Use Newtons method to find the roots of the equation f(x)=x^3-3x-3 when x1=2
Could you solve the question and provide the answer
df/dx = 3 x^2 -3
X1= 2
then df/dx =12-3 = 9 =slope
f = 8-6-3= -1
so we go up from (2,-1) at slope 9
9 = change in f change in x = +1/change in x
change in x = 1/9
so
X2 = 2 1/9 = 19/9
etc
y = x^3 - 3x - 3
y' = 3 x^2 - 3
new x = old x + needed change in y / slope
X1 = 2, y = 8-6-3 = -1
so we need to go up 1
slope = 3(4)-3 = 9
---------------------
new x = 2 + 1/9
X2 = 19/9
start at X2 = 19/9 and do it again for the next guess, until you get there
In google paste:
Newton Raphson method calculator - AtoZmath.com
When page be open in rectangle type:
x^3-3x-3
Click option:
Initial solution x0 2
and click option:
Find
Four time click on option:
Click here to display next solution steps
You will see solution step by step.
I usually follow this algorithm:
f(x) = x^3-3x-3
f'(x) = 3x^2 - 3
Newton said: newx = x - f(x)/f'(x)
newx = x - (x^3 - 3x - 3)/(3x^2 - 3)
= (3x^3 - 3x - x^3 + 3x + 3)/(3x^2 - 3) = (2x^3 + 3)/(3x^2 - 3)
newx <--- x , start with x = 2
2.1111.... , keep replacing x with newx
2.1038359...
2.103803403
2.103803403 , correct to 9 decimals places
To use Newton's method to find the roots of the equation f(x) = x^3 - 3x - 3, we start with an initial guess x₁ = 2.
To begin, we need to calculate the derivative of the function f(x). The derivative of f(x) = x^3 - 3x - 3 is obtained by differentiating each term separately:
f'(x) = 3x^2 - 3.
Next, we use the formula for Newton's method to find the updated guess x₂:
x₂ = x₁ - f(x₁)/f'(x₁).
Let's calculate the value of f(x₁):
f(2) = (2)^3 - 3(2) - 3 = 8 - 6 - 3 = -1.
Now, let's calculate the value of the derivative f'(x₁):
f'(2) = 3(2)^2 - 3 = 12 - 3 = 9.
Using these values, we can apply Newton's method:
x₂ = x₁ - f(x₁)/f'(x₁)
= 2 - (-1)/9
= 2 + 1/9
= 2 + 0.1111... (to 4 decimal places)
≈ 2.1111.
So, the updated guess is x₂ ≈ 2.1111.
To refine the guess, we repeat the process by calculating the value of f(x₂) and f'(x₂). We substitute x₂ into the equation f(x) = x^3 - 3x - 3 to obtain f(x₂), and differentiate to get f'(x₂). We then use these values to calculate x₃ using the formula for Newton's method.
By repeating this process iteratively, we can continue to refine the guess and approach the actual root of the equation f(x) = 0.