Using integrals to figure out tangent lines

Question

I came across this problem in my calculus HW and I couldn't figure it out.

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

Suppose the marginal cost at a production level of 500 units is 10 and the cost of producing 5 $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.

I figured this out: if I plug in P(500)=12,000, I get 11,000 = (integral from 0 to 500)(M(s)ds). But I can't figure out how to use the M(500) = 10 into the question.

Answer 1

$P(x) = 1000 + \int_0^x M(s) ds\\ P(500) = 12000\\ M(500) = 10$

Approximate $P(497)$

$P(500) - P(497) = \int_{497}^{500} M(s) ds$

To do this we are going to assume that $M(s)$ is constant in a small neighborhood around $500$

$P(500) - P(497) = 30\\ P(497) = 11,970$