6. The function $f$ is defined by the power series $$f ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 2 n } } { ( 2 n + 1 ) ! } = 1 - \frac { x ^ { 2 } } { 3 ! } + \frac { x ^ { 4 } } { 5 ! } - \frac { x ^ { 6 } } { 7 ! } + \cdots + \frac { ( - 1 ) ^ { n } x ^ { 2 n } } { ( 2 n + 1 ) ! } + \cdots$$ for all real numbers $x$. (a) Find $f ^ { \prime } ( 0 )$ and $f ^ { \prime \prime } ( 0 )$. Determine whether $f$ has a local maximum, a local minimum, or neither at $x = 0$. Give a reason for your answer. (b) Show that $1 - \frac { 1 } { 3 ! }$ approximates $f ( 1 )$ with error less than $\frac { 1 } { 100 }$. (c) Show that $y = f ( x )$ is a solution to the differential equation $x y ^ { \prime } + y = \cos x$.
END OF EXAMINATION
6. The function $f$ is defined by the power series
$$f ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 2 n } } { ( 2 n + 1 ) ! } = 1 - \frac { x ^ { 2 } } { 3 ! } + \frac { x ^ { 4 } } { 5 ! } - \frac { x ^ { 6 } } { 7 ! } + \cdots + \frac { ( - 1 ) ^ { n } x ^ { 2 n } } { ( 2 n + 1 ) ! } + \cdots$$
for all real numbers $x$.\\
(a) Find $f ^ { \prime } ( 0 )$ and $f ^ { \prime \prime } ( 0 )$. Determine whether $f$ has a local maximum, a local minimum, or neither at $x = 0$. Give a reason for your answer.\\
(b) Show that $1 - \frac { 1 } { 3 ! }$ approximates $f ( 1 )$ with error less than $\frac { 1 } { 100 }$.\\
(c) Show that $y = f ( x )$ is a solution to the differential equation $x y ^ { \prime } + y = \cos x$.
\section*{END OF EXAMINATION}