- Show that is an approximation of the nonzero root of the equation
using the cubic Taylor polynomial approximation to .
- Given that
prove that the number from part (a) satisfies
Determine if is positive or negative and prove the result.
- Proof. From a previous exercise (Section 7.8, Exercise #3) we know
So, to approximate the nonzero root of we have
- We know from the same previous exercise we used in part (a) that the error term for satisfies the inequality
Using the values for and given we have
In case someone wants to know how to prove that arctan(r)-r^2 >or x-x^3/3 for x>0
Something got messed up in my comment, what I said is: If you want to prove that the difference is positive or negative first prove that for x>0
arctan(x)>x-x^3/3