An object moving along a curve in the $x y$-plane is at position $( x ( t ) , y ( t ) )$ at time $t$ with $$\frac { d x } { d t } = \arctan \left( \frac { t } { 1 + t } \right) \text { and } \frac { d y } { d t } = \ln \left( t ^ { 2 } + 1 \right)$$ for $t \geq 0$. At time $t = 0$, the object is at position $( - 3 , - 4 )$. (Note: $\tan ^ { - 1 } x = \arctan x$ ) (a) Find the speed of the object at time $t = 4$. (b) Find the total distance traveled by the object over the time interval $0 \leq t \leq 4$. (c) Find $x ( 4 )$. (d) For $t > 0$, there is a point on the curve where the line tangent to the curve has slope 2 . At what time $t$ is the object at this point? Find the acceleration vector at this point.
& : \text { integral for section of limaçon }
An object moving along a curve in the $x y$-plane is at position $( x ( t ) , y ( t ) )$ at time $t$ with
$$\frac { d x } { d t } = \arctan \left( \frac { t } { 1 + t } \right) \text { and } \frac { d y } { d t } = \ln \left( t ^ { 2 } + 1 \right)$$
for $t \geq 0$. At time $t = 0$, the object is at position $( - 3 , - 4 )$. (Note: $\tan ^ { - 1 } x = \arctan x$ )
(a) Find the speed of the object at time $t = 4$.
(b) Find the total distance traveled by the object over the time interval $0 \leq t \leq 4$.
(c) Find $x ( 4 )$.
(d) For $t > 0$, there is a point on the curve where the line tangent to the curve has slope 2 . At what time $t$ is the object at this point? Find the acceleration vector at this point.