Tide data for a coastal inlet are modeled by the differentiable
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Tide data for a coastal inlet are modeled by the differentiable, increasing function H, where H(t) is the water depth (in meters) at t hours after midnight. Selected measurements are provided.
t (h)
0
1
2
3
4
5
H(t) (m)
2.70
3.00
3.20
3.45
3.63
3.77
H′(t) (m/h)
0.28
0.25
0.24
0.18
0.14
0.12
Let T=H−1 so that T(d) gives the time (in hours after midnight) when the water depth is d meters.
(a) Using the data in the table, determine T′(3.20) and state its units. Show all supporting calculations.
(b) Provide a practical interpretation of the value of T′(3.20) within the context of the tidal motion of the inlet.
(c) Using a local linear approximation based at d=3.20 m, estimate how many minutes it will take for the depth to rise from 3.20 m to 3.30 m. Clearly communicate your reasoning and include units throughout your work.