The wave heights in the open sea depend on the speed v of the wind and the length of time that the...

Question

The wave heights in the open sea depend on the speed v of the wind and the length of time that the wind has been blowing at that speed. Values of the function h = ƒ(v, t) are recorded in feet in the following table.

Duration (hours)

Wind speed (knots)	v t	5	10	15	20	30	40	50
10	2	2	2	2	2	2	2
15	4	4	5	5	5	5	5
20	5	7	8	8	9	9	9
30	9	13	16	17	18	19	19
40	14	21	25	28	31	33	33
50	19	29	36	40	45	48	50
60	24	37	47	54	62	67	69

(a) What are the meanings of the partial derivatives ∂h/∂v and ∂h/∂t?

(b) Estimate the values of ƒ_v(40, 15) and ƒ_t(40, 50). What are the practical interpretations of these values?

(c) What appears to be the value of the following limit?

David · Accepted Answer

represents the rate of change of h when we fix t and consider h as a function of v which 
describes how quickly the wave heights change when the wind speed changes for a fixed time 
duration.
  
 represents the rate of change of h when we fix v and consider h as a function of t which describes 
how quickly the wave heights change when the duration of time changes,

The wave heights in the open sea depend on the speed v of the wind and the length of time that the wind has been blowing at that speed. Values of the function h = ƒ(v, t) are recorded in feet in the...

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