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All Textbook Solutions for Calculus and Its Applications (11th Edition)
30E31E32E33EReliability of a Machine. The reliability of the machine (the probability that it will work) in Exercise 33, is defined as R(T)=1.0T0.01e0.01tdt, where R(T) is the reliability at time T. Write R(T) without using an integral.35. Wait time for 911 calls. The wait time before a 911 call is answered in the state of California has a probability density function for where t is in seconds. (Source: California Government Code.)
a. The state standard is that 90% of 911 calls are to be answered within 10 sec. Verify that this standard is met using the probability density function f.
b. What is the probability that a 911 call is answered within 15 to 25 sec after being made?
36E37E38EUse your answer to Exercise 37 to find the probability that a rat requires more than 150 sec to learn its way through the maze.40E41E42E43E44E45E46E47E48E49-60. Verify Property 2 of the definition of a probability density function for each of the functions in Exercises 1-12. Check using the answers to Exercises 1-12.
49.
49-60. Verify Property 2 of the definition of a probability density function for each of the functions in Exercises 1-12. Check using the answers to Exercises 1-12.
50.
Verify Property 2 of the definition of a probability density function for each of the functions in Exercises 1-12. Check using the answers to Exercises 1-12. f(x)=3,[0,13]49-60. Verify Property 2 of the definition of a probability density function for each of the functions in Exercises 1-12. Check using the answers to Exercises 1-12.
52.
49-60. Verify Property 2 of the definition of a probability density function for each of the functions in Exercises 1-12. Check using the answers to Exercises 1-12.
53.
Verify Property 2 of the definition of a probability density function for each of the functions in Exercises 1-12. Check using the answers to Exercises 1-12. f(x)=364x2,[0,4]Verify Property 2 of the definition of a probability density function for each of the functions in Exercises 1-12. Check using the answers to Exercises 1-12. f(x)=1x,[1,e]Verify Property 2 of the definition of a probability density function for each of the functions in Exercises 1-12. Check using the answers to Exercises 1-12. f(x)=1e1ex,[0,1]49-60. Verify Property 2 of the definition of a probability density function for each of the
57.
49-60. Verify Property 2 of the definition of a probability density function for each of the
58.
59E60EFor each probability density function, over the given interval, find E(x),E(x2), the mean, the variance, and the standard deviation. f(x)=15,[3,8]For each probability density function, over the given interval, find the mean, the variance, and the standard deviation.
2.
For each probability density function, over the given interval, find the mean, the variance, and the standard deviation.
3.
For each probability density function, over the given interval, find E(x),E(x2), the mean, the variance, and the standard deviation. f(x)=29x,[0,3]5EFor each probability density function, over the given interval, find E(x),E(x2), the mean, the variance, and the standard deviation. f(x)=14x,[1,3]7E8EFor each probability density function, over the given interval, find the mean, the variance, and the standard deviation.
9.
For each probability density function, over the given interval, find E(x),E(x2), the mean, the variance, and the standard deviation. f(x)=1ln51x,[1.5,7.5]Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(0x0.36)Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(0x2.13)Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(1.37x0)Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(2.01x0)Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(2.94x2.00)Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(1.89x0.45)Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following.
17.
18ELet x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(2.45x1.24)20ELet x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(x3.01)22E23E24ELet x be a continuous random variable that is distributed with mean and standard deviation. Using Table A, find each of the following.
25.
26E27E28E29E30E31E32EUse a graphing calculator to verify the solutions to Exercises 11-32. Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(0x0.36)33-54. Use a graphing calculator to verify the solutions to Exercises 11-32.
Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following.
34.
33-54. Use a graphing calculator to verify the solutions to Exercises 11-32.
Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following.
35.
Use a graphing calculator to verify the solutions to Exercises 11-32. Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(2.01x0)Use a graphing calculator to verify the solutions to Exercises 11-32. Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(2.94x2.00)33-54. Use a graphing calculator to verify the solutions to Exercises 11-32.
Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following.
38.
33-54. Use a graphing calculator to verify the solutions to Exercises 11-32.
Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following.
39.
Use a graphing calculator to verify the solutions to Exercises 11-32. Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(0.76x1.45)Use a graphing calculator to verify the solutions to Exercises 11-32. Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(2.45x1.24)Use a graphing calculator to verify the solutions to Exercises 11-32. Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(1.27x0.58)Use a graphing calculator to verify the solutions to Exercises 11-32. Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. P(x3.01)33-54. Use a graphing calculator to verify the solutions to Exercises 11-32.
Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following.
44.
33-54. Use a graphing calculator to verify the solutions to Exercises 11-32.
Let x be a continuous random variable with a standard normal distribution. Using Table A, find each of the following.
45.
a.
b. What percentage of the area is from –2 to 2?
46EUse a graphing calculator to verify the solutions to Exercises 11-32. Let x be a continuous random variable that is distributed with mean =22 and standard deviation =5. Using Table A, find each of the following. P(22x27)33-54. Use a graphing calculator to verify the solutions to Exercises 11-32.
Let x be a continuous random variable that is distributed with mean and standard deviation. Using Table A, find each of the following.
48.
Use a graphing calculator to verify the solutions to Exercises 11-32. Let x be a continuous random variable that is distributed with mean =22 and standard deviation =5. Using Table A, find each of the following. P(18x26)33-54. Use a graphing calculator to verify the solutions to Exercises 11-32.
Let x be a continuous random variable that is distributed with mean and standard deviation. Using Table A, find each of the following.
50.
Use a graphing calculator to verify the solutions to Exercises 11-32. Let x be a continuous random variable that is distributed with mean =22 and standard deviation =5. Using Table A, find each of the following. P(20.3x27.5)33-54. Use a graphing calculator to verify the solutions to Exercises 11-32.
Let x be a continuous random variable that is distributed with mean and standard deviation. Using Table A, find each of the following.
52.
33-54. Use a graphing calculator to verify the solutions to Exercises 11-32.
Let x be a continuous random variable that is distributed with mean and standard deviation. Using Table A, find each of the following.
53.
Use a graphing calculator to verify the solutions to Exercises 11-32. Let x be a continuous random variable that is distributed with mean =22 and standard deviation =5. Using Table A, find each of the following. P(x22.5)55. Find the z-value that corresponds to each percentile for a standard normal distribution.
a. 30th percentile
b. 50th percentile
c. 95th percentile
56. In a normal distribution with and, find the x-value that corresponds to the
a. 35th percentile
b. 75th percentile
57. In a normal distribution with and, find the x-value that corresponds to the
a. 46th percentile
b. 92nd percentile
58. In a normal distribution with and, find the x-value that corresponds to the
a. 50th percentile
b. 84th percentile
59EBread Baking. The number of loaves of bread, N baked each day by Fireside Bakers is normally distributed with mean 1000 and standard deviation 50. The bakery pays bonus to its employees on days when at least 1100 loaves are baked. On what percentage of days will the bakery have to pay a bonus?61EIn an automotive body-welding line, delays encountered during the process can be modeled by various probability distributions. (Source: R.R. Inman, Empirical Evaluation of Exponential and Independence Assumptions in Queueing Models of Manufacturing Systems, Production and Operations Management, Vol. 8, 409-432 (1999). The processing time for the automatic piercing station has a normal distribution with mean 36.2 sec and standard deviation 2.108 sec. Find the probability that the next operation of the piercing station will take between 35 and 40 sec.In an automotive body-welding line, delays encountered during the process can be modeled by various probability distributions. (Source: R.R. Inman, “Empirical Evaluation of Exponential and Independence Assumptions in Queueing Models of Manufacturing Systems, “Production and Operations Management, Vol. 8, 409-432 (1999).
63. Test Score Distribution. In 2013, combined SAT scores were normally distributed with mean 1498 and standard deviation 348. Find the combined SAT scores that correspond to these percentiles. (Source: www.collegeboard.com.)
a. 35th percentile
b. 60th percentile
c. 92nd percentile
64. Test Score Distribution. The scores on a biology test are normally distributed with mean 65 and standard deviation 20. A score from 80 to 89 is a B. What is the probability of getting a B?
Test Score Distribution. In a large class, student test scores had a mean of =76 and a standard deviation =7. a. The top 12% of students got an A. Find the minimum score needed to get an A (round to the appropriate integer). b. The top 75% of students passed. Find the minimum score needed to pass (round to the appropriate integer).66. Average Temperature. Las Vegas, Nevada, has an average daily high temperature of 104 degrees in July, with a standard deviation of 4.5 degrees. (Source: www.weatherspark.com.)
a. In what percentile is a temperature of 112 degrees?
b. What temperature would be at the 67th percentile?
c. What temperature would be in the top 0.5% of all July temperature for this location?
67. Heights of Basketball Players. Players in the National Basketball Association have a mean height of 6ft 7 in (79 in.) (Source: www.weatherspark.com.). If a basketball player who is 7ft 2 in. (86 in.) tall is in the top 1% of players by height, in what percentile is a 6ft 11 in. (83 in.) player?
68. Bowling Scores. At the time this book was written, the bowling scores, S of author Marv Bittinger (shown below) were normally distributed with mean 201 and standard deviation 23.
a. Find the probability that one of Marv’s scores is from 185 to 215.
b. Find the probability that one of his scores is from 160 to 175.
c. Find the probability that one of his scores is greater than 200.
d. Marv’s best score is 299. Find the percentile that corresponds to this score, and explain what that number represents.
69EFor each probability density function, over the given interval, find E(x),E(x2), the mean, the variance, and the standard deviation. f(x)=3a3x4, over [a,)71E72E73E74. Business: Coffee Production. Suppose the amount of coffee beans loaded into a vaccum-packed bag has a mean weight of ounces, which can be adjusted on the filling machine. Also, the amount dispensed is normally distributed with. What should be set at to ensure that only 1 bag in 50 will have less than 16 oz?
75. Business: Does thy cup overflow? Suppose the mean amount of cappuccino, , dispensed by a vending machine can be set. If a cup holds 8.5 oz and the amount dispensed is normally distributed with , what should be set at to ensure that only 1 bag in 50 will have less than 16 oz?
76. Explain why a normal distribution may not apply if you are analyzing the distribution of weights of students in a classroom.
A professor gives an easy test worth 100 points. The mean is 94, and the standard deviation is 5. Is it possible to apply a normal distribution to this situation? Why or why not?78. Approximate the integral
.
Find the volume generated by rotating the area bounded by the graphs of each set of equations
around the x-axis.
1.
2EFind the volume generated by rotating the area bounded by the graphs of each set of equations
around the x-axis.
3.
Find the volume generated by rotating the area bounded by the graphs of each set of equations
around the x-axis.
4.
Find the volume generated by rotating the area bounded by the graphs of each set of equations around the x-axis. y=ex,x=2,x=5Find the volume generated by rotating the area bounded by the graphs of each set of equations
around the x-axis.
6.
Find the volume generated by rotating the area bounded by the graphs of each set of equations
around the x-axis.
7.
Find the volume generated by rotating the area bounded by the graphs of each set of equations around the x-axis. y=1x,x=1,x=3Find the volume generated by rotating the area bounded by the graphs of each set of equations around the x-axis. y=2x,x=4,x=910EFind the volume generated by rotating the area bounded by the graphs of each set of equations
around the x-axis.
11.
12E13EFind the volume generated by rotating the area bounded by the graphs of each set of equations around the x-axis. y=x+1,x=1,x=2Find the volume generated by rotating the area bounded by the graphs of each set of equations around the x-axis. y=2x,x=1,x=2Find the volume generated by rotating the area bounded by the graphs of each set of equations
around the x-axis.
16.
Find the volume generated by rotating the area bounded by the graphs of each set of equations around the x-axis. y=4x2,x=2,x=2Find the volume generated by rotating the area bounded by the graphs of each set of equations
around the x-axis.
18.
Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis.
19.
Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis.
20.
Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis. y=x2,x=0,x=3Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis. y=x3,x=0,x=3Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis. y=1x,x=1,x=5Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis. y=1x2,x=2,x=4Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis. y=x2+3,x=1,x=2Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis.
26.
Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis. y=x,x=4,x=9Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis.
28.
Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis.
29.
Find the volume generated by rotating the area bounded by the graphs of each set of equations around the y-axis. y=1x2+1,x=1,x=7
31. Let R be the area bounded by the graph of and the x-axis over .
a. Find the volume of the solid of revolution generated by rotating R around the x-axis.
b. Find the volume of the solid of revolution generated by rotating R around they y-axis.
c. Explain why the solids in parts (a) and (b) do not have the same volume.
Let R be the area bounded by the graph of y=9x and the x-axis over [0,9]. a. Find the volume of the solid of revolution generated by rotating R around the x-axis. b. Find the volume of the solid of revolution generated by rotating R around the y-axis. c. Explain why the solids in parts (a) and (b) have the same volume.33. Cooling Tower Volume. Cooling towers at nuclear power plants have a “pinched” chimney shape (which promotes cooling within the tower) formed by rotating a hyperbola around an axis. The function
for
where x and y are in feet, describes the shape of such a tower (laying on its side). Determine the volume of the tower by rotating the area bounded by the graph of y around x-axis.
34. Volume of a football. A regulation football used in the National Football League (NFL) is 11 in. from tip to tip and 7 in. in diameter at its thickest (the regulations allow for slight variation in these dimensions). (Source: NFL.) The shape of a football can be modeled by the function
for
where x is in inches. Find the volume of an NFL football by rotating the area bounded by the graph of around the x-axis.
Volume of a Hogan. A Hogan is a circular shelter used by Native Americans in the Four Corners regions of the southwestern United States. The volume of Hogan can be approximated if the graph of 0.02x2+12, for 0x15, where x and y are in feet and the x-axis represents group level, is rotated around the y-axis. Find the volume.Volume of a domed stadium. The volume of a stadium with a domed roof can be approximated if the graph of y=0.00025x2+130, for 0x400, where x and y are in feet and the x-axis represents grouped level, is rotated around the y-axis. Find the volume.37. Using volume by disks, prove that volume of a sphere of radius r is .
38EFind the volume generated by rotating about the x-axis the area bounded by the groups of each set of equations and the x-axis.
39.
Find the volume generated by rotating about the x-axis the area bounded by the groups of each set of equations and the x-axis. y=xex,x=1,x=2In Exercises 41 and 42, the first quadrant is the region of the xy-plane in which and
41. Find the volume of the solid of revolution generated by rotating the area in the first quadrant between and the x-axis around the y-axis.
In Exercises 41 and 42, the first quadrant is the region of the xy-plane in which x0 and y0 Find the volume of the solid of revaluation generated by rotating the area in the first quadrant between y=xx3 and the x-axis around the y-axis.Let R be the area between y=x+1 and the x-axis over [0,a], where a0. 73,53; the volume around the x-axis is larger. a. Find the volumes of the solids generated by rotating R, with a=1, around the x-axis and around the y-axis. Which volume is larger? b. Find the volumes of the solids generated by rotating R, with a=2, around the x-axis and the y-axis. Which volume is larger? c. Find the value of a for which the two solids of revolution have the same volume.44. Let R be the area between the x-axis, and the line . Assume .
a. Find the value of k for which the volumes of the solids formed by rotating R around the x-axis and around the y-axis are the same.
b. Show that the answer to part (a) is true if the limits of integration are where .
45EParadox of Gabriels horn or the infinite paint can. Through we cannot prove it here, the surface area of Gabriels horn (see Exercise 45) is given by S=12x1+1x4dx. Show that the surface area of Gabriels horn is infinite. The paradox is that the volume of the horn is finite, but the surface area is infinite. This is like a can of paint that has a finite volume but, when full, does not hold enough paint to paint the outside of the can. Using the fnlnt feature on a graphing calculator, with successively larger values for the upper limit, you will find that the integral diverges.In Exercise 1-6, find the general solution and three particular solutions.
1.
In Exercise 1-6, find the general solution and three particular solutions. y=5x6In Exercise 1-6, find the general solution and three particular solutions. y=2ex+xIn Exercise 1-6, find the general solution and three particular solutions.
4.
5EIn Exercise 1-6, find the general solution and three particular solutions. y=3x+x2x47EShow that y=xlnx5x+7 is a solution of y1x=0.9E10E11E12E13E14. Let .
a. Show that is a solution of this differential equation.
b. Show that is a solution.
c. Show that is a solution, where and are constants.
15EIn Exercises 15-22, (a) find the general solution of each differential equation, and (b) check the solution by substituting into the differential equation.
16.
17EIn Exercises 15-22, (a) find the general solution of each differential equation, and (b) check the solution by substituting into the differential equation. dVdt=1.33V19EIn Exercises 15-22, (a) find the general solution of each differential equation, and (b) check the solution by substituting into the differential equation. dhdt=0.023h21EIn Exercises 15-22, (a) find the general solution of each differential equation, and (b) check the solution by substituting into the differential equation.
22.
23EIn Exercises 23-34, (a) find the particular solution of each differential equation as determined by the initial condition, and (b) check the solution by substituting into the differential equation. y=3x2x+5; y=6 when x=0In Exercises 23-34, (a) find the particular solution of each differential equation as determined by the initial condition, and (b) check the solution by substituting into the differential equation. f(x)=x2/3x;f(1)=626EIn Exercises 23-34, (a) find the particular solution of each differential equation as determined by the initial condition, and (b) check the solution by substituting into the differential equation.
27. where
28E29E30EIn Exercises 23-34, (a) find the particular solution of each differential equation as determined by the initial condition, and (b) check the solution by substituting into the differential equation. dTdt=0.015T, where T=50 when t=0In Exercises 23-34, (a) find the particular solution of each differential equation as determined by the initial condition, and (b) check the solution by substituting into the differential equation.
32. where when
33EIn Exercises 23-34, (a) find the particular solution of each differential equation as determined by the initial condition, and (b) check the solution by substituting into the differential equation. dNdt=3N, where N=3.5 when t=0Solve by separating variables.
35.
Solve by separating variables.
36.
Solve by separating variables.
37.
Solve by separating variables.
38.
39E40ESolve by separating variables. dydx=6y42E43E44E45E46E47E48EIn Exercises 47-52, (a) write a differential equation that models the situation, and (b) find the general solution. If an initial condition is given, find the particular solution. Recall that when y is directly proportional to x, we have y=kx, and when y is inversely proportional to x we have y=k/x, where k is constant of proportionally. In these exercises, let k=1. The rate of change of y with respect to x is inversely proportional to the cube of y.50E51E52E53. Growth of an Account. Debra deposits into an account that earns interest at a rate of 3.75%, compounded continuously.
a. Write the differential equation that represents the value of Debra’s account after t years.
b. Find the particular solution of the differential equation from part (a).
c. Find and .
d. Find , and explain what this number represents.
Growth of an Account. Jennifer deposits A0=1200 into an account that earns 4.2% compounded continuously. a. Write the differential equation that represents A(t), the value of Jennifers account after t years. b. Find the particular solution of the differential equation from part (a). c. Find A(7) and A(7). d. Find A(7)/A(7) and explain what this number represents.Capital Expansion. Domars capital expansion model is dIdt=hkI, where I is the investment, h is the investment productivity (constant), k is the marginal productivity to the consumer (constant), and t is the time. a. Use separation of variables to solve the differential equation. b. Rewrite the solution in terms of the condition I0=I(0).56E57E58. Utility. The reaction R in pleasure units by a consumer receiving S units of a product can be modeled by the differential equation
a. Use separation of a variables to solve the differential equation.
b. Rewrite the solution in terms of the initial condition .
c. Explain why the condition is a reasonable.
Find the demand function given each set of elasticity conditions.
59. when
60E61E62E63. Population Growth. The City of New River had a population of 17,000 in 2002 with a continuous growth rate of 1.75% per year.
a. Write the differential equation that represents the population of New River after t years.
b. Find the particular solution of the differential equation from part (a).
c. Find and .
d. Find and explain what this number represents.
Population Growth. An initial population of 70 bacteria is growing continuously at a rate of 2.5% per hour. a. Write the differential equation that represents P(t), the population of bacteria after t hours. b. Find the particular solution of the differential equation from part (a). c. Find P(24) and P(24). d. Find P(24)/P(24), and explain what this number represents.Population Growth. Before 1859, rabbits did not exist in Australia. That year, a settler released 24 rabbits into the wild. Without natural predators, the growth of the Australian rabbit population can be modeled by the uninhibited growth model dP/dt=kP, where P(t) is the population of rabbits t years after 1859. (Source: www.dpi.vic.gov.au/agriculture.) a. When the rabbit population was established to be 8900, its rate of growth was about 2630 rabbits per year. Use this information to find k, and then find the particular solution of the differential equation. b. Find the rabbit population in 1900 (t=41) and the rate at which it was increasing in that year. c. Without using a calculator, find P(41)/P(41).Population Growth. Suppose 30 sparrows are released into a region where they have no natural predators. The growth of the regions sparrow population can be modeled by the uninhibited growth model dP/dt=kP, where P(t) is the population of sparrows t years after their initial release. a. When the sparrow population is estimated at 12,500, its rate of growth is about 1325 sparrows per year. Use this information to find k, and then find the particular solution of the differential equation. b. Find the number of sparrows after 70 yr. c. Without using a calculator, find P(70)/P(70).Exponential Growth. a. Use separation of variables to solve the differential equation model of uninhibited growth, dPdt=kP. b. Rewrite the solution of part (a) in terms of the condition P0=P(0).The Brentano-Stevens Law. The validity of the Weber-Fechner Law has been the subject of great debate among psychologists. An alternative model, dRdS=kRS, where k is a positive constant, has been proposed. Find the general solution of this equation. (This model has also been referred to as the Power Law of Stimulus-Response).69. The amount of money, in Ina’s saving account after t years is modeled by the differential equation .
a. What is the continuous growth rate?
b. Find the particular solution, if Ina’s account is worth $3479.02 after 2 yr.
c. Find the amount that Ina deposited initially.
70. The amount of money, in John’s savings account after t years is modeled by the differential equation .
a. What is the continuous growth rate?
b. Find the particular solution, if John’s account is worth $2582.58 after 1 yr.
c. Find the amount that John deposited initially.
Solve.
71.
Solve.
72.
Explain the difference between a constant rate of growth and a constant percentage rate of growth.74. What function is also its own derivative? Write a differential equation for which this function is a solution. Are there any other solutions to this differential equation? Why or why not?
75E76. Solve . Graph the particular solutions for , and .
77EMatch each expression in column A with an equivalent expression in column B. Assume that z=f(x,y). [6.2,6.6] Column A Column B zx a. 23 12 xdx b. fyx c. fxy d. 23 y3 dy e. fx f. 15x2 y7 g. 35x3 y6 h. fyGiven f(x,y)=ey+3xy3+2y, find each of the following [6.1,6.2] f(2,0)Given, find each of the following
10.
Given f(x,y)=ey+3xy3+2y, find each of the following [6.1,6.2] fyGiven, find each of the following
12.
Given, find each of the following
13.
Given f(x,y)=ey+3xy3+2y, find each of the following [6.1,6.2] fxxGiven, find each of the following
15.
16. State the domain of
Given, find each of the following
17.
Given z=2x3Iny+xy2, find each of the following [6.2] zyGiven, find each of the following
19.
Given, find each of the following
20.
Given, find each of the following
21.
Given, find each of the following
22.
Find the relative maximum and minimum values [6.3] f(x,y)=x36xy+y3+6x+3y15Find the relative maximum and minimum values [6.3] f(x,y)=x2xy+y22x+4yFind the relative maximum and minimum values [6.3] f(x,y)=3x6yx2y226RE27. Consider the data in the following table regarding enrollment in colleges and universities during a recent 3-year period
Year, x Enrollment, y.(millions)
1 7.2
2 8.0
3 8.4
[6.4]
(a) Find the regression line,
(b) Use the regression line to predict enrollment in the fourth year.
Consider the data in the table below regarding workers average monthly out-of-pocket premium for health insurance for a family. [6.4] (a) Find the regression line, y=mx+b (b) Use the regression line to predict workers average monthly. Out-of-pocket premium for health insurance for a family in 2020 (c) Find the exponential regression curve y=aekx (d) Use the exponential regression curve to estimate the average monthly out-of-pocket premium in 2020 Number of years, x, since Workers Average Monthly Out-of-pocket Premium for Health Insurance for a Family 0 $129 6 $226 12 $360 13 $38029. Find the extremum of Minimum
subject to the constraint . State whether it is a maximum or a minimum.
Find the extremum of f(x,y)=6xy subject to the constraint 2x+y=20. State whether it is a maximum or a minimum.Find the absolute maximum and minimum values of f(x,y)=x2y2 subject to the constraints 1x3and1y2.Evaluate [6.6] 0112x2y3dydxEvaluate
[6.6]
33.
Business: demographics. The density of students living in a region near a university is modeled by p(x,p)=9x2y2, where x and y are in miles and p is the number of students per square mile, in hundreds Assume the university is located at (0, 0) in the following graph representing the region. Find the number of students who live in the region. (b) Find the average number of students per square mile of the region.35. Evaluate
.
36RE37RE38REGiven, find each of the following
1.
2TGiven f(x,y)=ex+2x3y+y, find each of the following fy4TGiven, find each of the following
5.
Given, find each of the following
6.
Given f(x,y)=ex+2x3y+y, find each of the following 2fy2Find the relative maximum and minimum values. f(x,y)=x2xy+y3x9T10T11T12T13T14. Business: maximizing production. Southwest Appliances has the following Cobb-Douglas production function for a certain product:
,
Where x is labor and y is capital, both measured in dollars. Suppose Southwest can make a total investment in labor and capital of $600,000. How should it allocate the investment between labor and capital in order to maximize production?
Find fxandft: f(x,t)=x22tx3+2t.Use a 3D graphics program to graph f(x,y)=x12y213x3.Forf(x,y)=x23xy,find(0,2),f(2,3),andf(10,5).2. .
3. .
4. .
Forf(x,y)=Inx+y3,findf(e,2),f(e2,4),andf(e3,5).6. .
Forf(x,y,z)=x2y2+z2,findf(1,2,3)andf(2,1,3).8. .
In Exercises 9-14, determine the domain of each function of two variables. f(x,y)=x2+4x+y2In Exercises 9-14, determine the domain of each function of two variables.
10.
In Exercises 9-14, determine the domain of each function of two variables.
11.
In Exercises 9-14, determine the domain of each function of two variables.
12.
In Exercises 9-14, determine the domain of each function of two variables.
13.
In Exercises 9-14, determine the domain of each function of two variables. k(x,y)=1x+yx115. Price-earnings ratio. The price-earnings ratio of stock is given by
,
where P is the price of the stock and E is the earnings per share. For the quarter ending in January 2014, the price per share of Hewlett-Packard stock was $29.00, and the earnings per share were $0.74. (Source.yahoo.finance.com.) Find the price-earnings ratio. Use decimal notation rounded to the nearest hundredth.
Yield. The yield of a stock is given by Y(D,P)=DP, Where D is the dividend per share of stock and P is the price per share. On April 1, 2014, the price per share of Taxas Instruments stock was $47.53, and the dividend per share was $1.20. (Source: yahoo.finance.com.) Find the yield. Use percent notation rounded to the nearest hundredth of a percent.17. Cost of storage equipment. Consider the cost model in Example 5. For $100,000, Tonopah Storage buys a storage tank that has a capacity of 80,000 gal. Later it replaces the tank with a new one that has triple that capacity. Estimate the cost of the new tank.
Savings and interest. A sum of $1000 is deposited in a savings account for which interest is compounded monthly. Future value A is a function of the annual percentage rate r and the term t, in months, and is given by A(r,t)=1000(1+r12)12t. Determine A(0.05,10) What is the interest earned for the rate and term in part (a)? How much more interest can be earned over the same term as in part (a) if the APR is increased to 5.75%?Monthly car payments. Ashley wants to buy a 2014 Nissan Leaf hybrid and finance $20,000 of the cost through a loan. Use the table in Example 3 to answer the following questions. a. SouthBank will lend Ashley $20,000 at an APR of 5.0% for 7 yr. What will her monthly payment be? b. Find Ashleys total payments, assuming that she pays the amount found in part (a) every month for the full term of the loan. c. Valley Credit Union offers a 5-yr term on a loan of $20,000. What is the highest APR that Ashley can accept if she wants to pay less overall than what she would with the loan from SouthBank? d. If she accepts Valley credit unions offer of a 5-yr term with an APR of 6.5% and makes every monthly payment, how much less overall will she pay?20E21. Poiseuille’s Law. The speed of blood in a vessel is given by
,
where R is the radius of the vessel, r is the distance of the blood from the center of the vessel, L is the length of the blood vessel, p is the blood pressure, and v is the viscosity of the blood. Find