A function $f$ has derivatives of all orders at $x = 0$. Let $P _ { n } ( x )$ denote the $n$th-degree Taylor polynomial for $f$ about $x = 0$. (a) It is known that $f ( 0 ) = - 4$ and that $P _ { 1 } \left( \frac { 1 } { 2 } \right) = - 3$. Show that $f ^ { \prime } ( 0 ) = 2$. (b) It is known that $f ^ { \prime \prime } ( 0 ) = - \frac { 2 } { 3 }$ and $f ^ { \prime \prime \prime } ( 0 ) = \frac { 1 } { 3 }$. Find $P _ { 3 } ( x )$. (c) The function $h$ has first derivative given by $h ^ { \prime } ( x ) = f ( 2 x )$. It is known that $h ( 0 ) = 7$. Find the third-degree Taylor polynomial for $h$ about $x = 0$.
A function $f$ has derivatives of all orders at $x = 0$. Let $P _ { n } ( x )$ denote the $n$th-degree Taylor polynomial for $f$ about $x = 0$.
(a) It is known that $f ( 0 ) = - 4$ and that $P _ { 1 } \left( \frac { 1 } { 2 } \right) = - 3$. Show that $f ^ { \prime } ( 0 ) = 2$.
(b) It is known that $f ^ { \prime \prime } ( 0 ) = - \frac { 2 } { 3 }$ and $f ^ { \prime \prime \prime } ( 0 ) = \frac { 1 } { 3 }$. Find $P _ { 3 } ( x )$.
(c) The function $h$ has first derivative given by $h ^ { \prime } ( x ) = f ( 2 x )$. It is known that $h ( 0 ) = 7$. Find the third-degree Taylor polynomial for $h$ about $x = 0$.