Question

The cost of producing x units of a certain commodity is given by P(x) = 1000 + integral(x,0) MC(s)ds, where P is in dollars and M is marginal cost in dollars per unit.

A. Suppose the marginal cost at a production level of 500 units is $10 per unit, and the cost of producing 500 units is $12,000 (that is, M(500)=10 and P(500)=12000). Use a tangent line approximation to estimate the cost of producing only 497 units. (Answer: $11,970)
B. Suppose the production schedule is such that the company produces five units each day. That is, the number of units produced is x=5t, where t is in days, and t = 0 corresponds to the beginning of production. Write an equation for the cost of production P as a function of time t. (My Answer: 1000+ M(5t)C(5t) - M(0)C(0)
C. Use your equation for P(t) from part B to find dP/dT. Be sure to indicate units and describe what dP/dT represents, practically speaking. (NEED HELP)

Answer 1

P(x) = ∫[0,x] M(s) ds
so
dP/dx = M(x)
and then the linear approximation is
∆P = dP/dx ∆x
see what you can do with that.