### Home > APCALC > Chapter 12 > Lesson 12.3.1 > Problem12-91

12-91.

Write the equation of the fourth-degree Taylor polynomial centered at $x = 0$ for $f(x) = e^{–2x}$. Use it to approximate $f(0.5) = e^{–1 }=\frac { 1 } { e }$, and calculate a bound for the error in this approximation. Then approximate $f ^ { \prime \prime \prime } ( 0.5 )$ using your polynomial approximation.

$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}$

Substitute $–2x$ for $x$ to get the polynomial for $e^{–2x}$.

To bound the error, determine the $x^{5}$ term in the polynomial and evaluate it for $x = 0.5$.