wold that not be just the integral of C' from 3500 to 8500?
Just use the power rule for each term.
Increase in cost =
Just use the power rule for each term.
The total cost of producing x yards of fabric is given by the integral of the marginal cost function C'(x). We can find the total cost function C(x) by integrating C'(x) with respect to x.
C(x) = ∫(3 - 0.01x + 0.000006x^2) dx
To integrate, we add the antiderivatives of each term:
C(x) = ∫3 dx - ∫0.01x dx + ∫0.000006x^2 dx
Simplifying, we get:
C(x) = 3x - 0.01/2 * x^2 + 0.000006/3 * x^3 + C
Now we can calculate the total cost at each production level:
C(3500) = 3(3500) - 0.01/2 * (3500)^2 + 0.000006/3 * (3500)^3 + C
C(8500) = 3(8500) - 0.01/2 * (8500)^2 + 0.000006/3 * (8500)^3 + C
Subtracting the total cost at 3500 yards from the total cost at 8500 yards will give us the increase in cost:
Increase in cost = C(8500) - C(3500)
Now you can plug in the values and calculate the increase in cost.