ap-calculus-ab 1999 Q7

ap-calculus-ab · USA · free-response_formB Differentiation from First Principles Limit Involving Derivative Definition of Composed Functions

For a differentiable function $f$, let $f ^ { * }$ be the function defined by $f ^ { * } ( x ) = \lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x - h ) } { h }$. (a) Determine $f ^ { * } ( x )$ for $f ( x ) = x ^ { 2 } + x$. (b) Determine $f ^ { * } ( x )$ for $f ( x ) = \cos x$. (c) Write an equation that expresses the relationship between the functions $f ^ { * }$ and $f ^ { \prime }$, where $f ^ { \prime }$ denotes the usual derivative of $f$.

For a differentiable function $f$, let $f ^ { * }$ be the function defined by $f ^ { * } ( x ) = \lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x - h ) } { h }$.
(a) Determine $f ^ { * } ( x )$ for $f ( x ) = x ^ { 2 } + x$.
(b) Determine $f ^ { * } ( x )$ for $f ( x ) = \cos x$.
(c) Write an equation that expresses the relationship between the functions $f ^ { * }$ and $f ^ { \prime }$, where $f ^ { \prime }$ denotes the usual derivative of $f$.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7