4. A moving particle has position $( x ( t ) , y ( t ) )$ at time $t$. The position of the particle at time $t = 1$ is $( 2,6 )$, and the velocity vector at any time $t > 0$ is given by $\left( 1 - \frac { 1 } { t ^ { 2 } } , 2 + \frac { 1 } { t ^ { 2 } } \right)$. (a) Find the acceleration vector at time $t = 3$. (b) Find the position of the particle at time $t = 3$. (c) For what time $t > 0$ does the line tangent to the path of the particle at $( x ( t ) , y ( t ) )$ have a slope of 8 ? (d) The particle approaches a line as $t \rightarrow \infty$. Find the slope of this line. Show the work that leads to your conclusion.
4. A moving particle has position $( x ( t ) , y ( t ) )$ at time $t$. The position of the particle at time $t = 1$ is $( 2,6 )$, and the velocity vector at any time $t > 0$ is given by $\left( 1 - \frac { 1 } { t ^ { 2 } } , 2 + \frac { 1 } { t ^ { 2 } } \right)$.\\
(a) Find the acceleration vector at time $t = 3$.\\
(b) Find the position of the particle at time $t = 3$.\\
(c) For what time $t > 0$ does the line tangent to the path of the particle at $( x ( t ) , y ( t ) )$ have a slope of 8 ?\\
(d) The particle approaches a line as $t \rightarrow \infty$. Find the slope of this line. Show the work that leads to your conclusion.\\