A particle moving along a curve in the plane has position $( x ( t ) , y ( t ) )$ at time $t$, where $$\frac { d x } { d t } = \sqrt { t ^ { 4 } + 9 } \text { and } \frac { d y } { d t } = 2 e ^ { t } + 5 e ^ { - t }$$ for all real values of $t$. At time $t = 0$, the particle is at the point $( 4,1 )$. (a) Find the speed of the particle and its acceleration vector at time $t = 0$. (b) Find an equation of the line tangent to the path of the particle at time $t = 0$. (c) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 3$. (d) Find the $x$-coordinate of the position of the particle at time $t = 3$.
A particle moving along a curve in the plane has position $( x ( t ) , y ( t ) )$ at time $t$, where
$$\frac { d x } { d t } = \sqrt { t ^ { 4 } + 9 } \text { and } \frac { d y } { d t } = 2 e ^ { t } + 5 e ^ { - t }$$
for all real values of $t$. At time $t = 0$, the particle is at the point $( 4,1 )$.\\
(a) Find the speed of the particle and its acceleration vector at time $t = 0$.\\
(b) Find an equation of the line tangent to the path of the particle at time $t = 0$.\\
(c) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 3$.\\
(d) Find the $x$-coordinate of the position of the particle at time $t = 3$.