The cost (in dollars) of producing x units of a certain commodity is C(x) = 6000

Question

The cost (in dollars) of producing x units of a certain commodity is C(x) = 6000 + 5x + 0.1x^2. Find the average rate of change of C with respect to x when the production level is changed from x = 100 to the given values. (Round your answers to the nearest cent.) x = 104 $ per unit x = 101 $ per unit Find the instantaneous rate of change of C with respect to x when x = 100. (This is called the marginal cost.) $ per unit Please To find the average rate of change, simply divide the difference in cost by the difference in production level. To find the rate of change, take the limit of the function for the average rate of change as the production level approaches the desired production level. Recall that lim_x rightarrow infinity f(x) - f(a)/x - a gives the limit as a of the average rate of change of f on the interval (a, x).

Explanation / Answer

given cost C(x)=6000+5x+0.1x2

a)

i)

average rate of change =[C(104)-C(100)]/(104-100)

average rate of change =[6000+5*(104)+0.1*(104)2-6000-5*(100)-0.1*(100)2]/(104-100)

average rate of change =[101.6]/4

average rate of change =25.4$ per unit

ii)

average rate of change =[C(101)-C(100)]/(101-100)

average rate of change =[6000+5*(101)+0.1*(101)2-6000-5*(100)-0.1*(100)2]/(101-100)

average rate of change =[25.1]/1

average rate of change =25.1$ per unit

b)C(x)=6000+5x+0.1x2

differentiate with respect to x

instantaneous rate of change dC/dx =0+5+0.1*2x

instantaneous rate of change dC/dx =5+0.2x

when x =100

instantaneous rate of change dC/dx =5+(0.2*100)

instantaneous rate of change dC/dx =25

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