At time $t$, a particle moving in the $x y$-plane is at position $( x ( t ) , y ( t ) )$, where $x ( t )$ and $y ( t )$ are not explicitly given. For $t \geq 0 , \frac { d x } { d t } = 4 t + 1$ and $\frac { d y } { d t } = \sin \left( t ^ { 2 } \right)$. At time $t = 0 , x ( 0 ) = 0$ and $y ( 0 ) = - 4$. (a) Find the speed of the particle at time $t = 3$, and find the acceleration vector of the particle at time $t = 3$. (b) Find the slope of the line tangent to the path of the particle at time $t = 3$. (c) Find the position of the particle at time $t = 3$. (d) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 3$.
: \text { integral }
At time $t$, a particle moving in the $x y$-plane is at position $( x ( t ) , y ( t ) )$, where $x ( t )$ and $y ( t )$ are not explicitly given. For $t \geq 0 , \frac { d x } { d t } = 4 t + 1$ and $\frac { d y } { d t } = \sin \left( t ^ { 2 } \right)$. At time $t = 0 , x ( 0 ) = 0$ and $y ( 0 ) = - 4$.
(a) Find the speed of the particle at time $t = 3$, and find the acceleration vector of the particle at time $t = 3$.
(b) Find the slope of the line tangent to the path of the particle at time $t = 3$.
(c) Find the position of the particle at time $t = 3$.
(d) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 3$.