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

Question

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

anonymous · Answer

Where is A.?

anonymous · Answer

$$
P(x) = 1000+\int _0^xMC(s)ds
$$
B. Just wants you to substitute $x=5t$.  So $$
P(t) = 1000+\int _0^{5t}MC(s)ds
$$(This is an abuse of notation, we should not be using the same function name here).

C. Wants you to use the fundamental theorem of Calculus.

anonymous · Answer

$$
\frac{d}{dx} \int_a^{g(x)}f(t)dt = (f\circ g)(x)\cdot \frac{dg}{dx} = f(g(x))\cdot g'(x)
$$