1) The total revenue received from the sale of x items is R(x) = 30 ln (2x + 1) and the total cost to produce x items is C(x) = x/2. (a) Find the marginal revenue. (b) Find the marginal profit P'(x). (c) What value of x makes the marginal profit zero? 2) Suppose that at time t = 0, an olive oil distributor found that the demand for olive oil, sold at $15 dollars a quart, was 500 quarts each month. At the same time, the price was going up at a rate of $0.50 per month, and the demand was also going up at a rate of 30 quarts per month due to increased advertising. How fast was the revenue increasing at time t = 0? Note: The revenue function is given by R(t) = D(t)P(t) , where D(t) is the demand at time t, and P(t) is the price at time t. 3) Let F(x) = Find F'(1). 4+ 3 f ( g( x )) 4) Find the derivative of f(x) = , where f(2) = 4, f '(2) = 1, g(1) = 2, f(1) = 5, and g'(1) = 6. x e 2 5) According to work by the psychologist C. L. Hull, the strength of a habit is a function of the number of times the habit is repeated. If N is the number of repetitions and H(N) is the strength of the habit, then H(N) = 1000(1 - e^kN) , where k is a constant. (a) Find H'(N) if k = 0.1. (b) Explain why H'(N) is always positive. What does this mean in the context of habits? 6) Suppose a certain drug is administered to a patient, with the percent of concentration in the bloodstream t hours later given by K(t) = 5t/t^2 +1. On what time interval is the concentration of the drug increasing