1. ## 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!!

2. 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)