Question

Define g(v) to be the fuel efficiency in miles per gallon of a vehicle moving at v miles per hour. What are the units of g'(90)? What does it mean in 'real world' terms if g'(55)= -0.54?

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Answer 1

By "g'(v)", I presume you mean dg(v)/dv. If so, then the units of g' are (miles per gallon)/(miles per hour), and g' expresses the sensitivity of fuel consumption to vehicle speed. g'(55) = -0.54 means that if a vehicle traveling at 55 miles per hour accelerates to an incrementally higher speed, the subsequent steady state mileage will decrease by a proportionality factor of 0.54.
lim delta y / delta x -> dy/dx as delta x ->0.so then dy/dx * delta x , where delta x = 1 , we havedy ~ dy