3. Given the curve \( r(t)=\left\langle 4 t,-\frac{2 t^{3}}{3}, 2 t^{2}\right\rangle \), find the followings. (a) ( 5 pts) Find \( \mathrm{a}(t) \) and \( \mathrm{T}(1) \). (b) ( 8 pts) Find the decomposition of a \( (t) \) into tangential component, \( a_{T} \), and normal com- ponents, \( a_{N} \), at \( t=1 \). (c) ( 5 pts) Find the curvature \( \kappa(1) \).

Find the area bounded by the graphs of the indicated equations over the given interval. \[ y=-x^{2}+29 ; y=0 ;-3 \leq x \leq 3 \] The area is \( \square \) square units.

Find \( f^{\prime}(x) \). \( f(x)=\left(3 x^{5}+4\right)^{3} \) \( f^{\prime}(x)=\square \)

Calculate \( \int_{9}^{7} 3 x^{2} d x \), given the following. \( \int_{4}^{7} x d x=16.5 \int_{4}^{7} x^{2} d x=93 \int_{7}^{9} x^{2} d x=\frac{386}{3} \) \( \int_{9}^{7} 3 x^{2} d x=\square \) (Type an integer or a simplified fraction.)

\( \left. \begin{array} { l } { y = \cos ^ { 3 } ( 5 + 8 x ) } \\ { y ^ { \prime } - ? } \end{array} \right. \)

Use L'Hôpital's rule to find the following limit and check the answer using an algebraic simplification. \( \lim _{x \rightarrow 8} \frac{x^{2}-x-56}{x-8} \) \( \lim _{x \rightarrow 8} \frac{x^{2}-x-56}{x-8}=\square \) (Simplify your answer.)