Let $f$ be the function given by $f ( x ) = \sin \left( 5 x + \frac { \pi } { 4 } \right)$, and let $P ( x )$ be the third-degree Taylor polynomial for $f$ about $x = 0$. (a) Find $P ( x )$. (b) Find the coefficient of $x ^ { 22 }$ in the Taylor series for $f$ about $x = 0$. (c) Use the Lagrange error bound to show that $\left| f \left( \frac { 1 } { 10 } \right) - P \left( \frac { 1 } { 10 } \right) \right| < \frac { 1 } { 100 }$. (d) Let $G$ be the function given by $G ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$. Write the third-degree Taylor polynomial for $G$ about $x = 0$.
Let $f$ be the function given by $f ( x ) = \sin \left( 5 x + \frac { \pi } { 4 } \right)$, and let $P ( x )$ be the third-degree Taylor polynomial for $f$ about $x = 0$.
(a) Find $P ( x )$.
(b) Find the coefficient of $x ^ { 22 }$ in the Taylor series for $f$ about $x = 0$.
(c) Use the Lagrange error bound to show that $\left| f \left( \frac { 1 } { 10 } \right) - P \left( \frac { 1 } { 10 } \right) \right| < \frac { 1 } { 100 }$.
(d) Let $G$ be the function given by $G ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$. Write the third-degree Taylor polynomial for $G$ about $x = 0$.