For $0 \leq t \leq 5$, a particle is moving along a curve so that its position at time $t$ is $( x ( t ) , y ( t ) )$. At time $t = 1$, the particle is at position $( 2 , - 7 )$. It is known that $\frac { d x } { d t } = \sin \left( \frac { t } { t + 3 } \right)$ and $\frac { d y } { d t } = e ^ { \cos t }$. (a) Write an equation for the line tangent to the curve at the point $( 2 , - 7 )$. (b) Find the $y$-coordinate of the position of the particle at time $t = 4$. (c) Find the total distance traveled by the particle from time $t = 1$ to time $t = 4$. (d) Find the time at which the speed of the particle is 2.5. Find the acceleration vector of the particle at this time.
For $0 \leq t \leq 5$, a particle is moving along a curve so that its position at time $t$ is $( x ( t ) , y ( t ) )$. At time $t = 1$, the particle is at position $( 2 , - 7 )$. It is known that $\frac { d x } { d t } = \sin \left( \frac { t } { t + 3 } \right)$ and $\frac { d y } { d t } = e ^ { \cos t }$.\\
(a) Write an equation for the line tangent to the curve at the point $( 2 , - 7 )$.\\
(b) Find the $y$-coordinate of the position of the particle at time $t = 4$.\\
(c) Find the total distance traveled by the particle from time $t = 1$ to time $t = 4$.\\
(d) Find the time at which the speed of the particle is 2.5. Find the acceleration vector of the particle at this time.