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College Mathematics for Business Economics Life Sciences and Social Sciences 12th edition Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen - Solutions
An oil tanker aground on a reef is losing oil and producing an oil slick that is radiating outward at a rate given approximately bywhere R is the radius (in feet) of the circular slick after t minutes. Find the radius of the slick after 16 minutes if the radius is 0 when t = 0.
An average student enrolled in a steno typing class progressed at a rate of N'(t)=12e -0.06t words per minute per week t weeks after enrolling in a 15-week course. If, at the beginning of the course, a student could stenotype at zero words per minute, how many words per minute N(t)would the student
Problems 13-18 refer to the following slope fields: a. Which slope field is associated with the differential equation dy/dx = -x? Briefly justify your answer.
In Problems 1-12, find the general or particular solution, as indicated, for each differential equation. a. b. c. d. e.
In Problems 19-26, find the general or particular solution, as indicated, for each differential equation. a. b. c.
Problems 27-34 refer to the following slope fields: a. Which slope field is associated with the differential equation dy/dx=y+1? Briefly justify your answer.
Show that y = √x2 + C is a solution of the differential equation dy/dx = x/y for any real number C. Find the particular solution that passes through (-6,7).
a. Show that y = C/x is a solution of the differential equation dy/dx = - y/x for any real number C. Find the partic solution that passes through (2,5). b. Show that y = 2/(1 + ce-6t) is a solution of the differential equation dy/dt= 3y(2 - y) for any real number c. Find the particular solution
In Problems 41-48, use a graphing calculator to graph the given examples of the various cases in Table 1 on page 727.a. Unlimited growth:y = 5,250e012t0 ‰¤ t ‰¤ 100 ‰¤ y ‰¤ 20,000b. Exponential decay:p = 1,000e - 0.08x0 ‰¤ x
Find the value of t for which the logistic functionIs equal to M/2
Continuous compound interest. Find the amount A in an account after t years if
The marginal price dp/dx at x units of supply per day is proportional to the price p. There is no supply at a price of $10 per unit [p(0)=10], and there is a daily supply of 50 units at a price of $12.84 per unit [p(50) = 12.84]. (A) Find the price-supply equation. (B) At a supply of 100 units per
Advertising. Suppose that the differential equation for Problem 59 is(A) Explain what the equation N(10)= 0.1L means. (B) Solve the differential equation. (C) How many days will it take to expose 50% of L?
A community of 1,000 people is homogeneously mixed. One person who has just returned from another community has influenza. Assume that the home community has not had influenza shots and all are susceptible. One mathematical model assumes that influenza tends to spread at a rate in direct proportion
Many countries have banned the use of the insecticide DDT because of its long-term adverse effects. Five years after a particular country stopped using DDT, the amount of DDT in the ecosystem had declined to 75% of the amount present at the time of the ban. Find the continuous compound rate of
For a person learning to type, the number N of words per minute that the person could type after / hours of practice was given by N = 100(1 - e - 0.02t) What is the rate of improvement after 10 hours of practice? After 40 hours of practice?
The Weber-Fechner law concerns a person's sensed perception of various strengths of stimulation involving weights, sound, light, shock, taste, and so on. One form of the law states that the rate of change of sensed sensation S with respect to stimulus R is inversely proportional to the strength of
In Problem 71, how long (to the nearest minute) will it take for half of the group of 400 to have heard the rumor?
Problems 9-16 involve estimating the area under the curves in Figures A-D from x = 1 to x = 4. For each figure, divide the interval [1,4] into three equal subintervals.a. Draw in left and right rectangles for Figures C and D.12b. Using the results of Problem 10, compute L3 and R3 for Figure C and
a. Partition [0,12] into four subintervals of equal length, and for each subinterval b. Partition [-5,5] into five subintervals of equal length, and for each subinterval [xk-1, xk], let ck = (x k -1 + xk)/3
a. Partition [0, 3] into three subintervals of equal length, and let c1, = 0.2, c2 = 1.5, and c3 = 2.8. b. Partition [1,7] into six subintervals of equal length, and let c1 = 2, c2 = 2, c3 = 4, c4 = 4, c5 = 6, and c6 = 6.
In Problem 25 - 36 calculate the definite integral by referring to the figure the indicated areasa. b. c. d. e.
In Problems 37-48, calculate the definite integral, given thata.b. c. d.
a. In Problems 49-54, discuss the validity of each statement. If the statement is always true, explain why. If it is not always true, give a counter example. b. If /is a decreasing function on [a,b],then the area under the graph of f is greater than the left sum LR and less than the right sum Rn,
Refer to Problem 55. Use R10 estimate the combined area of both parcels, and calculate an error bound for this estimate. How many subdivisions of the baseline would be required so that the error incurred in using Rn would not exceed 1.000 square feet?
Use L5 and R5 to approximate ∫52 (0.25x2 - 4) dx. Compute error bounds for each. (Round answers to two decimal places.) Describe in geometric terms what the definite integral over the interval [1,6] represents.
For Problems 59-62, use a graphing calculator to determine the intervals on which each function is increasing or decreasing.a.b. f (x) = ex2
In Problems 63-66, the left sum Ln or the right sum Rn is used to approximate the definite integral to the indicated accuracy. How large must n be chosen in each case? (Each function is increasing over the indicated interval.)a.b.
For a new employee in Problem 67, use left and right sums to estimate the area under the graph of N (t)from t = 20 to t = 100. Use four equal subintervals for each. Replace the question marks with the values of L4 or R4as appropriate:
Refer to Problem 69. Use left and right sums over five equal subintervals to approximate the area under the graph of A'(t) from t = 5 to t = 10. Calculate an error bound for this estimate.
For the data in Problem 71, use left and right sums over three equal subintervals to approximate the area under the graph of N'(x) from x = 0 to x = 6. Replace the question marks with values of L3 and R3as appropriate:
Evaluate the integrals in Problems 5-24.a.b. c. d. e.
In Problems 1-4, (A) Calculate the change in F(x) from x = 10 to x = 15. (B) Graph F'(x) and use geometric formulas (see Appendix C) to calculate the area between the graph of F'(x) and the x = axis from x = 10 to x = 15. (C) Verify that your answers to (A) and (B) are equal, as is guar- anteed by
Evaluate the integrals in Problems 25-40.a.b. c. d. e.
In Problems 41-48, (A) Find the average value of each function over the indicated interval. (B) Use a graphing calculator to graph the function and its average value over the indicated interval in the same viewing window. a. g(x) = 2x + 7; [0,5] b. g(t) = 4t - 3t2; [-2,2] c. g(x) = √x + 1;
Evaluate the integrals in Problems 49-54.a.b. c.
Use a numerical integration routine to evaluate each definite integral in Problems 55-58 (to three decimal places).a.b.
Cost. Referring to Problem 61, compute the increase in cost going from a production level of 0 bikes per month to 600 bikes per month. Set up a definite integral and evaluate it.
Maintenance costs for an apartment house generally increase as the building gets older. From past records, the rate of increase in maintenance costs (in dollars per year) for a particular apartment complex is given approximately by M'(x) = f{x) = 90x2 + 5,000 where x is the age of the apartment
Refer to Problem 65. (A) Find a cubic regression equation for the data, and graph it and the data set in the same viewing window. (B) Use the regression equation and a numerical integration routine on a graphing calculator to approximate the number of units assembled by a new employee during the
The total accumulated costs C(i)and revenues R(t)(in thousands of dollars), respectively, for a coal mine satisfy C'(t) = 3 and R' (t) = 15e-01t where t is the number of years that the mine has been in operation. Find the useful life of the mine, to the nearest year. What is the total profit
The total cost (in dollars) of printing x dictionaries is C(x) = 20,000 + 10x. (A) Find the average cost per unit if 1.000 dictionaries are produced. (B) Find the average value of the cost function over the in-terval [0,1,000]. (C) Discuss the difference between parts (A) and (B).
Cost. Refer to Problem 71. (A) Find a cubic regression equation for the data, and graph it and the data set in the same viewing window. (B) Use the regression equation and a numerical integration routine on a graphing calculator to approximate (to the nearest dollar) the increased cost in going
Demand function. Given the demand functionfind the average price (in dollars) over the demand interval [400.600].
If the rate of labor use in Problem 75 is g(x) = 2,000x -1/3 then approximately how many labor-hours will be required to assemble the 9th through the 27th control units? [Hint. Let a = 8 and b = 21.)
Repeat Problem 77 with an order of 1,200 units every 4 months.
In Problem 79, if the rate is found to bethen approximately how many barrels of oil will the field produce during the first 5 years of production? The second 5 years of production?
The rate of healing of a skin wound (in square centimeters per day) is given approximately by A'(t) = - 0.9e -0.1tThe initial wound has an area of 9 square centimeters. How much will the area change during the first 5 days? The second 5 days?
A drug is injected into the bloodstream of a patient through her right arm. The drug concentration in the hours after the injection is given byWhat is the average drug concentration in the bloodstream of the left arm during the first hour after the injection? During the first 2 hours after the
The number of children in a large city was found to increase and then decrease rather drastically. If the number of children over a 6-year period was given bywhat was the average number of children in the city over the 6-year period? [Assume that N-N(t)is continuous.]
In Problems 7-20, find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places y = x2 + 2; y = 0; 0 ≤ x ≤ 3
In Problems 7-20, find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places. y = -2x2; y = 0; - 6 ≤ x ≤ 2
In Problems 7-20, find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places. y = - x3 + 3 ; y = 0; 0 ≤ x ≤ 2
In Problems 7-20, find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places. y = - x(3 - x); y = 0; -2 ≤ x ≤ 0
In Problems 7-20, find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places. y = -e x; y = 0; 0 ≤ x ≤ 1
In Problems 7-20, find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
In Problems 21-26, use a definite integral to find the area bounded by the graphs of the indicated equations over the given interval. Then check your answer by finding the area without using a definite integral. [Hint: Partition the region into triangles and/or rectangles]. Y = - x ; y = 8; 0 ≤ x
In Problems 21-26, use a definite integral to find the area bounded by the graphs of the indicated equations over the given interval. Then check your answer by finding the area without using a definite integral. [Hint: Partition the region into triangles and/or rectangles].
In Problems 21-26, use a definite integral to find the area bounded by the graphs of the indicated equations over the given interval. Then check your answer by finding the area without using a definite integral. [Hint: Partition the region into triangles and/or rectangles]. y = - 2x -3 ; y = x - 3;
Referring to Figure A, explain how you would use definite integrals to find the area between the graph of y = f(x) and the x axis from x = a to x = d.
In Problems 37-52, find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places. y = -x + 1; y = 0; -1 ≤ x ≤ 2
In Problems 37-52, find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places. Y = 4 - x2; y = 0; 0 ≤ x ≤ 4
In Problems 37-52, find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places. y = - x2 - 2x; y = 0; - 2 ≤ x ≤ 1
In Problems 37-52, find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places. y = 2x + 6; y = 3; -1 ≤ x ≤ 2
In Problems 37-52, find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places. y = x2; y = 9
In Problems 37-52, find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places. y = x2 - 1; y = 3
In Problems 37-52, find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places. y = x2 - 1; y = x - 2; - 2 ≤ x ≤ 1
In Problems 37-52, find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
Problems 53-58, set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places. [Hint: A circle of radius r, with center at the origin, has equation x2 + y2 = r2 and area πr2] y = √ 25 - x2; y
Problems 53-58, set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places. [Hint: A circle of radius r, with center at the origin, has equation x2 + y2 = r2 and area πr2] y = - √36 - x2;
Problems 53-58, set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places. [Hint: A circle of radius r, with center at the origin, has equation x2 + y2 = r2 and area πr2] y = - √100 - x2;
Explain why ˆ« ba[- h(x)] dx represents the area between the graph of y = h(x) and the x axis from x = a to x = b in Figure C.In Problems 7-20, find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places
Problems 59-62, use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places. y = 3 - 2x2; y = 2x2 - 4x
Problems 59-62, use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
In Problems 63-68, find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places. y = ex; y = - e -x; 1 ≤ x ≤ 2
In Problems 63-68, find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places. y = x3 + 1; y = x + 1
In Problems 63-68, find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places. y = x3 - 6x2 + 9x; y = x
In Problems 69-74, use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places. y = 2x3 + 2x2 - x; y = - 2x3 - 2x2 + 2x
In Problems 69-74, use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places y = 2 - (x + l)2;y = ex+1
In Problems 69-74, use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places. y = 2 - ex; y = x3 + 3x2
Jin Problems 75-78, use a numerical integration routine on a graphing calculator to find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places. y = x2 + 3x + 1; y = eex - 3 ≤ x ≤ 0
in Problems 75-78, use a numerical integration routine on a graphing calculator to find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places. y = ln(ln x) ; y = 0.01 x
In Problems 7-20, find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places. y = - x + 10; y = 0; -2 ≤ x ≤ 2
Oil production. In Problem 79, if the rate is found to beThen find the area between the graph of R and the t axis over the interval [5,15] and interpret the results.
Useful life. Repeat Problem 81 if C'(t)= 2t and R'(t)=5te -0.1t2
Income distribution. Using data from the U.S. Census Bureau, an economist produced the following Lorenz curves for the distribution of U.S. income in 1962 and in 1972:Find the Gini index of income concentration for each Lorenz curve and interpret the results.
Income distribution. The government of a small country is planning sweeping changes in the tax structure in order to provide a more equitable distribution of income. The Lorenz curves for the current income distribution and for the projected income distribution after enactment of the tax changes
Distribution of wealth. Refer to Problem 87. (A) Use cubic regression to find the equation of a Lorenz curve for the data. (B) Use the cubic regression equation you found in Part (A) and a numerical integration routine to approximate the Gini index of income concentration.
Natural resource depletion. The instantaneous rate of change in demand for U.S. lumber since 1970 (t= 0). in billions of cubic feet per year, is given by Q'(t)=12 + 0.006t2 0 ≤ t ≤ 50 Find the area between the graph of Q' and the t axis over the interval [15,20], and interpret the results.
Learning. Repeat Problem 91 if V'(t)=13/tl/2 and the interval is changed to [1,4].
In Problems 1-10, evaluate each definite integral to two decimal places.
In Problems 11 and 12, explain which of (A),(B), and (C) are equal before evaluating the expressions. Then evaluate each expression to two decimal places.(A)(B) (C)
The shelf life (in years) of a laser pointer battery is a continuous random variable with probability density function(A) Find the probability that a randomly selected laser pointer battery has a shelf life of 3 years or less. (B) Find the probability that a randomly selected laser pointer battery
In Problem 14, find d so that the probability of a randomly selected laser pointer battery lasting d years or less is .5.
In a certain city, the daily use of water (in hundreds of gallons) per household is a continuous random variable with probability density functionFind the probability that a household chosen at random will use (A) At most 400 gallons of water per day (B) Between 300 and 600 gallons of water per day
In Problem 18, what is the probability that a household will use more than 400 gallons of water per day? [See the hint in Problem 19.1
Interpret the results of Problem 22 with both a graph and a description of the graph.
Find the total income produced by a continuous income stream in the first 2 years if the rate of flow is f(t)= 600e0.06t.
Interpret the results of Problem 26 with both a graph and a description of the graph.
Suppose in Problem 29 that you start the IRA deposits at age 30, but the account earns 6%, compounded continuously. Treat the yearly deposits into the account as a continuous income stream. How much will be in the account 35 years later when you retire at age 65? How much of the final amount is
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