A small hovercraft travels back and forth along a straight test track. Its velocity (in m/s) is described by the piecewise–defined function
v(t)=⎩⎨⎧10−5t,−2(t−2)2+2,3t−14,0≤t≤2,2<t≤4,4<t≤6,
where t is measured in seconds.
(a) For each sub-interval of the domain on which v is defined, compute the corresponding acceleration function a(t).
(b) Create a combined sign analysis that shows the signs of v(t) and a(t) on every open sub-interval of [0,6] where both are continuous. Include all points where either function is zero or where the acceleration is not defined.
(c) Using your analysis from part (b), determine all intervals on which the hovercraft’s speed is increasing and all intervals on which it is decreasing. Justify your answer completely.
(d) Find every time in [0,6] at which the speed attains a local maximum or a local minimum. For each such time, state the speed value and classify it as a local maximum or minimum, providing a calculus-based justification.
(e) On your own paper, sketch a qualitative graph of the speed ∣v(t)∣ on 0≤t≤6. Clearly label the key points found in part (d) and indicate the intervals of increase and decrease identified in part (c).