Consider the function f(x) = x4 - 4x3 + x. We want to know at what values of x t
ID: 1947180 • Letter: C
Question
Consider the function f(x) = x4 - 4x3 + x. We want to know at what values of x this function crosses the x-axis, i.e., we want to solve f(x) = 0. Carry out 2 iterations of the Newton-Raphson method to find where the function approximately crosses the x-axis. Use an initial value of x0 = 4 and carry out all working to 3 decimal places: if you are unsure what this means then please ask your tutor. Use your working out to answer the following question. What is the value of x after 2 iterations of the Newton-Raphson method (i.e., when n = 2)?Explanation / Answer
f(x) = x4 -4x3+x
f '(x) = 4x3 - 12x2 + 1
Now as given if we use x0 = 4 as first assumption of f(x) = 0
f(x0) = f(4) = 256 - 4*64 + 4 = 4
f ' (x0) = f ' (4) = 4*64 - 12*16 +1 = 65
By 1st iteration we get
x1 = x0 - f(x0)/f ' (x0)
= 4 - 4/65 = 3.938
Hence by 1st iteration x1 = 3.938
Now we do the second iteration
f(x1) = f(3.938) = 0.151667
f ' (x1) = f ' (3.938) = 59.185431
Hence by the second iteration
x2 = x1 - f(x1)/f ' (x1) = 3.938 - 0.151667/59.185431 = 3.935
Hence after two iterations we get the value of the root as 3.935