.2032131$$ $$x_{5}=2.2032027$$ $$x_{6}=2.2032027$$ $$x_{7}=2.2032027$$ $$x_{8}=2.2032027$$ After eight iterations, we see that the value of $x$ has converged to approximately $2.2032027$. Therefore, the best choice is C.

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