Question

use newton’s method to find an approximate solution of ln(x) = 9 - x. start with x₀ = 2 and find x₂?
(do not round until the final answer. then round to six decimal places as needed.)
x₂ = \square

Explanation:

Step1: Define the function and its derivative

Let \( f(x)=\ln(x)-(9 - x)=\ln(x)+x - 9 \). Then the derivative \( f^{\prime}(x)=\frac{1}{x}+1 \).

Step2: Newton's method formula

Newton's method formula is \( x_{n + 1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})} \).

Step3: Find \( x_1 \)

Given \( x_0 = 2 \). First, calculate \( f(x_0)=f(2)=\ln(2)+2 - 9=\ln(2)-7\approx0.693147 - 7=-6.306853 \).
\( f^{\prime}(x_0)=f^{\prime}(2)=\frac{1}{2}+1 = 1.5 \).
Then \( x_1=x_0-\frac{f(x_0)}{f^{\prime}(x_0)}=2-\frac{-6.306853}{1.5}=2 + 4.204569=6.204569 \).

Step4: Find \( x_2 \)

Now, calculate \( f(x_1)=f(6.204569)=\ln(6.204569)+6.204569 - 9\approx1.825329+6.204569 - 9=-0.970102 \).
\( f^{\prime}(x_1)=f^{\prime}(6.204569)=\frac{1}{6.204569}+1\approx0.161172 + 1 = 1.161172 \).
Then \( x_2=x_1-\frac{f(x_1)}{f^{\prime}(x_1)}=6.204569-\frac{-0.970102}{1.161172}=6.204569 + 0.835472=7.039941 \).

Answer:

\( 7.039941 \)