Use Newton’s method to approximate the root of x^2 + 4x + 2 = 0 between x = -4
and x = -3 .
Starting at -3:
h(-3.00000) = 23.00000 -- next x: -0.70000
h(-0.70000) = 5.29000 -- next x: 0.27963
h(0.27963) = 0.95967 -- next x: 0.55854
h(0.55854) = 0.07779 -- next x: 0.58553
h(0.58553) = 0.00073 -- next x: 0.58579
h(0.58579) = 0.00000 -- next x: 0.58579
Starting at -4:
h(-4.00000) = 34.00000 -- next x: -1.16667
h(-1.16667) = 8.02778 -- next x: 0.10088
h(0.10088) = 1.60667 -- next x: 0.52388
h(0.52388) = 0.17893 -- next x: 0.58449
h(0.58449) = 0.00367 -- next x: 0.58579
h(0.58579) = 0.00000 -- next x: 0.58579
My bad. Misplaced a + sign.
h(-4.00000) = 2.00000 -- next x: -3.50000
h(-3.50000) = 0.25000 -- next x: -3.41667
h(-3.41667) = 0.00694 -- next x: -3.41422
h(-3.41422) = 0.00001 -- next x: -3.41421
To use Newton's method, we need an initial approximation of the root. Let's start by choosing x = -3 since it is within the given interval.
Step 1: Find the derivative of the function.
Let's find the derivative of f(x) = x^2 + 4x + 2.
f'(x) = 2x + 4.
Step 2: Set up the Newton's iteration formula.
The Newton's iteration formula is given by:
x[n+1] = x[n] - f(x[n])/f'(x[n])
Step 3: Apply the Newton's iteration formula.
Let's do the calculations:
x[0] = -3 (our initial approximation)
x[1] = x[0] - f(x[0])/f'(x[0])
= -3 - ( (-3)^2 + 4(-3) + 2 ) / ( 2(-3) + 4 )
= -3 - ( 9 - 12 + 2 ) / ( -6 + 4 )
= -3 - (-1) / (-2)
= -3 + 1/2
= -2.5
Repeat the steps until we reach the desired accuracy or number of iterations.
Step 4: Further iterations.
Let's continue the iteration:
x[2] = x[1] - f(x[1])/f'(x[1])
= -2.5 - ( (-2.5)^2 + 4(-2.5) + 2 ) / ( 2(-2.5) + 4 )
= -2.5 - ( 6.25 - 10 + 2 ) / ( -5 + 4 )
= -2.5 - (-1.75) / (-1)
= -2.5 + 1.75
= -0.75
Repeat the process until the desired accuracy is achieved or the number of iterations is reached.