Chapter 13

Exercises 55 and 56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int_{2}^{5} \int_{1}^{6} x d y d x=\int_{1}^{6} \int_{2}^{5} x d x d y $$

Evaluate the second partial derivatives \(f_{x x^{\prime}} f_{x y^{\prime}} f_{y y^{\prime}}\) and \(f_{y x}\) at the point. $$ f(x, y)=\sqrt{x^{2}+y^{2}} $$

Sketch the trace of the intersection of each plane with the given sphere. \(x^{2}+y^{2}+z^{2}=169\) (a) \(x=5\) (b) \(y=12\)

Evaluate the second partial derivatives \(f_{x x^{\prime}} f_{x y^{\prime}} f_{y y^{\prime}}\) and \(f_{y x}\) at the point. $$ f(x, y)=\ln (x-y) $$

Sketch the trace of the intersection of each plane with the given sphere. \(x^{2}+y^{2}+z^{2}-4 x-6 y+9=0\) (a) \(x=2\) (b) \(y=3\)

Evaluate the second partial derivatives \(f_{x x^{\prime}} f_{x y^{\prime}} f_{y y^{\prime}}\) and \(f_{y x}\) at the point. $$ f(x, y)=x^{2} e^{y} $$

Sketch the trace of the intersection of each plane with the given sphere. \(x^{2}+y^{2}+z^{2}-8 x-6 z+16=0\) (a) \(x=4\) (b) \(z=3\)

A company manufactures two models of bicycles: a mountain bike and a racing bike. The cost function for producing \(x\) mountain bikes and \(y\) racing bikes is given by \(C=10 \sqrt{x y}+149 x+189 y+675\) (a) Find the marginal costs \((\partial C / \partial x\) and \(\partial C / \partial y)\) when \(x=120\) and \(y=160\) (b) When additional production is required, which model of bicycle results in the cost increasing at a higher rate? How can this be determined from the cost model?

A pharmaceutical corporation has two plants that produce the same over-the- counter medicine. If \(x_{1}\) and \(x_{2}\) are the numbers of units produced at. plant 1 and plant 2 , respectively, then the total revenue for the product is given by \(R=200 x_{1}+200 x_{2}-4 x_{1}^{2}-8 x_{1} x_{2}-4 x_{2}^{2}\) When \(x_{1}=4\) and \(x_{2}=12\), find (a) the marginal revenue for plant \(1, \partial R / \partial x_{1}\). (b) the marginal revenue for plant \(2, \partial R / \partial x_{2}\).

Consider the Cobb-Douglas production function \(f(x, y)=200 x^{0.7} y^{0.3}\). When \(x=1000\) and \(y=500\), find (a) the marginal productivity of labor, \(\partial f / \partial x\). (b) the marginal productivity of capital, \(\partial f / \partial y\).

A spherical building has a diameter of 165 feet. The center of the building is placed at the origin of a three-dimensional coordinate system. What is the equation of the sphere?

Determine whether the demand functions describe complementary or substitute product relationships. Using the notation of Example 4, let \(x_{1}\) and \(x_{2}\) be the demands for products \(p_{1}\) and \(p_{2}\), respectively. $$ x_{1}=150-2 p_{1}-\frac{5}{2} p_{2}, \quad x_{2}=350-\frac{3}{2} p_{1}-3 p_{2} $$

Determine whether the demand functions describe complementary or substitute product relationships. Using the notation of Example 4, let \(x_{1}\) and \(x_{2}\) be the demands for products \(p_{1}\) and \(p_{2}\), respectively. $$ x_{1}=150-2 p_{1}+1.8 p_{2}, \quad x_{2}=350+\frac{3}{4} p_{1}-1.9 p_{2} $$

A model for the per capita consumptions (in gallons) of different types of plain milk in the United States from 1999 through 2004 is \(z=1.25 x-0.125 y+0.95 .\) Consumption of reduced-fat \((1 \%)\) and skim milks, reducedfat milk ( \(2 \%\) ), and whole milk are represented by variables \(x, y\), and \(z\), respectively. (Source: U.S. Department of Agriculture) (a) Find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). (b) Interpret the partial derivatives in the context of the problem.

The shareholder's equity \(z\) (in billions of dollars) for Wal-Mart Corporation from 2000 through 2006 can be modeled by \(z=0.205 x-0.073 y-0.728\) where \(x\) is net sales (in billions of dollars) and \(y\) is the total assets (in billions of dollars). (Source: Wal-Mart Corporation) (a) Find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). (b) Interpret the partial derivatives in the context of the problem.

Early in the twentieth century, an intelligence test called the Stanford-Binet Test (more commonly known as the \(I Q\) test was developed. In this test, an individual's mental age \(M\) is divided by the individual's chronological age \(C\) and the quotient is multiplied by 100 . The result is the individual's \(I Q\). \(I Q(M, C)=\frac{M}{C} \times 100\) Find the partial derivatives of \(I Q\) with respect to \(M\) and with respect to \(C\). Evaluate the partial derivatives at the point \((12,10)\) and interpret the result.

The value of an investment of $$ 1000\( earning \)10 \%\( compounded annually is \)V(I, R)=1000\left[\frac{1+0.10(1-R)}{1+I}\right]^{10}\( where \)I\( is the annual rate of inflation and \)R\( is the tax rate for the person making the investment. Calculate \)V_{I}(0.03,0.28)\( and \)V_{R}(0.03,0.28) .$ Determine whether the tax rate or the rate of inflation is the greater "negative" factor on the growth of the investment.

Let \(N\) be the number of applicants to a university, \(p\) the charge for food and housing at the university, and \(t\) the tuition. Suppose that \(N\) is a function of \(p\) and \(t\) such that \(\partial N / \partial p<0\) and \(\partial N / \partial t<0 .\) How would you interpret the fact that both partials are negative?

To 1996, twin sisters Izzy and Coco Tihanyi started Surf Diva, a surf school and apparel company for women and girls, in La Jolla, California. To advertise their business, they would donate surf lessons and give the surf report on local radio stations in exchange for air time. Today, they have schools in Japan and Costa Rica, and their clothing line can be found in surf and specialty shops, sporting goods stores, and airport gift shops. Sales from their surf schools have increased nearly \(13 \%\) per year, and product sales are expected to double each year. Use your school's library, the Internet, or some other reference source to research a company that increased the demand for its product by creative advertising. Write a paper about the company. Use graphs to show how a change in demand is related to a change in the marginal utility of a product or service.