Someone analyzed data for 10 weeks of production and now estimates marginal cost for producing x units in a week (in thousands of dollars) to be C'(x)=50e^(-x/100). Given that fixed costs for a week are 3 million dollars, find the total cost of producing 100 units. I thought I was meant to find the antiderivative of this function and then plug in the two other values to find the total cost but when I did that I was wrong and now I'm confused.

Question

Someone analyzed data for 10 weeks of production and now estimates marginal cost for producing x units in a week (in thousands of dollars) to be C'(x)=50e^(-x/100).  Given that fixed costs for a week are 3 million dollars, find the total cost of producing 100 units.  I thought I was meant to find the antiderivative of this function and then plug in the two other values to find the total cost but when I did that I was wrong and now I'm confused.

oobleck · Answer

too bad you didn't show your work, since you clearly want us to show ours.
C'(x) = 50e^(-x/100)
C(x) = -5000 e^(-x/100) + k
since C(0) = 3000 (the cost of producing 0 units is the fixed cost)
-5000 + k = 3000
k = 8000
C(x) = -5000e^(-x/100) + 8000
C(100) = -5000/e + 8000 = 6160

Anonymous · Answer

I see what I did wrong. I used the fixed cost as if it were k rather than plugging it in as a point. So the best way to find K is checking C(0) when having fixed cost?