4. A squirrel starts at building $A$ at time $t = 0$ and travels along a straight, horizontal wire connected to building $B$. For $0 \leq t \leq 18$, the squirrel's velocity is modeled by the piecewise-linear function defined by the graph above.
(a) At what times in the interval $0 < t < 18$, if any, does the squirrel change direction? Give a reason for your answer.
(b) At what time in the interval $0 \leq t \leq 18$ is the squirrel farthest from building $A$ ? How far from building $A$ is the squirrel at that time?
(c) Find the total distance the squirrel travels during the time interval $0 \leq t \leq 18$.
(d) Write expressions for the squirrel's acceleration $a ( t )$, velocity $v ( t )$, and distance $x ( t )$ from building $A$ that are valid for the time interval $7 < t < 10$.
WRITE ALL WORK IN THE EXAM BOOKLET.
- Consider the differential equation $\frac { d y } { d x } = \frac { x + 1 } { y }$.
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and for $- 1 < x < 1$, sketch the solution curve that passes through the point $( 0 , - 1 )$. (Note: Use the axes provided in the exam booklet.) [Figure]
(b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the $x y$-plane for which $y \neq 0$. Describe all points in the $x y$-plane, $y \neq 0$, for which $\frac { d y } { d x } = - 1$.
(c) Find the particular solution $y = f ( x )$ to the given differential equation with the initial condition $f ( 0 ) = - 2$. - Two particles move along the $x$-axis. For $0 \leq t \leq 6$, the position of particle $P$ at time $t$ is given by $p ( t ) = 2 \cos \left( \frac { \pi } { 4 } t \right)$, while the position of particle $R$ at time $t$ is given by $r ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 9 t + 3$.
(a) For $0 \leq t \leq 6$, find all times $t$ during which particle $R$ is moving to the right.
(b) For $0 \leq t \leq 6$, find all times $t$ during which the two particles travel in opposite directions.
(c) Find the acceleration of particle $P$ at time $t = 3$. Is particle $P$ speeding up, slowing down, or doing neither at time $t = 3$ ? Explain your reasoning.
(d) Write, but do not evaluate, an expression for the average distance between the two particles on the interval $1 \leq t \leq 3$.
WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM
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