Approximate the real root of cosx+x=2 to four decimal place. (Using Newton's method)

I need help with this! Thank you

f(x) = cosx+x-2

f'(x) = 1-sinx

start with, say, x=3
x -> 3 - f(3)/f'(3) = 3-.01/.8588 = 2.98835
and proceed as needed

I will start you ought. A neat way to get to near the solution is to graph

cosx + x -2

look at where the graph crosses the x axis. Then change the limits on the graph to expand the area just around that intercept. Keep doing that until you can read it to 5 significant figures....

That is faster than the old method of using the calculator (non grahic) to iterate until you get the solution.

To approximate the real root of the equation cos(x) + x = 2 using Newton's Method, you'll need to follow these steps:

Step 1: Choose an initial guess, let's call it x_0. This initial guess should be reasonably close to the actual root, so that the method converges quickly.

Step 2: Calculate the derivative of the function f(x) = cos(x) + x.

Step 3: Use the following iteration formula to update the guess:
x_(n+1) = x_n - (f(x_n) / f'(x_n))

Repeat this iteration process until the result converges to the desired accuracy.

Now, let's apply these steps to approximate the root:

Step 1: A good initial guess for this problem would be x_0 = 0, since it's relatively close to the actual root between 0 and 1.

Step 2: The derivative of f(x) = cos(x) + x is f'(x) = -sin(x) + 1.

Step 3: Now, let's perform the iteration:
x_(n+1) = x_n - (f(x_n) / f'(x_n))

For the first iteration (n=0):
x_1 = x_0 - (cos(x_0) + x_0 - 2) / (-sin(x_0) + 1)

Using the initial guess x_0 = 0:
x_1 = 0 - (cos(0) + 0 - 2) / (-sin(0) + 1)
x_1 ≈ 0.5

For the second iteration (n=1):
x_2 = x_1 - (cos(x_1) + x_1 - 2) / (-sin(x_1) + 1)

Continue this process until you reach the desired accuracy. Keep in mind that Newton's Method can converge quickly, but it's always a good idea to check for convergence and stop iterating if the desired accuracy is achieved.

Repeat the iteration until you reach the desired number of decimal places of accuracy. In this case, you're aiming for four decimal places, so you may need to repeat the process a few more times until you achieve that level of accuracy.