Chapter 7

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=\sqrt{9-x^{2}-y^{2}} $$

Sketch the \(y z\) -trace of the sphere. $$ (x+2)^{2}+(y-3)^{2}+z^{2}=9 $$

Identify the quadric surface. $$ 4 y=x^{2}+z^{2} $$

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{2} \int_{\sqrt{4-x^{2}}}^{4-x^{2} / 4} \frac{x y}{x^{2}+y^{2}+1} d y d x $$

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=\frac{1}{x-y} $$

Sketch the \(y z\) -trace of the sphere. $$ x^{2}+y^{2}+z^{2}-4 x-4 y-6 z-12=0 $$

Identify the quadric surface. $$ 3 z=-y^{2}+x^{2} $$

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{4} \int_{0}^{y} \frac{2}{(x+1)(y+1)} d x d y $$

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=\frac{x}{x+y} $$

Sketch the \(y z\) -trace of the sphere. $$ x^{2}+y^{2}+z^{2}-6 x-10 y+6 z+30=0 $$

Identify the quadric surface. $$ z^{2}=2 x^{2}+2 y^{2} $$

In 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. $$ \int_{-1}^{1} \int_{-2}^{2} y d y d x=\int_{-1}^{1} \int_{-2}^{2} y d x d y $$

Evaluate the second partial derivatives \(f_{x x}, f_{x y}, f_{y y},\) and \(f_{y x}\) at the point. $$ \text { Function } \quad \text { Point } $$ $$ f(x, y)=x^{4}-3 x^{2} y^{2}+y^{2} \quad(1,0) $$

In Exercises \(55-58,\) sketch the trace of the intersection of each plane with the given sphere. $$ \begin{array}{l}{x^{2}+y^{2}+z^{2}=25} \\ {\text { (a) } z=3 \quad \text { (b) } x=4}\end{array} $$

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 \(f_{x x}, f_{x y}, f_{y y},\) and \(f_{y x}\) at the point. $$ \text { Function } \quad \text { Point } $$ $$ f(x, y)=\sqrt{x^{2}+y^{2}} \quad(0,2) $$

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

Evaluate the second partial derivatives \(f_{x x}, f_{x y}, f_{y y},\) and \(f_{y x}\) at the point. $$ \text { Function } \quad \text { Point } $$ $$ f(x, y)=\ln (x-y) \quad(2,1) $$

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

Evaluate the second partial derivatives \(f_{x x}, f_{x y}, f_{y y},\) and \(f_{y x}\) at the point. $$ \text { Function } \quad \text { Point } $$ $$ f(x, y)=x^{2} e^{y} \quad(-1,0) $$

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

Marginal cost 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 \((\partial C / \partial x \text { and } \partial C / \partial y)\) 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?

Modeling Data Per capita consumptions (in gallons) of different types of plain milk in the United States from 1999 through 2004 are shown in the table. Consumption of reduced-fat ( 1 \(\%\) ) and skim milks, reduced-fat milk (2 \(\%\) ), and whole milk are represented by the variables \(x, y,\) and \(z,\) respectively. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline \text { Year } & {1999} & {2000} & {2001} & {2002} & {2003} & {2004} \\ \hline x & {6.2} & {6.1} & {5.9} & {5.8} & {5.6} & {5.5} \\ \hline y & {7.3} & {7.1} & {7.0} & {7.0} & {6.9} & {6.9} \\\ \hline z & {7.8} & {7.7} & {7.4} & {7.3} & {7.2} & {6.9} \\ \hline\end{array} $$ A model for the data in the table is given by \(-1.25 x+0.125 y+z=0.95 .\) (a) Complete a fourth row of the table using the model to approximate \(z\) for the given values of \(x\) and \(y .\) Compare the approximations with the actual values of \(z .\) (b) According to this model, increases in consumption of milk types \(y\) and \(z\) would correspond to what kind of change in consumption of milk type \(x ?\)

Marginal Revenue 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 \(2, \partial R / \partial x_{2}\)

Physical Science Because of the forces caused by its rotation, Earth is actually an oblate ellipsoid rather than a sphere. The equatorial radius is 3963 miles and the polar radius is 3950 miles. Find an equation of the ellipsoid. Assume that the center of Earth is at the origin and the \(x y\) -trace \((z=0)\) corresponds to the equator.

Marginal Productivity 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, \(\partial f / \partial y .\)

Architecture 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. $$ x_{1}=150-2 p_{1}+1.8 p_{2}, \quad x_{2}=350+\frac{3}{4} p_{1}-1.9 p_{2} $$

Milk Consumption 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, reduced-fat 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.

Shareholder's Equity 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.

Psychology Early in the twentieth century, an intelligence test called the Stanford-Binet Test (more commonly known as the IQ 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. (Source: Adapted from Bernstein/Clark-Stewart/Roy/Wickens, Psychology, Fourth Edition)

Investment The value of an investment of 1000 dollars 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.

Think About It 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?

Research Project 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.