The marginal cost of manufacturing x yards of a certain fabric is C′(x)=3−0.01x+0.000006x2 (in dollars per yard). Find the increase in cost if the production level is raised from 3500 to 8500 yards.

wold that not be just the integral of C' from 3500 to 8500?

Just use the power rule for each term.

The marginal cost of manufacturing x yards of a certain fabric is C′(x)=3−0.01x+0.000006x2 (in dollars per yard). Find the increase in cost if the production level is raised from 2500 to 6500 yards

621000

To find the increase in cost, we need to calculate the difference between the total cost of producing 8500 yards of fabric and the total cost of producing 3500 yards of fabric.

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.