a) (1.25 points) Let $f$ and $g$ be two differentiable functions for which the following data are known: $$f ( 1 ) = 1 ; f ^ { \prime } ( 1 ) = 2 ; g ( 1 ) = 3 ; g ^ { \prime } ( 1 ) = 4 :$$ Given $h ( x ) = f \left( ( x + 1 ) ^ { 2 } \right)$, use the chain rule to calculate $h ^ { \prime } ( 0 )$. Given $k ( x ) = \frac { f ( x ) } { g ( x ) }$, calculate $k ^ { \prime } ( 1 )$. b) (1.25 points) Calculate the integral $\int ( \operatorname { sen } x ) ^ { 4 } ( \cos x ) ^ { 3 } d x$. (You can use the change of variables $t = \operatorname { sen } x$.)
a) (1.25 points) Let $f$ and $g$ be two differentiable functions for which the following data are known:
$$f ( 1 ) = 1 ; f ^ { \prime } ( 1 ) = 2 ; g ( 1 ) = 3 ; g ^ { \prime } ( 1 ) = 4 :$$
Given $h ( x ) = f \left( ( x + 1 ) ^ { 2 } \right)$, use the chain rule to calculate $h ^ { \prime } ( 0 )$. Given $k ( x ) = \frac { f ( x ) } { g ( x ) }$, calculate $k ^ { \prime } ( 1 )$.
b) (1.25 points) Calculate the integral $\int ( \operatorname { sen } x ) ^ { 4 } ( \cos x ) ^ { 3 } d x$. (You can use the change of variables $t = \operatorname { sen } x$.)