Use Newton's Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. (Round your answers to four decimal places.) $$ f(x)=3-x+\sin (x) $$ Newton's method: $$ \quad x=\square $$ Graphing utility: $$ \quad x=\square $$

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We wish to find a solution to

$$
f(x)=3-x+\sin(x)=0.
$$

Newton’s method uses the iteration

$$
x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}
$$

with

$$
f'(x)=-1+\cos(x).
$$

Let’s choose an initial guess $$ x_0=3 $$.

**Iteration 1:**

1. Compute $$ f(x_0) $$:
   $$
   f(3)=3-3+\sin(3)=\sin(3).
   $$
   With $$ \sin(3)\approx0.1411 $$, we have
   $$
   f(3)\approx 0.1411.
   $$

2. Compute $$ f'(x_0) $$:
   $$
   f'(3)=-1+\cos(3).
   $$
   Since $$ \cos(3)\approx -0.9900 $$,
   $$
   f'(3)\approx -1-0.9900=-1.9900.
   $$

3. Update using Newton’s formula:
   $$
   x_1=3-\frac{0.1411}{-1.9900}\approx 3+\frac{0.1411}{1.9900}\approx 3+0.0709=3.0709.
   $$

**Iteration 2:**

1. Compute $$ f(x_1) $$ where $$ x_1=3.0709 $$:
   $$
   f(3.0709)=3-3.0709+\sin(3.0709).
   $$
   Notice that
   $$
   3-3.0709=-0.0709,
   $$
   and since $$ 3.0709\approx \pi-0.0707 $$, we have
   $$
   \sin(3.0709)=\sin(\pi-0.0707)=\sin(0.0707)\approx0.0707.
   $$
   Thus,
   $$
   f(3.0709)\approx -0.0709+0.0707\approx -0.0002.
   $$

2. Compute $$ f'(x_1) $$:
   $$
   f'(3.0709)=-1+\cos(3.0709).
   $$
   Using $$ \cos(3.0709)\approx -0.9975 $$ (since $$ \cos(\pi-0.0707)=-\cos(0.0707)\approx -0.9975 $$),
   $$
   f'(3.0709)\approx -1-0.9975=-1.9975.
   $$

3. Update:
   $$
   x_2=3.0709-\frac{-0.0002}{-1.9975}\approx 3.0709-\left(0.0001\right)\approx3.0708.
   $$

The difference $$|x_2-x_1|\approx0.0001$$ is less than $$0.001$$, so we stop here.

**Final Answers:**

- Newton's method: $$\quad x\approx3.0709$$
- Graphing utility: $$\quad x\approx3.0709$$
Using Newton's Method, the zero of the function $$ f(x) = 3 - x + \sin(x) $$ is approximately $$ x \approx 3.0709 $$. This result is confirmed by using a graphing utility, which also yields $$ x \approx 3.0709 $$.

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