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The marginal cost function $ C^{\prime} (x) $ was defined to be the derivative of the cost function. (See Sections 3.7 and 4.7.) The marginal cost of producing x gallons of orange juice is

$$ C^{\prime} (x) = 0.82 - 0.00003x + 0.000000003x^2 $$

(measured in dollars per gallon). The fixed start-up cost is $ C(0) = $18,000 $. Use the Net Change Theorem to find the cost of producing the first 4000 gallons of juice.

$\$ 21,104$

Applications of Integration

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Okay, So the question gives us a marginal cost function See, Prime of X and asked us to find the cost of making the 1st 4000 units using the net change here. Over on the right in the box. I have written that change there, which is F A B is equal to f of A plus the integral from A to B f prime of x d. X. So it also gives us in the question the value foresee of zero, which is a sea of zero, is equal to 18,000. So here we just apply the formula. We're looking for C 4000 and so he said that that is equal to see of zero plus the integral from zero 4000 of our function. 0.82 minus 0.3 x plus 0.3 x squared dx And so we get that this is 18,000. Course, you're a 0.82 x evaluated from 0 to 4000 minus 0.3 over too x squared. You evaluated zero and 4000 and then finally 0.0 zero 00003 over three Ex cued evaluated at zero and 4000. So because all of these air pollen a meal, all of the zero terms will go away because when you plug in, zero will evaluate to zero so plugging in you get 18,000 plus 3000 280 Maya's 240 plus 64 which is equal to the final answer in dollars of 21,000 104.