Use Newton's method to find the positive fourth root of 6 by solving x 4 − 6 = 0 . Start with x 1 = 1 and find x 2 and x 3 . Answers should be accurate to 5 decimal places.

Question

Use Newton's method to find the positive fourth root of 6 by solving (. Start with  and find  and . Answers should be accurate to 5 decimal places.

Answer 1

F(x) = x^4 - 6 is the function whose roots we are looking

F' (x) = 4 x^3

X_n is the next approximation

X_(n-1) is the current approximation

X_1 = 1

F(1) = 1^4-6 = -5

F'(1) = 4(1)^3 = 4

x_2 = 1 - (-5)/4 = 1 + 5/4 = 9/4 = 2.25

X_3 = x_2 - F(x_2)/F'(x_2) = 2.25 - F(2.25)/F'(2.25)

= 2.25 - [(2.25)^4 - 6]/ [4*(2.25)^3]

= 2.25 - 19.62890625 / 45.5625

= 1.81918....

Here is the table:

n x accuracy f(x)

1 2.25 2 - 5

2 1.819187242798353909465 2 19.62890625

3 1.613539377697176816804 2 4.952407836378698392539

4 1.567223866994051411492 1.6 0.77826079692342279009

5 1.565088956332483365129 1.57 0.0328725055898177475607

6 1.565084580091642447875 1.56508 6.71086167280952913E-5

7 1.565084580073287316585 1.5650845801 2.814692295836E-10

8 1.565084580073287316585 1.565084580073287316585 5.E-21