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

The function $f$ is defined by the power series $$f(x) = -\frac{x}{2} + \frac{2x^{2}}{3} - \frac{3x^{3}}{4} + \cdots + \frac{(-1)^{n} n x^{n}}{n+1} + \cdots$$ for all real numbers $x$ for which the series converges. The function $g$ is defined by the power series $$g(x) = 1 - \frac{x}{2!} + \frac{x^{2}}{4!} - \frac{x^{3}}{6!} + \cdots + \frac{(-1)^{n} x^{n}}{(2n)!} + \cdots$$ for all real numbers $x$ for which the series converges.
(a) Find the interval of convergence of the power series for $f$. Justify your answer.
(b) The graph of $y = f(x) - g(x)$ passes through the point $(0, -1)$. Find $y'(0)$ and $y''(0)$. Determine whether $y$ has a relative minimum, a relative maximum, or neither at $x = 0$. Give a reason for your answer.

The function $f$, defined by $$f(x) = \begin{cases} \frac{\cos x - 1}{x^2} & \text{for } x \neq 0 \\ -\frac{1}{2} & \text{for } x = 0 \end{cases}$$ has derivatives of all orders. Let $g$ be the function defined by $g(x) = 1 + \int_{0}^{x} f(t)\,dt$.
(a) Write the first three nonzero terms and the general term of the Taylor series for $\cos x$ about $x = 0$. Use this series to write the first three nonzero terms and the general term of the Taylor series for $f$ about $x = 0$.
(b) Use the Taylor series for $f$ about $x = 0$ found in part (a) to determine whether $f$ has a relative maximum, relative minimum, or neither at $x = 0$. Give a reason for your answer.
(c) Write the fifth-degree Taylor polynomial for $g$ about $x = 0$.
(d) The Taylor series for $g$ about $x = 0$, evaluated at $x = 1$, is an alternating series with individual terms that decrease in absolute value to 0. Use the third-degree Taylor polynomial for $g$ about $x = 0$ to estimate the value of $g(1)$. Explain why this estimate differs from the actual value of $g(1)$ by less than $\frac{1}{6!}$.