Answer:
a. [tex]\frac{dC(q)}{dq} = 0.000012q^2 -0.06q + 100[/tex]
b. [tex]\frac{dR(q)}{dq}=-0.01q+200[/tex]
c.
[tex]U'(2000)=-0.000012(2000)^2+0.05(2000)+100 = 152[/tex]
[tex]U'(7000)=-0.000012(7000)^2+0.05(7000)+100 = -138[/tex]
Step-by-step explanation:
a) The marginal cost function is given by the derivative of the total cost function, in this way the marginal cost function for this company is:
[tex]\frac{dC(q)}{dq} = \frac{dC(q)}{dq} (0.000004q^ 3 - 0.03q ^ 2 + 100q + 75000) = 0.000012q^2 -0.06q + 100[/tex]
b) The income function is given by the relation [tex]R (q) = P (q) q = -0.005q^2 + 200q[/tex].
The marginal revenue function for the company is given by the derivative of the revenue function, in this way the marginal revenue function is:
[tex]\frac{dR(q)}{dq}= -0.01q+200[/tex]
(c) The profit function of the company is given by the relation [tex]U (q) = R (q) - C (q)[/tex], and the marginal utility function is given by the derivative of the utility function, in this way , the marginal utility function is:
[tex]\dfrac{dU(q)}{dq}=R'(q) - C'(q) = -0.01q+200 - (0.000012q^2-0.06q+100) = -0.000012q^2+0.05q+100[/tex]
When q = 2000, the marginal utility is:
[tex]U'(2000)=-0.000012(2000)^2+0.05(2000)+100 = 152[/tex]
When q = 7000, the marginal utility is:
[tex]U'(7000)=-0.000012(7000)^2+0.05(7000)+100 = -138[/tex]
