Let $f$ be the function given by $f ( x ) = 6 e ^ { - x / 3 }$ for all $x$. (a) Find the first four nonzero terms and the general term for the Taylor series for $f$ about $x = 0$. (b) Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$. Find the first four nonzero terms and the general term for the Taylor series for $g$ about $x = 0$. (c) The function $h$ satisfies $h ( x ) = k f ^ { \prime } ( a x )$ for all $x$, where $a$ and $k$ are constants. The Taylor series for $h$ about $x = 0$ is given by $$h ( x ) = 1 + x + \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 3 } } { 3 ! } + \cdots + \frac { x ^ { n } } { n ! } + \cdots .$$ Find the values of $a$ and $k$.
Let $f$ be the function given by $f ( x ) = 6 e ^ { - x / 3 }$ for all $x$.
(a) Find the first four nonzero terms and the general term for the Taylor series for $f$ about $x = 0$.
(b) Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$. Find the first four nonzero terms and the general term for the Taylor series for $g$ about $x = 0$.
(c) The function $h$ satisfies $h ( x ) = k f ^ { \prime } ( a x )$ for all $x$, where $a$ and $k$ are constants. The Taylor series for $h$ about $x = 0$ is given by
$$h ( x ) = 1 + x + \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 3 } } { 3 ! } + \cdots + \frac { x ^ { n } } { n ! } + \cdots .$$
Find the values of $a$ and $k$.