So I have a question that says I'm part of a group selling tickets for a cruise. There is a supposed profit function for the profit earned each day from selling x tickets that is P(x)=-900 + ∫[0,x] (400√(1+0.08t^2)) dt and I'm supposed to find the marginal profit when x=10 since that is the approximate profit by selling the 11th unit. I am confused on how to do this though with the ∫ being inside a function with x also as one of its limits.

Question

So I have a question that says I'm part of a group selling tickets for a cruise.  There is a supposed profit function for the profit earned each day from selling x tickets that is P(x)=-900 + ∫[0,x] (400√(1+0.08t^2)) dt and I'm supposed to find the marginal profit when x=10 since that is the approximate profit by selling the 11th unit.  I am confused on how to do this though with the ∫ being inside a function with x also as one of its limits.

oobleck · Accepted Answer

recall that if F(x) = ∫[0,x] f(t) then F'(x) = f(t) so, you have C'(x) = 400√(1+0.08x^2) This is just the chain rule in reverse If F'(t) = f(t) then the Fundamental Theorem of calculus says that ∫[a,x] f(t) = F(x)-F(a) so F'(x) = f(x) - 0