Problem
If the cost function for a commodity is \[ C(x)=\frac{1}{90} x^{3}+2 x^{2}+2 x+16 \text { dollars } \] find the marginal cost \( \overline{M C} \) at \( x=9 \) units. (Round your answer to two decimal places.) \( \overline{M C}=\$ 407 / 10 \) Tell what the marginal cost predicts about the cost of producing 1 additional unit. The cost of producing 1 additional unit is \( \$-139 / 30 \). Tell what the marginal cost predicts about the cost of producing 2 additional units. The cost of producing 2 additional units is \( \$ 1501 / 30 \). Need Help? Master It
Solution
Step 1 ：Differentiate the cost function: \( MC(x) = \frac{dC}{dx} = \frac{1}{30} x^{2} + 4x + 2\)
Step 2 ：Find the marginal cost at x=9: \( MC(9) = \frac{1}{30}(9)^{2} + 4(9) + 2 \)
Step 3 ：Calculate the cost of producing 1 additional unit and 2 additional units: \(\Delta C_{1} = MC(9)-MC(10), \Delta C_{2} = MC(9)-MC(11)
