`cos(x^2 - x) = x^4` Use Newton's method to find all roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.

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lemjay eNotes educator| Certified Educator

`cos(x^2-x) = x^4`

Set the left side equal to zero.


To solve using Newton's method, apply the formula:

`x_(n+1)=x_n - (f(x_n))/(f'(x_n))`

Let the function of the given equation be:

`f(x) =x^4-cos(x^2-x)`

And its derivative is:

`f'(x) = 4x^3 + (2x-1)sin(x^2-x)`

Plug-in f(x) and f'(x) to the formula of Newton's method.

`x_(n+1) = x_n - ((x_n)^4-cos((x_n)^2-x_n))/(4(x_n)^3+(2x_n-1)sin((x_n)^2-x_n))`

To get the initial value of x, refer to the graph of f(x). (See figure.)

Notice that when f(x) =0, the values of x are near -0.8 and 1. Use these two values of x to solve for the roots of the function to eight decimal places.

For the first root,  let the initial value be -0.8.


`x_2= x_1 - ((x_1)^4-cos((x_1)^2-x_1))/(4(x_1)^3+(2x_1-1)sin((x_1)^2-x_1))=-0.7396478896`

`x_3= x_2 - ((x_2)^4-cos((x_2)^2-x_2))/(4(x_2)^3+(2x_2-1)sin((x_2)^2-x_2))=-0.7348883415`

`x_4= x_3 - ((x_3)^4-cos((x_3)^2-x_3))/(4(x_3)^3+(2x_3-1)sin((x_3)^2-x_3))=-0.7348591049`

`x_5= x_4 - ((x_4)^4-cos((x_4)^2-x_4))/(4(x_4)^3+(2x_4-1)sin((x_4)^2-x_4))=-0.7348591038`

Notice that the two approximates have the same eight decimal places. So we stop the iteration here. Thus, one of the roots of f(x) is x=-0.73485910 .

For the second root,  let the initial value be 1.


`x_2= x_1 - ((x_1)^4-cos((x_1)^2-x_1))/(4(x_1)^3+(2x_1-1)sin((x_1)^2-x_1))=1`

Notice that the second root is an integer. It has an exact value which is x=1.

Therefore, the solution of the equation `cos(x^2-x) =x^4`   is   x={-0.73485910, 1}.

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