
Taylor's Inequality
This is part b. of the problem. I found T_3(x) but I really don't know the process to find the estimate of accuracy of the approximation.
HEre is the problem:
a) Approximate f by a Taylor polynomial with degree n at the number a.
$\displaystyle f(x) = e^{4x^2} $ $\displaystyle a=0$ $\displaystyle n=3$ on $\displaystyle 0\leq x \leq 0.3$
My Taylor polynomial:
$\displaystyle T_3 (x) = 1 + 4x^2$
b) Use Taylor's Inequality to estimate the accuracy of the approximation $\displaystyle f = T_n(x)$ when x lies in the given interval. Round to five decimal places.
The 4th derivative (not sure if this is correct):
$\displaystyle f^4(x) = 192e^{4x^2} + 3072x^2e^{4x^2} + 4096x^3e^{4x^2}$
$\displaystyle f^4(0) = 192$ *nevermind on this part, can't use zero because a=0
I really need someone to please take me through each step of how to find the accuracy using Taylor's Inequality, I am not having luck understanding it from my notes. Thanks!!

Rn(x) = Mx^(n+1)/(n+1)!
Where M is the max of the fourth derivative
f '''' (x) is increasing on (0,.3) so M is f''''(.3)
Rn(x)< f''''(.3) (.3)^4/(4!) for all x on (0.3)