- Using the cubic Taylor polynomial approximation of , show that the nonzero root of the equation
is approximated by .
- Using part (a) show that
given that . Determine whether is positive or negative and prove the result.
- Proof. The cubic Taylor polynomial approximation of is
This implies
Therefore, we can approximate the nonzero root by
- 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
(sin(r)-r^2) > 0 since r>0 and r^2 = r – r^3/3! < sin(r)
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!