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All Textbook Solutions for Calculus and Its Applications (11th Edition)

22E 23. Body surface area. The Mosteller formula for approximating the surface area S, in square meters , of a human is given by , where h is the person’s height in centimeters and w is the person’s weight in kilograms. (Source: www.halls.md.)Use the Mosteller approximation to estimate the surface area of a person whose height is 165 cm and whose weight is 80 kg. Body surface area. The Haycock formula for approximating the surface area S, in square meters (m2), of a human is given by S(h,w)=0.024265h0.3964w0.5378, where h is the persons height in centimeters and w is the persons weight in kilograms. (Source: www.halls.md.) Use the Haycock approximation to estimate the surface area of a person whose height is 165 cm and whose weight is 80 kg.25. Goals against average. A hockey goalie’s goals against average A is a function of the number of goals allowed g and the number of minutes played m and is given by . a. Find the goals against average of a goalie who allows 35 goals while playing 820 min. Round A to the nearest hundredth. b. A goalie gave up 124 goals during the season and had a goals against average of 3.75. How many minutes did he play? (Round to the nearest integer.) c. State the domain for A. 26. Dewpoint. The dewpoint is the temperature at which moisture in the air condenses into liquid (dew). It is a function of air temperature t and relative humidity h. The table below shows the dewpoints for select values of t and h. Air Temperature, Relative Humidity (%) 20 40 60 80 100 70 29 44 55 63 70 80 35 53 65 73 80 90 43 62 74 83 90 100 52 71 84 93 100 a. What is the dewpoint when the air temperature is 80°F with a relative humidity of 60%? b . What is the dewpoint when the air temperature is 90°F with a relative humidity of 40%? The air feels humid when the dewpoint reaches about 60. If the air temperature is 100°F at what approximate relative humidity will the air feel humid? Explain why the dewpoint is equal to the air temperature when the relative humidity is 100%. For the tornado described in Exercise 22, if the wind speed measures 200 mph, how far from the center was the measurement taken?According to the Mosteller formula in Exercise 23, if a persons weight drops 19%, by what percentage does his or her surface area change?Explain the difference between a function of two variables and a function of one variable.30. Find some examples of function of several variables not considered in the text, even some that may not have formulas. Wind Chill Temperature. Because wind speed enhances the loss of heat from the skin, we feel colder when there is wind than when there is not. The wind chill temperature is what the temperature would have to be with no wind in order to give the same chilling effect. The wind chill temperature, W, is given by W(,T)=91.4(10.45+6.680.447)(4575T)110, where T is the temperature measured by a thermometer, in degrees Fahrenheit, and v is the speed of the wind, in miles per hour. Find the wind chill temperature in each case. Round to the nearest degree. T=30F,=25mph Wind Chill Temperature. Because wind speed enhances the loss of heat from the skin, we feel colder when there is wind than when there is not. The wind chill temperature is what the temperature would have to be with no wind in order to give the same chilling effect. The wind chill temperature, W, is given by , where T is the temperature measured by a thermometer, in degrees Fahrenheit, and v is the speed of the wind, in miles per hour. Find the wind chill temperature in each case. Round to the nearest degree. 32. 33E Wind Chill Temperature. Because wind speed enhances the loss of heat from the skin, we feel colder when there is wind than when there is not. The wind chill temperature is what the temperature would have to be with no wind in order to give the same chilling effect. The wind chill temperature, W, is given by , where T is the temperature measured by a thermometer, in degrees Fahrenheit, and v is the speed of the wind, in miles per hour. Find the wind chill temperature in each case. Round to the nearest degree. 34. Use a 3D graphics program to generate the graph of each function. 36. Use a 3D graphics program to generate the graph of each function. 37. Use a 3D graphics program to generate the graph of each function. f(x,y)=(x416x2)ey2 Use a 3D graphics program to generate the graph of each function. f(x,y)=4(x2+y2)(x2+y2)2 40E Use a 3D graphics program to generate the graph of each function. 41. Find zx,zy,zx|(2,3),andzy|(0,5) z=7x5y Find zx,zy,zx|(2,3),andzy|(0,5) z=2z3y 3E Find zx,zy,zx|(2,3),andzy|(0,5) z=2x3+3xyx . 5. . 6. Find. 7. Find fx,fy,fz(2,1),andfy(3,2). f(x,y)=x2y2 Find 9. 10E 11E 12E 13E 14E 15E 16E 17E Find fxandfy f(x,y)=xy+y5x 19E Find 20. Find fbandfm f(b,m)=m3+4m2bb2+(2m+b5)2+(3m+b6)2 Find fbandfm f(b,m)=5m2mb23b+(2m+b8)2+(3m+b9)2 Find fx,fy,andf (The symbol is the Greek letter lambda) f(x,y,)=5xy(2x+y8)Find (The symbol is the Greek letter lambda) 24. Find fx,fy,andf (The symbol is the Greek letter lambda) f(x,y,)=x2+y2(10x+2y4)Find fx,fy,andf (The symbol is the Greek letter lambda) f(x,y,)=x2y2(4x7y10)27E 28E Find the four second-order partial derivatives. f(x,y)=7xy2+5xy2y Find the four second-order partial derivatives. f(x,y)=3x2y2xy+4y 31E 32E Find fxy,fxy,fyx,andfyy. (Remember, fyx means to differentiate with respect to y and then with respect to x) f(x,y)=2x3y Find. (Remember, means to differentiate with respect to y and then with respect to x) 34. Find. (Remember, means to differentiate with respect to y and then with respect to x) 35. Find. (Remember, means to differentiate with respect to y and then with respect to x) 36. Find. (Remember, means to differentiate with respect to y and then with respect to x) 37. Find. (Remember, means to differentiate with respect to y and then with respect to x) 38. Find. (Remember, means to differentiate with respect to y and then with respect to x) 39. Find fxy,fxy,fyx,andfyy. (Remember, fyx means to differentiate with respect to y and then with respect to x) f(x,y)=xIny The Cobb-Douglas model. Riverside Appliances has the following production function for a certain product: P(x,y)=1800x0.621y0.379, Where p is the number of units produced with x units of labor and y units of capital. a. Find the number of units produced with 2500 units of labor and 1700 units of capital b. Find the marginal productivities c. Evaluate the marginal productivities at x=2500andy=1700 d. d) Interpret the meanings of the marginal productivities found in part (c)The Cobb-Douglas model. Lincolnville Sporting Goods has the following production function for a certain product: p(x,y)=2400x2/5y3/5, where p is the number of units produced with x units of labor and y units of capital a. Find the number of units produced with 32 units of labor and 1024 units of capital. b. Find the marginal productivities c. Evaluate the marginal productivities at x=32andy=1024 d. d) Interpret the meanings of the marginal productivities found in part (c).A study of Texas nursing homes found that the annual profit P (in dollars) of profit-seeking, independent nursing homes in urban locations is modeled by P(w,r,s,t)=0.007955w0.638r1.038s0.873t2.468, where w is the average hourly wage of nurses and aides (in dollars), r is the occupancy rate (as a percentage), s is the total square footage of the facility, and t is the Texas Index of Level of Effort (TILE), a number between 1 and 11 that measures state Medicaid reimbursement. (Source: K. J. Knox, E. C. Blankmeyer, and J. R. Stutzman, Relative Economic Efficiency in Texas Nursing Facilities, Journal of Economics and Finance, Vol. 23, 199213 (1999).) Use this information for Exercises 43 and 44. A profit-seeking, independent Taxas nursing home in an urban setting has nurses and aides with an average hourly wage of $20 an hour, a TILE of 8, an occupancy rate of 70%, and 400,000ft2 of space. a. Estimate the nursing homes annual profit b. Find the four partial derivatives of P c. c) Interpret the meaning of the partial derivatives found in part (b).A study of Texas nursing homes found that the annual profit P (in dollars) of profit-seeking, independent nursing homes in urban locations is modeled by P(w,r,s,t)=0.007955w0.638r1.038s0.873t2.468, where w is the average hourly wage of nurses and aides (in dollars), r is the occupancy rate (as a percentage), s is the total square footage of the facility, and t is the Texas Index of Level of Effort (TILE), a number between 1 and 11 that measures state Medicaid reimbursement. (Source: K. J. Knox, E. C. Blankmeyer, and J. R. Stutzman, Relative Economic Efficiency in Texas Nursing Facilities, Journal of Economics and Finance, Vol. 23, 199213 (1999).) Use this information for Exercises 43 and 44. The change in P due to a change in w-when the other variables are held constant is approximately PPww. Use the values of w, r, s, and t in Exercise 43 and assume that the nursing home gives its nurses and aides a small raise so that the average hourly wage is now $20.25 an hour. By approximately how much does profit change?45E Temperaturehumidity Heat Index. In summer, higher humidity interacts with the outdoor temperature, making a person feel hotter because of reduced heat loss from the skin. The temperature-humidity index, Th, is what the temperature would have to be with no humidity in order to give the same heat effect. One index often used is given by Th=1.98T1.09(1H)(T58)56.9, where T is the air temperature, in degrees Fahrenheit, and H is the relative humidity, expressed as a decimal. Find the temperature-humidity index in each case. Round to the nearest tenth of a degree. T=90fandH=90 Temperaturehumidity Heat Index. In summer, higher humidity interacts with the outdoor temperature, making a person feel hotter because of reduced heat loss from the skin. The temperature-humidity index, Th, is what the temperature would have to be with no humidity in order to give the same heat effect. One index often used is given by Th=1.98T1.09(1H)(T58)56.9, where T is the air temperature, in degrees Fahrenheit, and H is the relative humidity, expressed as a decimal. Find the temperature-humidity index in each case. Round to the nearest tenth of a degree. T=90fandH=100 48E Use the equation for Th given above for Exercises 49 and 50. Find ThH, and interpret its meaning.Use the equation for Th given above for Exercises 49 and 50. Find ThT, and interpret its meaning 51E 52E Reading Ease The following formula is used by psychologists and educators to predict the reading ease, E, of a passage of words. Where w is the number of syllables in a 100-word section and s is the average number of words per sentence Use this information for Exercises 53-56. 53. Find E when Reading Ease The following formula is used by psychologists and educators to predict the reading ease, E, of a passage of words. Where w is the number of syllables in a 100-word section and s is the average number of words per sentence Use this information for Exercises 53-56. 54. Find E when 55E Reading Ease The following formula is used by psychologists and educators to predict the reading ease, E, of a passage of words. E=206.8350.846w1.015s, Where w is the number of syllables in a 100-word section and s is the average number of words per sentence Use this information for Exercises 53-56. FindEs 57E 58E 59E 60E 61E Find fxandft. f(x,t)=(x2+t2x2t2)5 In Exercises 63 and 64, find fxx,fxy,fyx,andfyy f(x,y)=xy2yx2 In Exercises 63 and 64, find fxx,fxy,fyx,andfyy f(x,y)=xyxy Do some research on the Cobb-Douglas production function, and explain how it was developed.66E Considerf(x,y)=In(x2+y2). Show that f is a solution of the partial differential equation 2fx2+2fy2=0.Consider f(x,t)=x35xy2 Show that f is a solution of the partial differential equation xfxyfy=0.69E Find the relative maximum and minimum values. 1. Find the relative maximum and minimum values. f(x,y)=x2+xy+y25y Find the relative maximum and minimum values. f(x,y)=2xyx3y2 Find the relative maximum and minimum values. f(x,y)=4xyx32y2 Find the relative maximum and minimum values. f(x,y)=x3+y36xy Find the relative maximum and minimum values. f(x,y)=x3+y33xy Find the relative maximum and minimum values. f(x,y)=x2+y24x+2y5 Find the relative maximum and minimum values. f(x,y)=x2+2xy+2y26y+2 Find the relative maximum and minimum values. f(x,y)=x2+y2+8x10y Find the relative maximum and minimum values. f(x,y)=4y+6xx2y2 Find the relative maximum and minimum values. f(x,y)=4x2y2 Find the relative maximum and minimum values. 12. Find the relative maximum and minimum values. f(x,y)=ex2+y2+1 Find the relative maximum and minimum values. 14. In Exercises 15-22, assume that relative maximum and minimum values are absolute maximum and minimum values. Maximizing profit. Safe Shades produces two kinds of sunglasses; one kind sells for $17, and the other for $21.The total revenue in thousands of dollars from the sale of x thousand sunglasses at $17 each and y thousand at $21 each is given by R(x,y)=17x+21y. The company determines that the total cost, in thousands of dollars, of producing x thousand of the $17 sunglasses and y thousand of the $21 sunglasses is given by C(x,y)=4x24xy+2y211x+25y3 How many of each type of sunglasses must be produced and sold to maximize profit?In Exercises 15-22, assume that relative maximum and minimum values are absolute maximum and minimum values. Maximizing profit. A concert promoter produces two kinds of souvenir shirt. Total revenue from the sale of x thousand shirts at $18 each and y thousand at $25 each is given by R(x,y)=18x+25y. The company determines that the total cost, in thousands of dollars, of producing x thousand of the $18 shirt and y thousand of the $25 shirt is C(x,y)=4x26xy+3y2+20x+19y12. How many of each type of shirt must be produced and sold to maximize profit?In Exercises 15-22, assume that relative maximum and minimum values are absolute maximum and minimum values. 17. Maximizing profit. McLeod Corp. finds that its profit, P, in millions of dollars, is given by where a is the amount spent on advertising, in millions of dollars, and p is the price charged per unit, in dollars. Find the maximum value of P and the values of a and p at which it occurs. In Exercises 15-22, assume that relative maximum and minimum values are absolute maximum and minimum values. 18. Maximizing profit. Humphery’s Medical supply finds that its profit, P, in millions of dollars, is given by where a is the amount spent on advertising, in millions of dollars, and n is the number of items sold, in thousands. Find the maximum value of P and the values of a and n at which it occurs. In Exercises 15-22, assume that relative maximum and minimum values are absolute maximum and minimum values. Minimizing the cost of a container. ProHauling Services is designing an open-top, rectangular container that will have a volume of 320 ft3. The cost of making the bottom of the container is $5 per square foot, and the cost of the sides is $4 per square foot. Find the dimensions of the container that will minimize total cost. (Hint: Make a substitution using the formula for volume.)In Exercises 15-22, assume that relative maximum and minimum values are absolute maximum and minimum values. Two-variable revenue maximization. Rad Designs sells two kinds of sweatshirts that compete with one another. Their demand function are expressed by the following relationships: q1=786p13p2, (1) q2=663p16p2, (2) Where p1 and p2 are the prices of the sweatshirts, in multiple of $10, and q1 and q2 are the quantities of the sweatshirts demanded, in hundreds of units. a. Find a formula for the total-revenue function, R, in terms of the variables p1 and p2. b. what prices p1 and p2 should be charged for each product in order to maximize total revenue? c. How many units will be demanded? d. What is the maximum total revenue?In Exercises 15-22, assume that relative maximum and minimum values are absolute maximum and minimum values. Two-variable revenue maximization. Repeat Exercise 20, using q1=644p12p2 and q2=562p14p2. a. R=64p14p124p1p2+56p24p22; b. p1=6, or $60, p2=4, $40; c. q1=32, 3200 units, q2=28, or 2800 units; d. $304,000 In Exercises 15-22, assume that relative maximum and minimum values are absolute maximum and minimum values. 22. Temperature. A flat metal plate is mounted on a coordinate plane. The temperature of the plate, in degrees Fahrenheit, at point (x, y) is given by Find the minimum temperature and where it occurs. Is there a maximum temperature? In Exercises 23-26, find the relative maximum and minimum values as well as any saddle points. 23. In Exercises 23-26, find the relative maximum and minimum values as well as any saddle points. f(x,y)=xy+2x+4y In Exercises 23-26, find the relative maximum and minimum values as well as any saddle points. 25. In Exercises 23-26, find the relative maximum and minimum values as well as any saddle points. S(b,m)=(m+b72)2+(2m+b73)2+(3m+b75)2 Is a cross-section of an anticlastic curve always a parabola? Why or why not?Explain the difference between a relative minimum and an absolute minimum of a function of two variables.Use a 3D graphics program to graph each of the following functions. Then estimate any relative extrema. f(x,y)=5x2+2y2+1 Use a 3D graphics program to graph each of the following functions. Then estimate any relative extrema. 30. Use a 3D graphics program to graph each of the following functions. Then estimate any relative extrema. f(x,y)=3xy(x2y2)x2+y2 Use a 3D graphics program to graph each of the following functions. Then estimate any relative extrema. f(x,y)=y+x2y28xxy In Exercises 1 4, find the regression line for each data set. x 1 2 4 5 y 1 3 3 4 In Exercises 1 – 4, find the regression line for each data set. 2. x 1 3 5 y 2 4 7 In Exercises 1 4, find the regression line for each data set. x 1 2 3 5 y 0 1 3 4 In Exercises 1 – 4, find the regression line for each data set. 4. x 1 2 4 y 3 5 8 In Exercises 5-8, find an exponential regression curve for each data set. x 0 1 2 y 10 19 42 In Exercises 5-8, find an exponential regression curve for each data set. x 1 2 3 4 y 8 25 72 225 7E In Exercises 5-8, find an exponential regression curve for each data set. x 2 4 6 8 y 13 7 3.7 1 All of the following exercises can be done with a graphing calculator if your instructor so directs. The calculator can also be used to check your work. 9. Labor force. The minimum hourly wage in the United States has grown over the years, as shown in the table below. Number of years, x, since 1997 Minimum Hourly Wage (dollars) 0 $5.15 10 5.85 11 6.55 12 7.25 18 10.10 (source: U.S. Dept. of Labor.) a. For the data in the table, find the regression line, b. Use the regression line to predict the minimum hourly wage in 2020 and 2025. All of the following exercises can be done with a graphing calculator if your instructor so directs. The calculator can also be used to check your work. Football ticket prices. Ticket prices for NFL football game have experienced steady growth, as shown in the table below. Number of years, x, since 2009 Average Ticket Price (dollars) 0 $73.18 1 76.47 2 77.36 3 79.09 4 81.54 a. Find the regression line, y=mx+b. b. Use the regression line to predict the average ticket price for an NFL game in 2020 and 2025.11E 12E All of the following exercises can be done with a graphing calculator if your instructor so directs. The calculator can also be used to check your work. 13. Grade predictions. A professor wants to predict students’ final examination scores on the basis of their midterm test scores. An equation was determined on the basis of data consisting of the scores of three students who took the same course with the same instructor the previous semester (see the following table). Midterm Score, x Final Exam Score, y 70% 75% 60 62 85 89 a. Find the regression line,. (Hint: The y-deviation are and so on.) b. The midterm score of a student was 81%. Use the regression line to the student’s final exam score. All of the following exercises can be done with a graphing calculator if your instructor so directs. The calculator can also be used to check your work. 14. Predicting the world record in the high jump. It has been established that most world records in track and field can be modeled by a linear function. The table below shows world high-jump records for various years. Number of years, x, since 1912 World Record In High Jump, y, (in inches) 0 (George home) 78.0 44 (Charles Dumas) 84.5 61 (Dwight Stones) 90.5 77 (Javier Sotomayer) 96.0 81 (Javier Sotomayer) 96.5 a. Find the regression line, . b. Use the regression line to predict the world record in the high jump in 2020 and in 2050. c. Does your answer in part (b) for 2050 seem realistic? Explain why extrapolating so far into the future could be a problem. All of the following exercises can be done with a graphing calculator if your instructor so directs. The calculator can also be used to check your work. Population. The data in the following table give the population of Detroit since 1970 (see Exercise 18 section R.6) Number of years, x, since 1970 Population (in millions) 0 1.5 10 1.2 20 1 30 0.95 40 0.71 a. Find the exponential regression curve, y=aekx b. Use the regression curve to estimate the population of Detroit in 2020 and 2025.All of the following exercises can be done with a graphing calculator if your instructor so directs. The calculator can also be used to check your work. Stock prices. The data in the following table give the price of one share of Starbucks stock on January 1of various years (see Exercise 20 Section R6) Number of years, x, since 2010 Price of one share of Starbucks stock on January 1 0 $20.59 1 $30.21 2 $46.61 3 $55.39 4 $76.17 a. Find the exponential regression curve, y=aekx b. Use the regression curve to estimate the price of one share of Starbucks stock on January 1 in 2016 and 2020.17E 18E Find the extremum of f(x,y) subject to given constraint, and state whether it is a maximum or a minimum. f(x,y)=xy;3x+y=10 2E 3E 4E Find the extremum of f(x,y) subject to given constraint, and state whether it is a maximum or a minimum. f(x,y)=4x2y2;x+2y=10 Find the extremum of f(x,y) subject to given constraint, and state whether it is a maximum or a minimum. f(x,y)=3x2y2;x+6y=37 Find the extremum of f(x,y) subject to given constraint, and state whether it is a maximum or a minimum. f(x,y)=2y26x2;2x+y=4 Find the extremum of subject to given constraint, and state whether it is a maximum or a minimum. 8. Find the extremum of f(x,y) subject to given constraint, and state whether it is a maximum or a minimum. f(x,y,z)=x2+y2+z2;y+2xz=3 Find the extremum of subject to given constraint, and state whether it is a maximum or a minimum. 10. 11E 12E 13E 14E 15E 16E 17E 18E 19. Maximizing typing area. A standard piece of printer has a perimeter of 39 in. Find the dimensions of the paper that will give the most area. What is that area? Does standard in. paper have maximum area? 20. Maximizing room area. A carpenter is building a rectangular room with a fixed perimeter of 80 ft. What are the dimensions of the largest room that can be built? What is its area? 21. Minimizing surface area. An oil drum of standard size has a volume of 200 gal, or 27. What dimensions yield the minimum surface area? Find the minimum surface area Do these drums appear to be made in such a way as to minimize surface area? Juice-can problem. A large juice can has a volume of 99 in3. What dimensions yield the minimum surface area? Find the minimum surface area.Maximizing total sales. Total sales, S, of Cre-Tech are given by S(L,M)=MLL2, Where M is the cost of materials and L is the cost of labor. Find the maximum value of this function subject to the budget constraint M+L=90.Maximizing total sales. Total sales, S, of Sea Change, Inc, are given by S(L,M)=MLL2, Where M is the cost of materials and L is the cost of labor. Find the maximum value of this function subject to the budget constraint M+L=70.25. Minimizing construction costs. Denney Construction is planning to build a warehouse whose interior volume is to be 252, 000 . Construction costs per square foot are estimated as follows; Walls $3.00 Floor $4.00 Ceiling: $3.00 a. The total cost of the building is where x is the length, y is the width, and z is the height, all in feet. Find a formula for. b. What dimensions of the building will minimize the total cost? What is the minimum cost? Minimizing the costs of container construction. North-side Nursery is designing a 12-ft3 shipping crate with a square bottom and top. The cost of the top and the sides is $2 per square foot, and the cost of the bottom is $3 per square foot. What dimensions will minimize the cost of the crate?Minimizing total cost. Each unit of a product can be made on either machine A or machine B. The nature of the machine makes their cost functions differ: Machine A: C(x)=10+x26,, Machine B: C(y)=200+y39.28. Minimizing distance and cost. A highway passes by the small town of Las Cienegas. From Las Cienegas, the highway is 5 miles to the north and 3 miles to the east. Assume that the highway is straight as it passes through this region. The town wants to build an access road at a cost of $250,000 per mile to connect to the highway. What is the shortest possible distance (to three decimal places) from Las Cienegas to the highway, and what would be the minimum cost, to the nearest dollar, of constructing such a road? 29. Minimizing distance and cost. From the center of Bridgeton, a water main runs northwest at a angle relative to due north and due west A new house is being built at a point 2 miles west and 1 mile north of the center of town. The cost to connect the house to the water main is $8,000 per mile. What is the shortest possible distance (to three decimal places) from the house to the water main, and what would be the minimum cost, to the nearest dollar, of connecting the house to the water main? In Exercises 30-33, find the absolute maximum and minimum values of each function, subject to the given constraints. 30. In Exercises 30-33, find the absolute maximum and minimum values of each function, subject to the given constraints. g(x,y)=x2+2y2;1x1and1y2 In Exercises 30-33, find the absolute maximum and minimum values of each function, subject to the given constraints. h(x,y)=x2+y24x2y+1;x0,y0andx+2y5 In Exercises 30-33, find the absolute maximum and minimum values of each function, subject to the given constraints. k(x,y)=x2y2+4x4y;0x3,y0,andx+y6 Business: minimizing costs with constraints. Farmer Frank grows two crops: celery and lettuce. He has determined that the cost of planting these crops is modeled by C(x,y)=x2+3xy+3.5y2775x1600y+250,000, where x is the number of acres of celery and y is the number of acres of lettuce. Suppose Farmer Frank has 300 acres available for planting and must plant more acres of lettuce than of celery. Find the number of acres of celery and of lettuce he should plant to minimize the cost, and state the cost.Business: maximizing profits with constraints. A manufacturer of decorative end tables produces two models, basic and large. Its weekly profit function is modeled by P(x,y)=x22y2xy+140x+210y4300, where x is the number of basic models sold each week and y is the number of large models sold each week. The warehouse can hold at most 90 tables. Assume that x and y must be nonnegative. How many of each models should be produce to maximize weekly profit, and what will the maximum profit be?Find the indicated maximum or minimum value of (x,y) subject to the given constraint. Minimum: f(x,y)=xy;x2+y2=9 Find the indicated maximum or minimum value of (x,y) subject to the given constraint. Minimum: f(x,y)=2x2+y2+2xy+3x+2y;y2=x+1 Find the indicated maximum or minimum value of (x,y) subject to the given constraint. Maximum: f(x,y,z)=x+y+z,x2+y2+z2=1 Find the indicated maximum or minimum value of (x,y) subject to the given constraint. Maximum: f(x,y,z)=x2y2z2;x2+y2+z2=2 Find the indicated maximum or minimum value of (x,y) subject to the given constraint. Maximum: f(x,y,z)=x+2y2z;x2+y2+z2=4 Find the indicated maximum or minimum value of (x,y) subject to the given constraint. Maximum: f(x,y,z,t)=x+y+z+t;x2+y2+z2+t2=1 Find the indicated maximum or minimum value of (x,y) subject to the given constraint. Maximum: f(x,y,z)=x2+y2+z2;x2y+5z=1 Economics: the Law of Equimarginal Productivity. Suppose p(x,y) represents the production of a two-product firm. The company produces x units of the first product at a cost of c1 each and y unit of the second product at a cost of c2 each. The budget constraint. B, is B=c1x+c2y. Use the method of Lagrange multipliers to find the value of in terms of px,py,c1,andc2. The resulting equation holds for any production function p and is called the Law of Equimarginal Productivity. =pxc1=pyc2 44. Business: maximizing production. A computer company has the following Cobb-Douglas production function for a certain product: , Where x is labor and y is capital, both measured in dollars. Suppose the company can make a total investment in labor and capital of $1,000,000. How should it allocate the investment between labor and capital in order to maximize production? 45. Discuss the difference between solving maximum-minimum problem using the method of Lagrange multipliers and the method of Section 6.3 Write a brief report on the life and work of the mathematician Joseph Louis Lagrange (1736-1813).47E 1E 2E 3E 4E Evaluate. 4113(x+5y)dxdy 6E 7E Evaluate. 8. Evaluate. 01x2x(x+y)dydx 10E 11E Evaluate. 12. Evaluate. 020x(x+y2)dydx 14E Find the volume of the solid capped by the surface z=1yx2 over the region bounded on the xy-plane by y=1x2,y=0,x=0andx=1, by evaluating the integral 0101x2(1yx2)dydx.16. Find the volume of the solid capped by the surface z = x + y over the region bounded on the xy-plane by, by evaluating the integral . 17. Find the average value of. 18. Find the average value of. 19. Find the average value of, where the region of integration is a triangle with vertices 20E 21. Life sciences: population. The population density of fireflies in a field is given by are in yards and p is the number of fireflies per square yard a. Determine the total population of fireflies in this field. b. Determine the average number of fireflies per square yard of the field. 22. Life sciences: population. The population density of Gladstone City is given by , where x and y are in miles and p is the number of people per square mile in thousands The city limits are as shown in the graph below. a. Determine the city’s population b. Determine the average number of people per square mile of the city 23E 24E 25E Is evaluated in much the same way as a double iterated integral. We first evaluate the inside x-integral, treating y and z as constants Then we evaluate three middle y-integral, treating z as a constant. Finally we evaluate the outside z-integral Evaluate the following triple integrals. 022y62y04y2zdzdydx 27. Describe the geometric meaning of the double integral of a function of two variables. 28. Explain how Exercise 1 can be answered without finding any antiderivatives. 29E 30E Use a calculator that does multiple integration to confirm your answers to Exercises 1, 5, 9, and 13.Write an equation of the line with slope 4 and containing the point (7,1).2E For f(x)=x25, find f(x+h). x2+2xh+h25 4. a. Graph: b. Find. c. Find. d. Is f continuous at 2? 5E Find each limit, if it exists. If a limit does not exist, state that fact. limx1x3+8 Find each limit, if it exists. If a limit does not exist, state that fact. limx4x216x+4 Find each limit, if it exists. If a limit does not exist, state that fact. limx34x3 Find each limit, if it exists. If a limit does not exist, state that fact. limx12x73x+2 Find each limit, if it exists. If a limit does not exist, state that fact. 10. Find each limit, if it exists. If a limit does not exist, state that fact. If f(x)=x2+3, find f(x) by determining limh0f(x+h)f(x)h.For Exercises 12-14, refer to the following graph of y=h(x). Identify the input values for which h has no limit.For Exercises 12-14, refer to the following graph of y=h(x). Identify the input values for which h is discontinuous.For Exercises 12-14, refer to the following graph of y=h(x). Identify the input values for which the derivatives of h does not exist.Differentiate. y=9x+3 Differentiate. y=x27x+3 Differentiate. y=x1/4 Differentiate. f(x)=x6 19E Differentiate. 22. 21E Differentiate. y=elnx 23E Differentiate. 24. Differentiate. 25. 26. For find. Business: average cost. Doubletake Clothing finds that the cost, in dollars, of producing x pairs of jeans is given by C(x)=320+x9. Find the rate at which the average cost is changing when 100 pairs of jeans have been produced. 28. Differentiate implicitly to find if . Find an equation of the tangent line to the graph of y=exx23 at the point (0,2).30. Find the x-value(s) at which the tangent line to has a slope of –1. Sketch the graph of each function. List the label the coordinates of any extrema and points of inflection. State where the function is increasing or decreasing, where it is concave down, and where any asymptotes occur. f(x)=x33x+1 32E Sketch the graph of each function. List the label the coordinates of any extrema and points of inflection. State where the function is increasing or decreasing, where it is concave down, and where any asymptotes occur. 33. 34E Find the absolute maximum and minimum values, if they exist, over the indicated. If no interval is indicated, consider the entire real number line. 35. 36E 37E 38E 39. Business: minimizing inventory costs. An electronics store sells 450 sets of earbuds each year. It costs $4 to store a set for a year. When placing an order, there is a fixed cost of $1 plus $0.75 for each set. How many times per year should the store recorder earbuds, and in what lot size, in order to minimize inventory costs? 40E Business: exponential growth. Friedas Frozen Yogurt is experiencing growth of 10% per year in the number, N, of franchises it owns; that is, dNdt=0.1N, where N is the number of franchises and t is the time, in years, from 2011. a. Given that there were 8000 franchises in 2011, find the solution of the equation, assuming that N0=8000 and k=0.1. N(t)=8000e0.1t b. Predict the number of franchises in 2019. c. What is doubling time for the number of franchises?42E 43. Business: approximating cost average. A square plot of ground measures 75 ft by 75 ft, with a tolerance of in. Landscapers are going to cover the plot with grass sod. Each square of sod costs $8 and measures 3 ft by 3 ft. a. Use differentials to estimate the change in area when the measurement tolerance is taken into account. b. How many extra squares of sod should the landscapers bring to the job, and how much extra will this cost? 44E 46E 47E Evaluate. 48. (Use Table 1 on pp. 431-432) 49E Evaluate. (x+3)lnxdx 51E 52E 53. Find the area under the graph of over the interval. Business: present value. Find the present value of $250,000 due in 30 yr at 6%, compounded continuously.55E Evaluate. 56. Business: contract buyout. An athlete has an 8-yr contract that pays her $200,000 per year. She invests the money at an APR of 4.85%, compounded continuously. After 5 yr, the team offers her a buyout of the contract. What is the lowest amount she should accept? 57E 58. Economic: supply and demand. Demand and supply functions are given by and where p is the price per unit, in dollars, when x units are sold. Find the equilibrium point and the consumer’s surplus. 59. Find the volume of the solid of revolution generated by rotating the region under the graph of , from to around the x-axis. 60. Find the volume of the solid of revolution generated by rotating the region under the graph of from to around the y-axis. Consider the data in the following table. Age of Business (in years) 1 3 5 Profit (in tens of thousands of dollars) 4 7 12 a. Find the regression line, y=mx+b. b. Use the regression line to predict the profit when the business is 10 years old. c. Find the exponential regression curve, y=aebx. d. Use the exponential regression curve to predict the profit when the business is 10 years old.62E Given find each of the following. 63. 64E 65. Maximize subject to the constraint. 66. Evaluate . 67E Solve the differential equation dy/dx=xy.Solve the differential equation y+4xy=3x, where y=3 when x=1.70E 71E Business: distribution of weights. The weight, in pounds, of a box of a certain cereal is uniformly distributed between 2 lb and 2.25 lb. Find the probability that a randomly chosen box of this cereal weighs between 2.07 lb and 2.1 lb.Business: wait times. The wait time t in minutes, between customers at Teris Book Nook can be modeled by the probability density function f(t)=0.45e0.45t, for 0t. a. What is the probability that the wait time is at most 2 min? b. What is the probability that the wait time is greater than 5 min?74E 75. Business: distribution of salaries. The salaries paid by a large corporation are normally distributed with a mean and a standard deviation. a. Find the probability that a randomly chosen employee earns between $72,000 and $85,000 per year. b. An executive of the corporation earns $90,000 per year. In what percentile of the salaries does this salary place him? c. A new employee insists on a salary that is in the top 2% of salaries. What is the minimum salary that this employee would accept? Express as an equivalent expression without exponents. 1. Express as an equivalent expression without exponents. 2. Express as an equivalent expression without exponents. (7)2 Express as an equivalent expression without exponents. 4. 5E 6E 7E Express as an equivalent expression without exponents. (14)3 Express as an equivalent expression without exponents. (6x)0 Express as an equivalent expression without exponents. 10.

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