Chapter 13

Identify the quadric surface. $$ x^{2}-y+z^{2}=0 $$

Find the sphere's center and radius. $$ 2 x^{2}+2 y^{2}+2 z^{2}-4 x-12 y-8 z+3=0 $$

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ y=x^{2}+2 x+1, y=3(x+1) $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Data that are modeled by \(y=-0.238 x+25\) have a negative correlation.

The number of grams of your favorite ice cream can be modeled by $$ G(x, y, z)=0.05 x^{2}+0.16 x y+0.25 z^{2} $$ where \(x\) is the number of fat grams, \(y\) is the number of carbohydrate grams, and \(z\) is the number of protein grams. Use Lagrange multipliers to find the maximum number of grams of ice cream you can eat without consuming more than 400 calories. Assume that there are 9 calories per fat gram, 4 calories per carbohydrate gram, and 4 calories per protein gram.

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

Find the sphere's center and radius. $$ 4 x^{2}+4 y^{2}+4 z^{2}-8 x+16 y+11=0 $$

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

A rancher plans to use an existing stone wall and the side of a barn as a boundary for two adjacent rectangular corrals. Fencing for the perimeter costs \(\$ 10\) per foot. To separate the corrals, a fence that costs \(\$ 4\) per foot will divide the region. The total area of the two corrals is to be 6000 square feet. (a) Use Lagrange multipliers to find the dimensions that will minimize the cost of the fencing. (b) What is the minimum cost?

Common blood types are determined genetically by the three alleles \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{O}\). (An allele is any of a group of possible mutational forms of a gene.) A person whose blood type is \(\mathrm{AA}, \mathrm{BB}\), or \(\mathrm{OO}\) is homozygous. A person whose blood type is \(\mathrm{AB}, \mathrm{AO}\), or \(\mathrm{BO}\) is heterozygous. The Hardy-Weinberg Law states that the proportion \(P\) of heterozygous individuals in any given population is modeled by \(P(p, q, r)=2 p q+2 p r+2 q r\) where \(p\) represents the percent of allele \(\mathrm{A}\) in the population, \(q\) represents the percent of allele \(\mathrm{B}\) in the population, and \(r\) represents the percent of allele \(\mathrm{O}\) in the population. Use the fact that \(p+q+r=1\) (the sum of the three must equal \(100 \%\) ) to show that the maximum proportion of heterozygous individuals in any population is \(\frac{2}{3}\).

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=x^{2}-2 x y+3 y^{2} $$

Identify the quadric surface. $$ x^{2}-y^{2}+z=0 $$

Sketch the \(x y\) -trace of the sphere. $$ (x-1)^{2}+(y-3)^{2}+(z-2)^{2}=25 $$

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{2} \int_{x^{2}}^{2 x}\left(x^{3}+3 y^{2}\right) d y d x $$

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=y^{3}-4 x y^{2}-1 $$

Identify the quadric surface. $$ z^{2}-x^{2}-\frac{y^{2}}{4}=1 $$

Sketch the \(x y\) -trace of the sphere. $$ (x+1)^{2}+(y+2)^{2}+(z-2)^{2}=16 $$

Use a symbolic integration utility to evaluate the double integral. $$ \int_{1}^{2} \int_{0}^{x} e^{x y} d y d x $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. A linear regression model with a positive correlation will have a slope that is greater than 0 .

An automobile manufacturer has determined that its annual labor and equipment cost (in millions of dollars) can be modeled by \(C(x, y)=2 x^{2}+3 y^{2}-15 x-20 y+4 x y+39\) where \(x\) is the amount spent per year on labor and \(y\) is the amount spent per year on equipment (both in millions of dollars). Find the values of \(x\) and \(y\) that minimize the annual labor and equipment cost. What is this cost?

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=\frac{e^{2 x y}}{4 x} $$

The earnings per share \(z\) (in dollars) for Starbucks Corporation from 1998 through 2006 can be modeled by \(z=0.106 x-0.036 y-0.005\), where \(x\) is sales (in billions of dollars) and \(y\) is the shareholder's equity (in billions of dollars). (Source: Starbucks Corporation) (a) Find the earnings per share when \(x=8\) and \(y=5\). (b) Which of the two variables in this model has the greater influence on the earnings per share? Explain.

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

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

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

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the correlation coefficient for a linear regression model is close to \(-1\), the regression line cannot be used to describe the data. If the correlation coefficient for a linear regression model is close to \(-1\), the regression line cannot be used to describe the data.

In order to treat a certain bacterial infection, a combination of two drugs is being tested. Studies have shown that the duration of the infection in laboratory tests can be modeled by \(D(x, y)=x^{2}+2 y^{2}-18 x-24 y+2 x y+120\) where \(x\) is the dosage in hundreds of milligrams of the first drug and \(y\) is the dosage in hundreds of milligrams of the second drug. Determine the partial derivatives of \(D\) with respect to \(x\) and with respect to \(y\). Find the amount of each drug necessary to minimize the duration of the infection.

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

The shareholder's equity \(z\) (in billions of dollars) for Wal-Mart Corporation from 2000 to 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 the shareholder's equity when \(x=300\) and \(y=130\). (b) Which of the two variables in this model has the greater influence on shareholder's equity? Explain.

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

Sketch the \(x y\) -trace of the sphere. $$ x^{2}+y^{2}+z^{2}-4 y+2 z-60=0 $$

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

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. A saddle point always occurs at a critical point.

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=x^{3}-4 y^{2} $$

You are taking out a home mortgage for \(\$ 120,000\), and you are given the options below. Which option would you choose? Explain your reasoning. (a) A fixed annual rate of \(8 \%\), over a term of 20 years. (b) A fixed annual rate of \(7 \%\), over a term of 30 years. (c) An adjustable annual rate of \(7 \%\), over a term of 20 years. The annual rate can fluctuate - each year it is set at \(1 \%\) above the prime rate. (d) A fixed annual rate of \(7 \%\), over a term of 15 years.

Identify the quadric surface. $$ z^{2}=9 x^{2}+y^{2} $$

Sketch the \(y z\) -trace of the sphere. $$ x^{2}+(y+3)^{2}+z^{2}=25 $$

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{3} \int_{0}^{x^{2}} \sqrt{x} \sqrt{1+x} d y d x $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x, y)\) has a relative maximum at \(\left(x_{0}, y_{0}, z_{0}\right)\), then \(f_{x}\left(x_{0}, y_{0}\right)=f_{y}\left(x_{0}, y_{0}\right)=0\)

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

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

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

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

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

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

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

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

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 \(f_{x x^{\prime}} f_{x y^{\prime}} f_{y y^{\prime}}\) and \(f_{y x}\) at the point. $$ f(x, y)=x^{4}-3 x^{2} y^{2}+y^{2} $$

Sketch the trace of the intersection of each plane with the given sphere. \(x^{2}+y^{2}+z^{2}=25\) (a) \(z=3\) (b) \(x=4\)