Let $f ( x ) = \sin \left( x ^ { 2 } \right) + \cos x$. The graph of $y = \left| f ^ { ( 5 ) } ( x ) \right|$ is shown above. (a) Write the first four nonzero terms of the Taylor series for $\sin x$ about $x = 0$, and write the first four nonzero terms of the Taylor series for $\sin \left( x ^ { 2 } \right)$ about $x = 0$. (b) Write the first four nonzero terms of the Taylor series for $\cos x$ about $x = 0$. Use this series and the series for $\sin \left( x ^ { 2 } \right)$, found in part (a), to write the first four nonzero terms of the Taylor series for $f$ about $x = 0$. (c) Find the value of $f ^ { ( 6 ) } ( 0 )$. (d) Let $P _ { 4 } ( x )$ be the fourth-degree Taylor polynomial for $f$ about $x = 0$. Using information from the graph of $y = \left| f ^ { ( 5 ) } ( x ) \right|$ shown above, show that $\left| P _ { 4 } \left( \frac { 1 } { 4 } \right) - f \left( \frac { 1 } { 4 } \right) \right| < \frac { 1 } { 3000 }$.
Let $f ( x ) = \sin \left( x ^ { 2 } \right) + \cos x$. The graph of $y = \left| f ^ { ( 5 ) } ( x ) \right|$ is shown above.
(a) Write the first four nonzero terms of the Taylor series for $\sin x$ about $x = 0$, and write the first four nonzero terms of the Taylor series for $\sin \left( x ^ { 2 } \right)$ about $x = 0$.
(b) Write the first four nonzero terms of the Taylor series for $\cos x$ about $x = 0$. Use this series and the series for $\sin \left( x ^ { 2 } \right)$, found in part (a), to write the first four nonzero terms of the Taylor series for $f$ about $x = 0$.
(c) Find the value of $f ^ { ( 6 ) } ( 0 )$.
(d) Let $P _ { 4 } ( x )$ be the fourth-degree Taylor polynomial for $f$ about $x = 0$. Using information from the graph of $y = \left| f ^ { ( 5 ) } ( x ) \right|$ shown above, show that $\left| P _ { 4 } \left( \frac { 1 } { 4 } \right) - f \left( \frac { 1 } { 4 } \right) \right| < \frac { 1 } { 3000 }$.