You are using Newton's method to solve x^3-5x-2=0. If your first guess is X1=2, what value will you calculate for the next approximation X2?
your equation would be
xnew = x - (x^3 - 5x - 2)/(3x^2 - 5)
so for x=2
xnew = 2 - (8-10-2)/(12-5)
= 2.5714
Incidentally, my next 4 answers were
2.4268029
2.4143045
2.4142136 and
2.4142136 which would be one of the answers correct up to 7 decimal places.
To calculate the next approximation, X2, using Newton's method, we can use the formula:
X2 = X1 - f(X1) / f'(X1)
Given the function f(x) = x^3 - 5x - 2, we need to find the derivative f'(x).
Differentiating f(x), we get:
f'(x) = 3x^2 - 5
Now, let's calculate X2:
X2 = X1 - f(X1) / f'(X1)
= 2 - (2^3 - 5(2) - 2) / (3(2)^2 - 5)
= 2 - (8 - 10 - 2) / (12 - 5)
= 2 - (-4) / 7
= 2 + 4/7
= 2.5714
Therefore, the next approximation X2 is approximately 2.5714.
To find the next approximation using Newton's method, you need to follow these steps:
1. Start with the initial guess, X1 = 2
2. Evaluate the function f(x) = x^3 - 5x - 2 at X1 to determine its value.
- f(X1) = (2)^3 - 5(2) - 2 = 8 - 10 - 2 = -4
3. Calculate the derivative of the function f'(x) = 3x^2 - 5.
4. Plug X1 and f(X1) into the formula to find the next approximation:
- X2 = X1 - f(X1)/f'(X1)
- X2 = 2 - (-4)/(3(2)^2 - 5) = 2 + 4/(12 - 5) = 2 + 4/7 = 2.57 (rounded to two decimal places)
Therefore, if your first guess is X1 = 2, the next approximation X2 would be 2.57.