ap-calculus-bc 2000 Q3

ap-calculus-bc · USA · free-response Taylor series Construct Taylor/Maclaurin polynomial from derivative values
3. The Taylor series about $x = 5$ for a certain function $f$ converges to $f ( x )$ for all $x$ in the interval of convergence. The $n$th derivative of $f$ at $x = 5$ is given by $f ^ { ( n ) } ( 5 ) = \frac { ( - 1 ) ^ { n } n ! } { 2 ^ { n } ( n + 2 ) }$, and $f ( 5 ) = \frac { 1 } { 2 }$.
(a) Write the third-degree Taylor polynomial for $f$ about $x = 5$.
(b) Find the radius of convergence of the Taylor series for $f$ about $x = 5$.
(c) Show that the sixth-degree Taylor polynomial for $f$ about $x = 5$ approximates $f ( 6 )$ with error less than $\frac { 1 } { 1000 }$.
END OF PART A OF SECTION II
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CALCULUS BC SECTION II, Part B Time-45 minutes Number of problems-3 No calculator is allowed for these problems.
: $P _ { 3 } ( f , 5 ) ( x )$ 
3. The Taylor series about $x = 5$ for a certain function $f$ converges to $f ( x )$ for all $x$ in the interval of convergence. The $n$th derivative of $f$ at $x = 5$ is given by $f ^ { ( n ) } ( 5 ) = \frac { ( - 1 ) ^ { n } n ! } { 2 ^ { n } ( n + 2 ) }$, and $f ( 5 ) = \frac { 1 } { 2 }$.\\
(a) Write the third-degree Taylor polynomial for $f$ about $x = 5$.\\
(b) Find the radius of convergence of the Taylor series for $f$ about $x = 5$.\\
(c) Show that the sixth-degree Taylor polynomial for $f$ about $x = 5$ approximates $f ( 6 )$ with error less than $\frac { 1 } { 1000 }$.

\section*{END OF PART A OF SECTION II}
Copyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved. AP is a registered trademark of the College Entrance Examination Board.

CALCULUS BC\\
SECTION II, Part B\\
Time-45 minutes\\
Number of problems-3\\
No calculator is allowed for these problems.\\
Paper Questions
Q2 Q3 Q4 Q5 Q6