Activity $4\mathrm{.}1\mathrm{.}4.$ Suppose that an object moving along a straight line path has its velocity $v($in meters per second$)$ at time $t($in seconds$)$ given by the piecewise linear function whose graph is pictured at left in Figure $4\mathrm{.}1\mathrm{.}10.$ We view movement to the right as being in the positive direction $($with positive velocity$),$ while movement to the left is in the negative direction.

Figure $4\mathrm{.}1\mathrm{.}10.$ The velocity function of a moving object.

Suppose further that the object's initial position at time $t=0$ is $s(0)=1.$

a$.$ Determine the total distance traveled and the total change in position on the time interval $0t2.$ What is the object's position at $t=2?$

b$.$ On what time intervals is the moving object's position function increasing? Why? On what intervals is the objects position decreasing? Why?

c$.$ What is the object's position at $t=8?$ How many total meters has it traveled to get to this point $($including distance in both directions$)?$ Is this different from the object's total change in position on $t=0$ to $t=8?$

d$.$ Find the exact position of the object at $t=1,2,3,$dots,$8$ and use this data to sketch an accurate graph of $y=s(t)$ on the axes provided at right in Figure $4\mathrm{.}1\mathrm{.}10.$ How can you use the provided information about $y=v(t)$ to determine the concavity of $s$ on each relevant interval?