Use Newton's method to approximate a root of the equation
\[ \cos(x^2 + 3) = x^3 \]
as follows. Let \( x_1 = 1 \) be the initial approximation.
What is the second approximation \( x_2 \)?
Answer (2)
Main Answer:
The second Newton's method approximation, x 2 = 0.775 , refines the initial guess using iterative refinement for the equation. ;
The second approximation x 2 for the root of the equation cos ( x 2 + 3 ) = x 3 using Newton's method with an initial guess of x 1 = 1 is approximately 0.8730 . This result is derived from calculating the function and its derivative at the initial guess, then applying the Newton's method formula. As a result, we refine our estimate of the root using iterative processes that provide improved approximations.
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