Question

In Exercises \(1-4,\) complete two iterations of Newton's Method for the function using the given initial guess. $$ f(x)=\sin x, \quad x_{1}=3 $$

Short answer

To find the answers, you need to calculate the two expressions of Step 3 and Step 5. The obtained values will be the output of the first and second iteration of Newton's Method for the function \(f(x) = \sin x\) with initial guess \(x_1 = 3\).

Step-by-step solution

Function Derivation

First, derive the function \(f(x) = \sin x\) to get \(f'(x) = \cos x\). This derivative will be used in Newton's Method formula.

First Iteration

Now, conduct the first iteration using the initial guess \(x_1 = 3\) and the Newton's Method formula \(x_{n+1} = x_{n} - \dfrac{f(x_n)}{f'(x_n)}\). Substituting our values into the formula gives us: \(x_2= x_1 - \dfrac{f(x_1)}{f'(x_1)} = 3 - \dfrac{\sin 3}{\cos 3}.\)

Calculation of First Iteration

Calculate the above expression to get the value after the first iteration.

Second Iteration

The second iteration uses the output of the first iteration as the new guess in Newton's Method formula. It is calculated in the same way as the first iteration: \(x_3 = x_2 - \dfrac{f(x_2)}{f'(x_2)}.\) We must now calculate this expression to get the value after the second iteration.

Calculation of Second Iteration

Calculate the above expression to get the value after the second iteration.