Please post the EXACT assignment (word for word) as it was presented to you. Please include reference to class topics with this assignment.How do I find a non-linear, polynomial approximation using natural log of the square root of the function ln (sqrt(1+x^2) near x=0?
Please help to find the answer, thank you.
Oops! I didn't see the "ln" in the problem. The first thing we can do is simplify: \(\displaystyle ln\left(\sqrt{x^2+ 1}\right)= ln\left((x^2+ 1\right)^{1/2})= \frac{1}{2}ln(x^2+1)\). Letting \(\displaystyle f(x)= \frac{1}{2}ln(x^2+1)\) we have \(\displaystyle f(0)= 0\), \(\displaystyle f'(0)= 0\), \(\displaystyle f''(0)= 1\) so a quadratic approximation toThat's basically what the MacLaurin polynomial is! Letting \(\displaystyle f(x)= \sqrt{x^2+ 1}\), f(0)= 1, f'(0)= 0. f''(0)= 1, f'''(0)= 0, etc. \(\displaystyle \sqrt{x^2+ 1}\) can be approximated around x= 0 by \(\displaystyle 1+ \frac{1}{2}x^2\).