Solve the optimization problems. Maximize \(P=x y\) with \(x+y=10\).

The demand curve for original Iguanawoman comics is given by $$q=\frac{(400-p)^{2}}{100} \quad(0 \leq p \leq 400)$$ where \(q\) is the number of copies the publisher can sell per week if it sets the price at p. a. Find the price elasticity of demand when the price is set at $$\$ 40$$ per copy. b. Find the price at which the publisher should sell the books in order to maximize weekly revenue. c. What, to the nearest \(\$ 1\), is the maximum weekly revenue the publisher can realize from sales of Iguanawoman comics?

If you know how fast one quantity is changing and need to compute how fast a second quantity is changing, what kind of information do you need?

The demand and unit price for your store's checkered T-shirts are changing with time. Show that the percentage rate of change of revenue equals the sum of the percentage rates of change of price and demand. (The percentage rate of change of a quantity \(Q\) is \(\left.Q^{\prime}(t) / Q(t) .\right)\)

If we regard position, \(s\), as a function of time, \(t\), what is the significance of the third derivative, \(s^{\prime \prime \prime}(t) ?\) Describe an everyday scenario in which this arises.

You have been hired as a marketing consultant to Big Book Publishing, Inc., and you have been approached to determine the best selling price for the hit calculus text by Whiner and Istanbul entitled Fun with Derivatives. You decide to make life easy and assume that the demand equation for Fun with Derivatives has the linear form \(q=m p+b\), where \(p\) is the price per book, \(q\) is the demand in annual sales, and \(m\) and \(b\) are certain constants you'll have to figure out. a. Your market studies reveal the following sales figures: when the price is set at \(\$ 50.00\) per book, the sales amount to 10,000 per year; when the price is set at \(\$ 80.00\) per book, the sales drop to 1000 per year. Use these data to calculate the demand equation. b. Now estimate the unit price that maximizes annual revenue and predict what Big Book Publishing, Inc.'s annual revenue will be at that price.