Suppose the demand price for selling out a production run of x DVDs is given by p(x) = -0.0005x2 +60 dollars per DVD. Further suppose the weekly cost of producing these DVDs is given by C(x) = -0.001x2 + 18x + 4000 dollars to produce that many DVDs. (i) Write the Revenue function R(x). (ii) Use (i) to write the profit function P(x). (iii) Draw a signs diagram for P(x), the marginal profit function. (iv) Use (iii) to find the production level x for which the profit is a maximum. (v) So what is the maximum profit the DVD manufacturer can hope to make? Show your work! (vi) Answer (v) in a complete English sentence.

Consider any function of the form f(x) = ln u(x). (i) If we wish to find the DOMAIN of our function, what must be true of our u(x)? Explain in a complete English sentence. (ii) Now suppose u(x) = x3 12x. Write the function f(x) now, and then find its domain, using (i). Give exact answer (NOT decimal approximations). HINT: You might start with factoring u(x), setting it equal to zero, and then making a signs diagram for u(x) [just as weve done for derivatives]. Write a complete sentence answer to the domain question, and give your answer in interval notation. (iii) Now find the derivative of our f(x)