The height H(t), measured in meters, of water in a reservoir t days after a heavy rainfall is modeled by a twice–differentiable function for t≥0. The following facts are known:
• H(5)=12
• H′(5)=−0.6
• H′′(t)>0 for 0<t<5 and H′′(t)<0 for t>5
• H′′ is continuous for all t≥0
(a) Write the equation of the tangent line to the graph of H at t=5.
(b) Use this tangent line to approximate the water height H(6). Show your calculation clearly.
(c) Decide whether the approximation obtained in part (b) is an underestimate or an overestimate of H(6). Give a thorough justification that relies on the information provided about H′′.
(d) Suppose instead that the tangent line at t=4 were used to approximate H(4.3). Without computing that line explicitly, determine whether such an approximation would be an underestimate or an overestimate. Support your answer with a complete explanation referencing concavity on the appropriate interval.
(e) Discuss how the change in sign of H′′ at t=5 affects the reliability of tangent‐line approximations for times before and after t=5. Include in your discussion any limitations that concavity imposes on the magnitude and direction of the approximation error.