Find dH/dv, and interpret its meaning in this situation.
This is like dy/dx -- the rate at which variable y changes with respect to variable x. Only, in this scenario, y is called H and x is called v. H is a function of v, and dH/dv is the derivative of H with respect to v.
Symbol dH/dv represents the rate at which H changes, with respect to v. Therefore, dH/dv can be interpreted as the rate at which heat-loss changes with respect to wind speed.
Did you happen to graph H (heat loss) as a function of v (wind speed)? In general, heat loss is greater when standing in faster winds, but how do fluctuations in wind speed affect heat loss? dH/dt gives us a measure, to quantify how a change in v affects H.
With no wind at all, heat loss is 10.5, but, let wind speed increase by just one meter per second, and heat loss goes from 10.5 up to 339.5 -- that is, H changes by 328 when v changes from 0 to 1.
Increasing v by 1 unit again (from 1 m/s to 2 m/s) results in H increasing by only 135.7 instead of by 328. So, H is not increasing by the same amount, for 1-unit increases in v; we see that subsequent heat losses go up by smaller amounts, for each v increase of 1. In other words, heat loss through each square meter of exposed skin continues to increase with increasing wind speeds, but those increases become less and less dramatic.
Therefore, the
rate of heat-loss change
decreases as wind speed increases. dH/dv gets smaller as v gets bigger.
These changes in H and v are smooth and continuous, so dH/dv changes smoothly, too. A small fluctuation in v leads to small fluctuations in H and dH/dt. I'm assuming that you have already seen derivative defined as a limit of a difference quotient. My examples above, where v increased by an entire unit, describe
average changes in H, over 1-unit jumps in v. Any particular value of the variable dH/dv is a limit of average changes, as fluctuations in the related v go to zero. This limiting value is dH/dv, so a specific value of dH/dv represents the
exact rate of change in H at the associated value of v.
Graphically, you can "see" values of dH/dv. They are the slopes of lines tangent to the graph of H at each v. For a particular example, draw a tangent line to the curve of H(v) where v=5. The slope of that line is the value of dH/dv when v=5.
Cheers
