Use Newton's method to find all the real roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. Use Newton's method to find all the roots of...
Use Newton's method to find all the real roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.
Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.
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First draw the graph using a calculator or computer
Approximate the four roots (where the graph cuts the x axis) giving something like
`x= -1.8,-1.1,1.1,2.8`
Newton's method involves refining our estimates using the recursive formula
`x_(n+1) = x_n - f(x_n)/(f'(x_n))`
Now, we have `f(x) = x^6-x^5 -5x^4 -x^2+x+6`
`implies` `f'(x) = 6x^5 -5x^4 -20x^3 -2x +1`
For each initial estimate we get a sequence of improving approximations:
1) ` `x0 -1.8
x1 -1.76907576
x2 -1.76617802
x3 -1.76615375
x4 -1.76615375
2) x0 -1.1
x1 -1.08064658
x2 -1.08057763
x3 -1.08057763
3) x0 1.1
x1 1.05017360
x2 1.04708065
x3 1.04706901
x4 1.04706901
4) x0 2.8
x1 2.78786233
x2 2.78761037
x3 2.78761026
x4 2.78761026
Therefore we have real 4 solutions (the other two are complex)
`x = -1.76615375`, `-1.08057763`, `1.04706901` and `2.78761026`
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