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]# ?
1 Answer
The answer is
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
#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,
#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.