Question

How do you use Newton's Method to approximate the root of the equation `x^4-2x^3+5x^2-6=0` on the interval `[1,2]` ?

Answer 1

The answer is `1.217562155`.

Recall that Newton's Method uses the formula:

`x_(n+1)=x_n−(f(x_n))/(f'(x_n))`

The equation is already a function, so:

`f(x)=x^4-2x^3+5x^2-6`

And we need the derivative:

`f'(x)=4x^3-6x^2+10x`

The easiest way to iterate is to program your calculator. Enter `f(x)` into `Y_1` and `f'(x)` into `Y_2`. Then enter a very short program that does this:

`A−(Y_1(A))/(Y_2(A))->A`

You can go to my website for specific instructions for the TI-83 or the Casio fx-9750 .

Finally, you need a starting value, `x_1`. Since the question is asking for a root in the interval `[1,2]`, we should start at `x=1.5`. So enter the following into your calculator,

`1->A`

Then execute the program until you get the desired accuracy:

`1.2625`
`1.218807774`
`1.217563128`
`1.217562155`
`1.217562155`

We get 3 digits of accuracy after 2 iterations, 6 after 3 iterations, and 10 after 4 iterations. So the answer converges very quickly for this root.