Home » Blog » Use Taylor polynomials to approximate the nonzero root of x2=sin x

Use Taylor polynomials to approximate the nonzero root of x2=sin x

1. Using the cubic Taylor polynomial approximation of , show that the nonzero root of the equation is approximated by .

2. Using part (a) show that given that . Determine whether is positive or negative and prove the result.

1. Proof. The cubic Taylor polynomial approximation of is This implies Therefore, we can approximate the nonzero root by 2. Proof. We know from this exercise (Section 7.8, Exercise #1) that for we have So, for , and using the given inequality , we have Furthermore, since with the absolute value of each term in the sum strictly less than the absolute value of the previous term (since and ). Thus, each pair is positive, so the whole series is positive One comment

1. Julio Vieira says:

There is a little error when you calculate x^2-x+x^3/6 = 0, the next step is x^2+6x-6 = 0 and not 6x^2+x-1=0

And thanks for solutions, it really helps me a lot!