Question

I'm not sure how to do this one, with the concept of an interval thrown in there. Can someone help me? How do you solve this?

"Use Newton's Method to approximate the indicated root of the equation to six decimal places: The root of x^4 - 2x^3 + 5x^2 - 6 = 0, on [1,2]."

Any and all help is greatly appreciated!

Answer 1

Do I find a root of the equation by solving for x, and then use that value as X1 when using the Linear Approximation formula?

Answer 2

yes you do

Answer 3

So I set f'(x) equal to zero, solve for x, and then use that as x1?

Answer 4

ok... su substitute x = 1 into the equation

then substitute x = 2 into the equation...

you need to get an initial approximation

Answer 5

So I make a "guess" as they call it?

Answer 6

What do I use the interval for?

Answer 7

call the initial approximation a

then to get the next approximation its

\[a_{1} = a - \frac{f(a)}{f'a}\]

Answer 8

so the question siays the interval is [1, 2}

so substitute them to see that there is a change of side between f(1) and f(2)

this will help get the initial approximation

Answer 9

so f(1) = -2 f(2) = 14

so the root is closer to x = 1 then x = 2

so perhaps choose a = 1.25 as the initial value

Answer 10

If x1 = 1.25 then wouldn't x2 be outside the interval? :/

Answer 11

well don't worry about the interval at the moment

f(1.25) = 0.347656

\[f'(x) = 4x^3 - 6x^2 + 10x \]

then

\[f'(1.25) = 10.9375\]

then

\[a_{1} = 1.25 - \frac{0.347656}{10.9375}\]

so I think the approximation will be within the interval

Answer 12

hope it helps.

Answer 13

Yes, it does.. thank you:)