ap-calculus-bc 2005 Q6

ap-calculus-bc · USA · free-response_formB Chain Rule Chain Rule with Composition of Explicit Functions
6. The twice-differentiable function $f$ is defined for all real numbers and satisfies the following conditions:
$$f ( 0 ) = 2 , f ^ { \prime } ( 0 ) = - 4 , \text { and } f ^ { \prime \prime } ( 0 ) = 3$$
(a) The function $g$ is given by $g ( x ) = e ^ { a x } + f ( x )$ for all real numbers, where $a$ is a constant. Find $g ^ { \prime } ( 0 )$ and $g ^ { \prime \prime } ( 0 )$ in terms of $a$. Show the work that leads to your answers.
(b) The function $h$ is given by $h ( x ) = \cos ( k x ) f ( x )$ for all real numbers, where $k$ is a constant. Find $h ^ { \prime } ( x )$ and write an equation for the line tangent to the graph of $h$ at $x = 0$.
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
END OF EXAM
© 2006 The College Board. All rights reserved. Visit \href{http://apcentral.collegeboard.com}{apcentral.collegeboard.com} (for AP professionals) and \href{http://www.collegeboard.com/apstudents}{www.collegeboard.com/apstudents} (for students and parents).
6. The twice-differentiable function $f$ is defined for all real numbers and satisfies the following conditions:

$$f ( 0 ) = 2 , f ^ { \prime } ( 0 ) = - 4 , \text { and } f ^ { \prime \prime } ( 0 ) = 3$$

(a) The function $g$ is given by $g ( x ) = e ^ { a x } + f ( x )$ for all real numbers, where $a$ is a constant. Find $g ^ { \prime } ( 0 )$ and $g ^ { \prime \prime } ( 0 )$ in terms of $a$. Show the work that leads to your answers.\\
(b) The function $h$ is given by $h ( x ) = \cos ( k x ) f ( x )$ for all real numbers, where $k$ is a constant. Find $h ^ { \prime } ( x )$ and write an equation for the line tangent to the graph of $h$ at $x = 0$.

\section*{WRITE ALL WORK IN THE PINK EXAM BOOKLET.}
\section*{END OF EXAM}
© 2006 The College Board. All rights reserved.\\
Visit \href{http://apcentral.collegeboard.com}{apcentral.collegeboard.com} (for AP professionals) and \href{http://www.collegeboard.com/apstudents}{www.collegeboard.com/apstudents} (for students and parents).
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