In Exercises 13, (a) plot the points on a three-dimensional coordinate system, (b) find

In Exercises 13, (a) plot the points on a three-dimensional coordinate system, (b) find the distance between the points, and (c) find the coordinates of the midpoint of the line segment joining the points.

In Exercises 13, (a) plot the points on a three-dimensional coordinate system, (b) find the distance between the points, and (c) find the coordinates of the midpoint of the line segment joining the points.

In Exercises 13, (a) plot the points on a three-dimensional coordinate system, (b) find the distance between the points, and (c) find the coordinates of the midpoint of the line segment joining the points.

In Exercises 4 and 5, find the standard equation of the sphere. Center: radius: 4

In Exercises 4 and 5, find the standard equation of the sphere. Endpoints of a diameter:0, 3, 1, 2, 5, 5)

Find the center and radius of the sphere whose equation is x2  y2  z2  8x  2y  6z  23  0.

In Exercises 79, find the intercepts and sketch the graph of the plane.

In Exercises 79, find the intercepts and sketch the graph of the plane.

In Exercises 79, find the intercepts and sketch the graph of the plane.

In Exercises 1012, identify the quadric surface.

In Exercises 1012, identify the quadric surface.

In Exercises 1012, identify the quadric surface.

In Exercises 1315, find and

In Exercises 1315, find and

In Exercises 1315, find and

The contour map shows level curves of equal temperature (isotherms), measured in degrees Fahrenheit, across North America on a spring day. Use the map to find the approximate range of temperatures in (a) the Great Lakes region,(b) the United States, and (c) Mexico.

In Exercises 17 and 18, find and and evaluate each at the point

In Exercises 17 and 18, find and and evaluate each at the point

In Exercises 19 and 20, find any critical points, relative extrema, and saddle points of the function.

In Exercises 19 and 20, find any critical points, relative extrema, and saddle points of the function.

A company manufactures two types of wood burning stoves: a freestanding model and a fireplace-insert model. The total cost (in thousands of dollars) for producing freestanding stoves and fireplace-insert stoves can be modeled by Find the values of that minimize the total cost. What is this cost?

Physical Science Assume that Earth is a sphere with a radius of 3963 miles. If the center of Earth is placed at the origin of a three-dimensional coordinate system, what is the equation of the sphere? Lines of longitude that run north-south could be represented by what trace(s)? What shape would each of these traces form? Why? Lines of latitude that run east-west could be represented by what trace(s)? Why? What shape would each of these traces form? Why?

In Exercises 1926, identify the surface. z  x2 9  y2

In Exercises 1926, identify the surface. 4x2  y2  z2  4

In Exercises 1926, identify the surface. z  x2  y2

In Exercises 1926, identify the surface. z  9x  3y  5

In Exercises 27 and 28, find the function values.

In Exercises 27 and 28, find the function values.

n Exercises 29 and 30, describe the region in the -plane that corresponds to the domain of the function. Then find the range of the function.

n Exercises 29 and 30, describe the region in the -plane that corresponds to the domain of the function. Then find the range of the function.

In Exercises 3134, describe the level curves of the function. Sketch the level curves for the given -values.

In Exercises 3134, describe the level curves of the function. Sketch the level curves for the given -values.

In Exercises 3134, describe the level curves of the function. Sketch the level curves for the given -values.

In Exercises 3134, describe the level curves of the function. Sketch the level curves for the given -values.

Meteorology The contour map shown below represents the average yearly precipitation for Iowa. (Source: U.S. National Oceanic and Atmospheric Administration) (a) Discuss the use of color to represent the level curves. (b) Which part of Iowa receives the most precipitation? (c) Which part of Iowa receives the least precipitation?

Population Density The contour map below represents the population density of New York. (Source: U.S. Bureau of Census) (a) Discuss the use of color to represent the level curves. (b) Do the level curves correspond to equally spaced population densities? (c) Describe how to obtain a more detailed contour map.

Chemistry The acidity of rainwater is measured in units called pH, and smaller pH values are increasingly acidic. The map shows the curves of equal pH and gives evidence that downwind of heavily industrialized areas, the acidity has been increasing. Using the level curves on the map, determine the direction of the prevailing winds in the northeastern United States.

Sales The table gives the sales (in billions of dollars), the shareholders equity (in billions of dollars), and the earnings per share (in dollars) for Johnson & Johnson for the years 2000 through 2005. (Source: Johnson & Johnson)A model for these data is z  fx, y  0.078x  0.008y  0.767. (a) Use a graphing utility and the model to approximate z for the given values of x and y. (b) Which of the two variables in this model has the greater influence on shareholders equity? (c) Simplify the expression for and interpret its meaning in the context of the problem.

Equation of Exchange Economists use an equation of exchange to express the relation among money, prices, and business transactions. This equation can be written as P  MV T where is the money supply, is the velocity of circulation, is the total number of transactions, and is the price level. Find when and

Biomechanics The Froude number defined as F  v2 gl where represents velocity, represents gravitational acceleration,and represents stride length,is an example of a similarity criterion. Find the Froude number of a rabbit for which velocity is 2 meters per second, gravitational acceleration is 3 meters per second squared, and stride length is 0.75 meter.

In Exercises 4150, find all first partial derivatives. fx, y  x2y  3xy  2x  5y

In Exercises 4150, find all first partial derivatives. fx, y  4xy  xy2  3x2y

In Exercises 4150, find all first partial derivatives. z  x2 y2

In Exercises 4150, find all first partial derivatives. z  xy  2x  4y2

In Exercises 4150, find all first partial derivatives. fx, y  ln2x  3y

In Exercises 4150, find all first partial derivatives. fx, y  ln 2x  3y

In Exercises 4150, find all first partial derivatives. fx, y  xey  yex

In Exercises 4150, find all first partial derivatives. fx, y  x2e2y

In Exercises 4150, find all first partial derivatives. w  xyz2

In Exercises 4150, find all first partial derivatives. w  3xy  5xz  2yz

In Exercises 5154, find the slope of the surface at the indicated point in (a) the -direction and (b) the -direction. z  3x  4y  9,3, 2, 10

In Exercises 5154, find the slope of the surface at the indicated point in (a) the -direction and (b) the -direction. z  4x2  y22, 4, 0)

In Exercises 5154, find the slope of the surface at the indicated point in (a) the -direction and (b) the -direction. z  8  x2  y21, 2, 3

In Exercises 5154, find the slope of the surface at the indicated point in (a) the -direction and (b) the -direction. z  x2  y2, 5, 4, 9

In Exercises 5558, find all second partial derivatives.

In Exercises 5558, find all second partial derivatives.

In Exercises 5558, find all second partial derivatives.

In Exercises 5558, find all second partial derivatives.

Marginal Cost A company manufactures two models of skis:cross-country skis and downhill skis. The cost function for producing pairs of cross-country skis and pairs of downhill skis is given by C  15xy13  99x  139y  2293. Find the marginal costs when and

Marginal Revenue At a baseball stadium, souvenir caps are sold at two locations. If and are the numbers of baseball caps sold at location 1 and location 2, respectively, then the total revenue for the caps is modeled by R  15x1  16x2  1 10 x12  1 10 x22  1 100 x1x2. Given that and find the marginal revenues at location 1 and at location 2.

Medical Science The surface area of an average human body in square centimeters can be approximated by the model where is the weight in pounds and is the height in inches. (a) Determine the partial derivatives of with respect to and with respect to (b) Evaluate at Explain your result.

Medicine In order to treat a certain bacterial infection, a combination of two drugs is being tested. Studies have shown that the duration (in hours) of the infection in laboratory tests can be modeled by Dx, y  x2  2y2  18x  24y  2xy  120 where is the dosage in hundreds of milligrams of the first drug and is the dosage in hundreds of milligrams of the second drug. Evaluate and and interpret your results.

In Exercises 6370, find any critical points and relative extrema of the function. fx, y  x2  2y2

In Exercises 6370, find any critical points and relative extrema of the function. fx, y  x3  3xy  y2

In Exercises 6370, find any critical points and relative extrema of the function. fx, y  1  x  22  y  32

In Exercises 6370, find any critical points and relative extrema of the function. fx, y  ex  x  y2

In Exercises 6370, find any critical points and relative extrema of the function. fx, y  x3  y2  xy

In Exercises 6370, find any critical points and relative extrema of the function. fx, y  y2  xy  3y  2x  5

In Exercises 6370, find any critical points and relative extrema of the function. fx, y  x3  y3  3x  3y  2

In Exercises 6370, find any critical points and relative extrema of the function. fx, y  x2  y2

Revenue A company manufactures and sells two products. The demand functions for the products are given by p 1  100  x1 and p2  200  0.5x2 (a) Find the total revenue function for and (b) Find and such that the revenue is maximized. (c) What is the maximum revenue?

Profit A company manufactures a product at two different locations. The costs of manufacturing units at plant 1 and units at plant 2 are modeled by and respectively. If the product sells for $10 per unit, find and such that the profit, is maximized.

In Exercises 7378, locate any extrema of the function by using Lagrange multipliers.

In Exercises 7378, locate any extrema of the function by using Lagrange multipliers.

In Exercises 7378, locate any extrema of the function by using Lagrange multipliers.

In Exercises 7378, locate any extrema of the function by using Lagrange multipliers.

In Exercises 7378, locate any extrema of the function by using Lagrange multipliers.

In Exercises 7378, locate any extrema of the function by using Lagrange multipliers.

In Exercises 79 and 80, use a spreadsheet to find the indicated extremum. In each case, assume that and are nonnegative.

In Exercises 79 and 80, use a spreadsheet to find the indicated extremum. In each case, assume that and are nonnegative.

Maximum Production Level The production function for a manufacturer is given by Assume that the total amount available for labor and capital is $2000 and that units of labor and capital cost $20 and $4, respectively. Find the maximum production level for this manufacturer.

Minimum Cost A manufacturer has an order for 1000 units of wooden benches that can be produced at two locations. Let and be the numbers of units produced at the two locations. Find the number that should be produced at each location to meet the order and minimize cost if the cost function is given by C  0.25x2 1  10x1  0.15x2 2  12x2.

In Exercises 83 and 84, (a) use the method of least squares to find the least squares regression line and (b) calculate the sum of the squared errors. 2, 3, 1, 1, 1, 2, 3, 2

In Exercises 83 and 84, (a) use the method of least squares to find the least squares regression line and (b) calculate the sum of the squared errors. 3, 1, 2, 1, 0, 0), 1, 1, 2, 1

Agriculture An agronomist used four test plots to determine the relationship between the wheat yield (in bushels per acre) and the amount of fertilizer (in hundreds of pounds per acre). The results are listed in the table. (a) Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data. (b) Estimate the yield for a fertilizer application of 20 pounds per acre.

Work Force The table gives the percents and numbers (in millions) of women in the work force for selected years. (Source: U.S. Bureau of Labor Statistics) (a) Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data. (b) According to this model, approximately how many women enter the labor force for each one-point increase in the percent of women in the labor force?

In Exercises 87 and 88, use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression quadratic for the given points. Plot the points and graph the least squares regression quadratic. 1, 9, 0, 7, 1, 5, 2, 6, 4, 23

In Exercises 87 and 88, use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression quadratic for the given points. Plot the points and graph the least squares regression quadratic. 0, 10, 2, 9, 3, 7, 4, 4, 5, 0

In Exercises 8992, evaluate the double integral.1 01x 0 4x  2y dy dx

In Exercises 8992, evaluate the double integral.3 34 0 x  y2 dx dy

In Exercises 8992, evaluate the double integral. 2 12y 1 x y2 dx dy

In Exercises 8992, evaluate the double integral.4 0 16x2 0 2x dy dx

In Exercises 9396, use a double integral to find the area of the region.

In Exercises 9396, use a double integral to find the area of the region.

In Exercises 9396, use a double integral to find the area of the region.

In Exercises 9396, use a double integral to find the area of the region.

Find the volume of the solid bounded by the graphs of and

Find the volume of the solid bounded by the graphs of and

Average Elevation In a triangular coastal area, the elevation in miles above sea level at the point is modeled by fx, y  0.25  0.025x  0.01y where x and y are measured in miles (see figure). Find the average elevation of the triangular area.

Real Estate The value of real estate (in dollars per square foot) for a rectangular section of a city is given by fx, y  0.003x23y34 where and are measured in feet (see figure). Find the average value of real estate for this section.