91Ó°ÊÓ

Marginal revenue, cost, and profit. Let \(R(x), C(x),\) and \(P(x)\) be, respectively, the revenue, cost, and profit, in dollars, Irom the production and sale of \(x\) items. If \(R(x)=5 x\) and \(C(x)=0.001 x^{2}+1.2 x+60\) find each of the following. a) \(P(x)\) b) \(R(100), C(100),\) and \(P(100)\) c) \(R^{\prime}(x), C^{\prime}(x),\) and \(P^{\prime}(x)\) d) \(R^{\prime}(100), C^{\prime}(100),\) and \(P^{\prime}(100)\) e) Describe in words the meaning of each quantity in parts (b) and (d).

Differentiate implicitly to find dy/dx. Then find the slope of the curve at the given point. $$x^{2}+y^{2}=1 ; \quad\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$

Marginal profit. For Sunshine Motors, the weekly profit, in dollars, of selling \(x\) cars is $$P(x)=-0.006 x^{3}-0.2 x^{2}+900 x-1200$$ and currently 60 cars are sold weekly. a) What is the current weekly profit? b) How much profit would be lost if the dealership were able to sell only 59 cars weekly? c) What is the marginal profit when \(x=60 ?\) d) Use marginal profit to estimate the weekly profit if sales increase to 61 cars weekly.

Maximize \(Q=x y,\) where \(x\) and \(y\) are positive numbers such that \(x+\frac{4}{3} y^{2}=1\).

Of all rectangles that have a perimeter of \(42 \mathrm{ft}\), find the dimensions of the one with the largest area. What is its area?

From a 50 -cm-by- 50 -cm sheet of aluminum, square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume?

Drum Tight Containers is designing an open-top, square-based, rectangular box that will have a volume of 62.5 in \(^{3} .\) What dimensions will minimize surface area? What is the minimum surface area?

Assume the function \(f\) is differentiable over the interval \((-\infty, \infty)\) : that is, it is smooth and continuous for all real numbers \(x\) and has no corners or vertical tangents. Classify each of the following statements as cither true or false. If you choose false, explain why. The function \(f\) can have a point of inflection at a critical value.

A soup company is constructing an open-top, square-based, rectangular metal tank that will have a volume of \(32 \mathrm{ft}^{3} .\) What dimensions will minimize surface area? What is the minimum surface area?

Ever Green Gardening is designing a rectangular compost container that will be twice as tall as it is wide and must hold \(18 \mathrm{ft}^{3}\) of \(\mathrm{com}-\) posted food scraps. Find the dimensions of the compost container with minimal surface area (include the bottom and top).