Let h (t) be the height of the tide at the Bay of Fundy in meters that has changed since midnight, with t measured in hours. We restrict the domain of h (t) to where it is a one-to-one function. Interpret the following in practical terms, giving units a. h (7) = 2.75 b. h' (7) = 0.21 c. h^-1 (-1.5) = 13.2 d. (h^-1) ' (-1.5) = -1.6

Question

Mathematics

Quinten Campos

11 April, 13:05

Let h (t) be the height of the tide at the Bay of Fundy in meters that has changed since midnight, with t measured in hours. We restrict the domain of h (t) to where it is a one-to-one function. Interpret the following in practical terms, giving units a. h (7) = 2.75 b. h' (7) = 0.21 c. h^-1 (-1.5) = 13.2 d. (h^-1) ' (-1.5) = -1.6

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Dirty Harry · Answer 1

Answer and Step-by-step explanation:

a) h (7) = 2.75 metre

This means that the change in height of tide at Bay of Fundy 7 hrs after midnight is 2.75 meters. That is, putting t=7 hours into the function h (t) will give the change in height of the tide at Bay of Fundy since midnight.

b) h' (7) = 0.21 metres/hour

The h' sign indicates the first derivative of the function h (t). Since h (t) gives the change in height of tide at Bay of Fundy since midnight, h' (t) will give the rate of change of the height of tide at Bay of Fundy since midnight with time. That is, how fast or how slow the height of tide at Bay of Fundy since midnight is changing.

h' (7) = 0.21 m/h means the rate of change of the height of tide at Bay of Fundy 7 hours after midnight is 0.21 metres/hour.

c) h⁻¹ (-1.5) = 13.2

The inverse of a function is defined as the function that completely undo the effect of the original function.

The original function takes a value of time and gives the height of tide at Bay of Fundy since midnight, but the inverse now would take the height of tide at Bay of Fundy since midnight and give the value of time that corresponds to such a height.

h⁻¹ (-1.5) = 13.2 hours means that the time, measured from midnight, that it takes the height of tide at Bay of Fundy to change by - 1.5 m since midnight is 13.2 hours.

d) (h⁻¹) ' (-1.5) = - 1.6 hour/metre

Since we have previously established that the inverse of the function, h⁻¹ is the function that gives the value of time that corresponds to a particular change in height of tide at Bay of Fundy since midnight.

So, the derivative of that inverse function would give the how much the time (Since midnight) is changing with respect to the change of the height of tide at Bay of Fundy since midnight.

(h⁻¹) ' (-1.5) = - 1.6 hour/metre means that, at change of height = - 1.5 m, the time for this change to occur is changing at a rate of - 1.6 hours/metre