Use Newton's Method with the function
f(x)=x^2-2 and initial value Xo=2 to calculate x1,x2,x3.
I have no idea how to do this
No problem! I'll guide you through the process of using Newton's Method to find the roots of the function f(x) = x^2 - 2 with an initial value X0 = 2.
Step 1: Set up the formula
Newton's Method uses the formula:
Xn+1 = Xn - (f(Xn) / f'(Xn))
where Xn is the current estimate of the root, f(Xn) is the value of the function at Xn, and f'(Xn) is the derivative of the function evaluated at Xn.
Step 2: Calculate f(Xn) and f'(Xn)
First, let's find the value of the function f(Xn) and its derivative f'(Xn) for each iteration.
For the function f(x) = x^2 - 2:
- f(Xn) = (Xn)^2 - 2
- f'(Xn) = 2(Xn)
Step 3: Iterate through the formula
To find x1, x2, and x3, we will follow these steps iteratively:
- Start with the initial value X0 = 2.
- Plug X0 into the formula to find X1:
X1 = X0 - (f(X0) / f'(X0))
calculate f(X0) and f'(X0) then substitute them into the formula.
- Repeat this process to find X2 and X3 by plugging the current Xn into the formula until you reach the desired number of iterations.
Step 4: Calculate X1, X2, X3
Let's calculate X1, X2, and X3 using the formula and the previously found values.
Starting with X0 = 2:
- f(X0) = (2)^2 - 2 = 2
- f'(X0) = 2(2) = 4
Using the formula Xn+1 = Xn - (f(Xn) / f'(Xn)):
For X1:
X1 = 2 - (2 / 4) = 1.5
For X2:
- f(X1) = (1.5)^2 - 2 = 0.25
- f'(X1) = 2(1.5) = 3
X2 = 1.5 - (0.25 / 3) = 1.4167
For X3:
- f(X2) = (1.4167)^2 - 2 = 0.0069
- f'(X2) = 2(1.4167) = 2.8334
X3 = 1.4167 - (0.0069 / 2.8334) = 1.4142
So, using Newton's Method, x1 is approximately 1.5, x2 is approximately 1.4167, and x3 is approximately 1.4142.