2. - / 1 points use newtons method with initial approximation ( x_1 = 1 ) to find ( x_2 ), the second…

2. - / 1 points use newtons method with initial approximation ( x_1 = 1 ) to find ( x_2 ), the second approximation to the root of the following equation. ( x^{4}-x - 4 = 0 ) ( x_2 = )

2. - / 1 points use newtons method with initial approximation ( x_1 = 1 ) to find ( x_2 ), the second approximation to the root of the following equation. ( x^{4}-x - 4 = 0 ) ( x_2 = )

Answer

Explanation:

Step1: Find the derivative of the function

Let (f(x)=x^{4}-x - 4). Then (f^{\prime}(x)=4x^{3}-1)

Step2: Apply Newton's method formula

Newton's method formula is (x_{n + 1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}). For (n = 1), (x_{1}=1). First, calculate (f(1)): (f(1)=1^{4}-1 - 4=-4). Then, calculate (f^{\prime}(1)): (f^{\prime}(1)=4\times1^{3}-1 = 3). Now, (x_{2}=x_{1}-\frac{f(x_{1})}{f^{\prime}(x_{1})}=1-\frac{-4}{3}) (x_{2}=1+\frac{4}{3}=\frac{3 + 4}{3}=\frac{7}{3}\approx2.333)

Answer:

(\frac{7}{3})