Question

exercises 1.4 integration formulas and the net change theorem
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the marginal cost for printing a paperback book at a small publishing company is
c(p) = $0.02 per page where p is the number of pages in the book
a 720 page book has a $20.77 production cost. find the production cost function c(p).
hint: marginal cost is the derivative of cost.
c(p) = $ 0.02p + 5.37
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Explanation:

Step1: Integrate marginal cost

Since marginal cost $c(p)$ is the derivative of $C(p)$, integrate $c(p)$ to find $C(p)$:
$$C(p) = \int 0.02 \, dp = 0.02p + K$$
where $K$ is the constant of integration (fixed cost).

Step2: Solve for constant $K$

Use the given condition: when $p=720$, $C(720)=20.77$. Substitute into the function:
$$20.77 = 0.02(720) + K$$
Calculate $0.02(720) = 14.4$, then:
$$K = 20.77 - 14.4 = 6.37$$

Step3: Define cost function

Substitute $K=6.37$ back into $C(p)$:
$$C(p) = 0.02p + 6.37$$

Answer:

$C(p) = 0.02p + 6.37$