The depth d (in meters) of water at the end of a pier is modeled for 0≤t≤12 hours by the following piecewise description, where t is measured from midnight.
• For 0≤t<3, the graph is a quadratic curve opening downward that joins the points (0,5) and (3,8) and has its vertex at (1.5,8.5).
• At t=3 hours there is a jump upward: an open point at (3,8) and a closed point at (3,9).
• For 3<t<7, the graph is a straight line segment from (3,9) to (7,3).
• For 7≤t≤12, the graph is a semicircle (concave down) with center (9.5,3) and radius 2.5 that meets the line segment smoothly at (7,3) and ends at (12,3).
a. Part 1: a. Identify each hour t in [0,12] at which the depth model d is not continuous. Provide evidence from the description.
b. Part 2: b. Identify each hour t in [0,12] at which d is continuous but not differentiable. For every such hour, explain what graphical feature causes the derivative to fail to exist.
c. Part 3: c. Interpret, in the context of the pier, the meaning of a point where d has a vertical tangent. Does such a point occur on this graph? Justify your answer.