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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
Problem, find the relative rate of change of f(x) at the indicated value of x. Round to three decimal places. 8. f{x)=580; A = 300 10.f{x)=500 - 6A; A = 40
In Problems 9 and 10, f(x) is continuous on (-00,00) and has critical values at x = a, b, c, and d. Use the sign chart for f'(x) to determine whether f has a local maximum, a local minimum, or neither at each critical value.
In Problem 19 - 26 , find (A) fʹ (x), (B) the critical values of f, and (C) the partition number for fʹ20. f(x) = x3 - 21x + 3022. f(x)=(x-9)2/324.26. f(x) = |x + 3|
Problems 27-40, find the intervals on which f(x) is increasing, the intervals on which f(x) is decreasing, and the local extrema. 28. f(x) = -3x2 - 12x 30. f(x) = -3x2 + 12x - 5 32. f(x) = -x3 - 4x + 8 34. f(x) = -2x3 + 3x2 + 120x 36. f(x) = x4 + 2x3 + 5 38. f(x) = x In x - x 40. f(x) = (x2 - 9)2/3
In Problems 41-46, use a graphing calculator to approximate the critical values off(x) to two decimal places. Find the intervals on42. f(x) = x4 + 5x3 - 15x44. f(x) = e -x - 3x246.
In Problems 47-54, find the intervals on which f(x) is increasing and the intervals on which f(x) is decreasing. Then sketch the graph. Add horizontal tangent lines. 48. f(x) = 2x2 - 8x + 9 50. f(x) = x3 - 12x + 2 52. f(x) = x3 + 3x2 + 3x 54. f(x) = -x4 + 50x2
In Problems 55 - 62, f(x) is continuous on (-ˆž,ˆž). Use the given information to sketch the graph of f.56.58. 60. f(-2) = -l,f(0) = 0,f(2) = 1; f(-2) = 0,f'(2) =0; f(x) > 0 on (-ˆž, -2), (-2,2), and (2, ˆž) 62. f(-1) = 2,f(0) = 0,f(2) = 2; f(-1) = 0,f'(1)
In Problem 69 - 74, use the given graph of y = fʹ (x) to find the intervals on which f is increasing the intervals on which f is decreasing and the local extrema. Sketch a possible graph of y = f(x)70.72. 74.
In Problems 75-78, use the given graph of y = f(x) to find the intervals on which f'(x) > 0, the intervals on which f'(x) 76.78.
In Problem 79 - 86, find the critical values, the intervals on which f (x) is increasing, the intervals on which f (x) is decreasing, and the local extrema. Do not graph. 80. f(x) = 9/x + x 82. f(x) = 3 - 4/x - 2/x2 84.f(x) = x2/x-2 86. f(x) = x3(x-5)2
Let f(x) = x4 + kx2, where k is a constant. Discuss the number of local extrema and the shape of the graph of f if (A) k > 0 (B) k < 0
The graph of the total revenue R(x) (in dollars) from the sale of x cordless electric screwdrivers is shown in the figure.(A) Write a brief description of the graph of the marginal revenue function y = R(x), including a discussion of any x intercepts. (B) Sketch a possible graph of y=R'(x).
The figure approximates the rate of change of the price of eggs over a 70-month period, where E(t) is the price of a dozen eggs (in dollars) and t is time (in months).(A) Write a brief description of the graph of y = E(t), including a discussion of any local extrema. (B) Sketch a possible graph of
A manufacturer incurs the following costs in producing x rain jackets in one day for 0 < x < 200: fixed costs, $450; unit production cost, $30 per jacket; equipment maintenance and repairs, 0.08x2 dollars. (A) What is the average cost (x)per jacket if x jackets are produced in one day? (B) Find
Show that profit will be decreasing over production intervals (a,b)for which marginal revenue is less than marginal cost.
The concentration C(t),in milligrams per cubic centimeter, of a particular drug in a patient's bloodstream is given bywhere t is the number of hours after the drug is taken orally. Find the critical values of C(t),the intervals on which the drug concentration is increasing, the intervals on which
Use the graph of y = g(x) to identify(A) Intervals on which the graph of g is concave upward(B) Intervals on which the graph of g is concave downward(C) Intervals on which g"(x) (D) Intervals on which g"(x) > 0(E) Intervals on which g'(x) is increasing(F) Intervals on which g'(x) is decreasing(G)
In Problem 19 - 24, find the x and y coordinates of all inflection points. 20. f (x) = x3 - 24x2 22. f (x) = 5 - x4/3 24. f (x) = x3/5 − 6x + 7
In Problems 25-34, find the intervals on which the graph of f is concave upward, the intervals on which the graph of f is concave downward, and the inflection points. 26. f(x)= x4 + 6x 28. f(x)= - x3 - 5x2 + 4x - 3 30. f(x)= x4 - 2x3 - 36x + 12 32. f(x)=In (x2 + 6x + 13) 34. f(x) = e3x - 9ex
In Problem 35 - 42 f(x) is continuous on ( - ˆž, ˆž). Use the given information to sketch the graph of f.36.38. 40 f(-2)= -2,f(0) = l,f(2) = 4: f'(-2) = 0,f'(2) =0; f'(x) > 0 on (-2,2); f(x) f'(0)= 0; f"(x) > 0 on (-ˆž,0): f(x) 42. f(0)= -2,f(1) = 0,f(2) = 4: f'(0)
Problems 43-64, summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of y=f(x). 44. f(x) = (x - 3)(x2 - 6x - 3) 46. f(x)=(1 - x)(x2+x+4) 48. f(x) = ().25x4 - 2x3 50.f(x) = -4x(x + 2) 52.f(x) = (x2+3)(x2-1-) 54. f(x)= (x2 - l)(x2- 5) 56. f(x) = 3x5 -
In Problems 65-68, use the graph of y = f'(x) to discuss the graph of y = f{x). Organize your conclusions in a table (see Example 4), and66.68.
In Problems 69-76, apply steps 1-3 of the graphing strategy to f(x). Use a graphing calculator to approximate (to two decimal places) x intercepts, critical values, and x coordinates of inflection points. Summarize all the pertinent information. 70. f(x) = x4 + 2x3 - 5x2 - 4x + 4 72. f(x) = x4 -
Explain how you can determine where f'(x)is increasing or decreasing by examining the graph of y=f{x). In Problem assume that f is a polynomial.
In Problem 7 - 18, find the indicated derivative for each function, 8. g"(x) for g (x) = -x3 + 2x2 - 3x + 9 10. k"(x) for k (x) = -6x-2 + 12x-3 12. d2y/dx2for y = x3 - 24x1/3 14. yʹʹ for y = (x2 - 16)5 16. f"(x) for f(x) = xe-x 18. y" for y = x2 In x
Explain how you can locate local maxima and minima for the graph of y = f(x) by examining the graph of y = f'(x) In problem, assume that f is a polynomial.
The company in Problem 83 produces the same camp stove at another plant. The total cost C(x)(in dollars) of producing x camp stoves per week at plant B is shown in the figure. Discuss the graph of the marginal cost function C'(x)and interpret the graph of C'(x)in terms of the efficiency of the
Suppose that the cost equation for the company in Problem 85 isC(x) = 830 + 396x(A) Find the local extrema for the profit function.(B) On which intervals is the graph of the profit function concave upward? Concave downward?
A national food service runs food concessions for sporting events throughout the country. The company's marketing research department chose a particular football stadium to test market a new jumbo hot dog. It was found that the demand for the new hot dog is given approximately by p = 8 - 2 In x
point of diminishing returns. A baseball cap manufacturer is planning to expand its workforce. It estimates that the number of baseball caps produced by hiring x new workers is given by T(x)=-0.25x4 + 6x3 0 ≤ x ≤ 18 When is the rate of change of baseball cap production increasing and when
point of diminishing returns. A company estimates that it will sell N(x)units of a product after spending $x thousand on advertising, as given by N(x)=-0.25x4 + 13x3 - 180x2 + 10,000 15 ≤ x ≤ 24 When is the rate of change of sales increasing and when is it decreasing? What is the point of
A sporting goods chain places TV ads to promote golf club sales. The marketing director used past records to determine the following data, where x is the number of ads placed monthly and y is the number of golf clubs sold that month.(A) Enter the data in a graphing calculator and find a cubic
One hour after x milligrams of a particular drug are given to a person, the change in body temperature T(x),in degrees Fahrenheit, is given byThe rate T'(x)at which T(x)changes with respect to the size of the dosage x is called the sensitivity of the body to the dosage. (A) When is T'(x)
In Problems 15-18, explain why L'Hopital's rule does not apply. If the limit exists, find it by other means.16.18.
Use L'Hopital's rule to find each limit in Problems 1-14.2.4.6.8.10.12.14.
In Problems 19-22, the limit can be found in two ways. Use L'Hopital's rule to find the limit and check your answer using an algebraic simplification.20.22.
Find each limit in Problem 23 - 46. L'Hôpital's rule does not apply to every problem, and some problems will require more than one application of L'Hôpital's rule.24.26.28.30.32.34.36.38.40.42.44.46.
Find
In problem 49 - 52, n is a positive integer. Find each limit.50.52.
In Problems 53-56, show that the repeated application of L'Hopitals rule does not lead to a solution. Then use algebraic manipulation to evaluate each limit. 54. 56.
In Problem 17 - 56, summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of y = f(x)18.20.22. f(x) = 3 + 7e-0.2x24. f(x) = 10xe-0.1x26. f(x) = ln(2x + 4)28. f(x) = ln(x2 + 4)30.32.34.36.38.40.42.44.46. f(x) = (x - 2)ex48.50.52.54.56.
Repeat Problem 1 for the following graph of f:Problem 1(A) The intervals on which f'(x) (B) The intervals on which f'(x) > 0(C) The intervals on which f(x) is increasing(D) The intervals on which f(x) is decreasing(E) The x coordinate(s) of the point(s) where f(x) has a local maximum(F) The x
In Problems 3-10, use the given information to sketch the graph off. Assume that f is continuous on its domain and that all intercepts are included in the table of values.4. Domain: All real x;6.Domain: All real x, except 8.Domain: All real x, except x = 1; f(O) = -2,f(2) =0; fʹ(x)
In Problems 57-64, show that the line y = x is an oblique asymptote for the graph of y = f(x), summarize all pertinent information obtained by applying the graphing strategy, and sketch the graph of y=f(x).58.60. 62. 64.
In Problems 65-72, summarize all pertinent information obtained by applying the graphing strategy, and sketch the graph of y=f(x).66.68. 70. 72.
Suppose that the cost function C(x) (in dollars) for the company in Problem 73 is C(x) = 830 + 396x (A) Write an equation for the profit P(x). (B) Graph the profit function P.
A company producing dive watches has established that, on average, a new employee can assemble N(t)dive watches per day after t days of on-the-job training, as given by(A) Where is Nit)increasing? Decreasing? (B) Where is the graph of N concave upward? Downward? (C) Find any horizontal and vertical
The management of a manufacturing plant wishes to add a fenced-in rectangular storage yard of 20,000 square feet, using a building as one side of the yard (see the figure). If A: is the distance (in feet) from the building to the fence, show that the length of the fence required for the yard is
The total daily cost (in dollars) of producing x city bikes is given by C{x) = 500 + 2x + 0.2x2 (A) Sketch the graphs of the average cost function and the marginal cost function on the same set of coordinate axes. Include any oblique asymptotes. (B) Find the minimum average cost.
The table gives the total daily costs y (in dollars) of producing x deluxe pizzas at various production levels.(A) Enter the data into a graphing calculator and find a quadratic regression equation for the total costs. (B) Use the regression equation from part (A) to find the minimum average cost
In a study on the speed of muscle contraction in frogs under various loads, researchers found that the speed of contraction decreases with increasing loads. More precisely, they found that the relationship between speed ofGraph S.
In Problem 11 - 16, find the absolute maximum and absolute minimum of each function on the give interval. 12. f (x) = −3x + 20, on = [−2,6] 14. f (x) = ln x, on = [1, 2] 16. f (x) = x2 − 6x + 7, on = [0,10]
In Problem 17 to 32, find the absolute maximum and minimum, if either exists, for each function.18. f(x) = x2 + 4x - 320. f(x) = -x2 + 2x + 422. f(x) = -x3 - 2x24. f(x) = x4 - 4x326. f(x) = x + 25/x28.30. 32.
In Problem 33 - 56, find the indicated extremum of each function on the given interval. 34. Absolute maximum value on [0, ∞) for F(x) = 6x - x2 + 4 36. Absolute minimum value on [0, ∞) for F(x)- x3 - 6x2 38. Absolute minimum value on [0, ∞) for F(x)=(2 - x)(x + l)2 40. Absolute maximum value
In Problem 58 to 62, find the absolute maximum and minimum, if either exists, for each function on the indicated intervals. 58.f(x) = 2x3 - 3x2 - 12x + 24 (A) [-3,4] (B) [-2,3] (C) [-2,1] 60.f(x) = x4 - 8x2 + 16 (A) [-1,3] (B) [0,2] (C) [-3,4] 62. f(x) = x4 - 18x2 + 32 (A) [-4,4] (B) [-1,1] (C)
Find the dimensions of a rectangle with a perimeter of 76 feet that has the maximum area.
A company manufactures and sells x digital cameras per week. The weekly price-demand and cost equations are, respectively, p=400 - 0.4x and C(x)=2,000 + 160x (A)What price should the company charge for the cameras, and how many cameras should be produced to maximize the weekly revenue?
Repeat Problem 13 forC(x) = 60,000 + 60xProblem 13 A company manufactures and sells x television sets per month. The monthly cost and price-demand equations are C(x) - 72,000 + 60x (A) Find the maximum revenue. (B) Find the maximum profit, the production level that will realize the maximum profit,
The following table contains price-demand and total cost data for the production of regular sleeping bags, where p is the wholesale price (in dollars) of a sleeping bag for an annual demand of x sleeping bags and C is the total cost (in dollars) of producing x sleeping bags:(A) Find a quadratic
A university student center sells 1,600 cups of coffee per day at a price of $2.40. (A) A market survey shows that for every $0.05reduction in price, 50 more cups of coffee will be sold. How much should the student center charge for a cup of coffee in order to maximize revenue? (B) A different
Find two numbers whose sum is 21 and whose product is a maximum.
A 300-room hotel in Las Vegas is filled to capacity every night at $80 a room. For each $1 increase in rent, 3 fewer rooms are rented. If each rented room costs $10 to service per day, how much should the management charge for each room to maximize gross profit? What is the maximum gross profit?
A commercial pear grower must decide on the optimum time to have fruit picked and sold. If the pears are picked now, they will bring 30? per pound, with each tree yielding an average of 60 pounds of salable pears. If the average yield per tree increases 6 pounds per tree per week for the next 4
A parcel delivery service will deliver a package only if the length plus girth (distance around) does not exceed 108 inches.(A) Find the dimensions of a rectangular box with square ends that satisfies the delivery service's restriction and has maximum volume. What is the maximum volume?(B) Find the
The owner of a retail lumber store wants to construct a fence to enclose an outdoor storage area adjacent to the store, using all of the store as part of one side of the area (see the figure). Find the dimensions that will enclose the largest area if(A) 240 feet of fencing material are used.(B) 400
A pharmacy has a uniform annual demand for 200 bottles of a certain antibiotic. It costs $10 to store one bottle for one year and $40 to place an order. How many times during the year should the pharmacy order the antibiotic in order to minimize the total storage and reorder costs?
The cost per hour for fuel to run a train is v2/4 dollars, where v is the speed of the train in miles per hour. Other costs, including labor, are $300 per hour. How fast should the train travel on a 360-mile trip to minimize the total cost for the trip?
The concentration C(t),in milligrams per cubic centimeter, of a particular drug in a patient's bloodstream is given bywhere t is the number of hours after the drug is taken. How many hours after the drug is taken will the concentration be maximum? What is the maximum concentration?
Some birds tend to avoid flights over large bodies of water during daylight hours. Suppose that an adult bird with this tendency is taken from its nesting area on the edge of a large lake to an island 5 miles offshore and is then released (see the figure).(A) If it takes 1.4 times as much energy to
Two heavily industrial areas are located 10 miles apart, as shown in the figure. if the concentration of particulate matter (in parts per million) decreases as the reciprocal of the square of the distance from the source, and if area A1 emits eight times the particulate matter as A2, then the
A large grocery chain found that, on average, a checker can recall P% of a given price list x hours after starting work, as given approximately by P(x) = 96x - 24x2 0 ≤ x ≤ 3 At what time x does the checker recall a maximum percentage? What is the maximum?
Find two numbers whose difference is 21 and whose product is a minimum.
Find two positive numbers whose product is 21 and whose sum is a minimum.
Find the dimensions of a rectangle with an area of 108 square feet that has the minimum perimeter.
In Problems 17-24, find all the anti derivatives for each derivative.(a)(b) (c)
In Problems 35-50, find each indefinite integral. (Check by differentiation.)a.b. c. d. e. f. g. h.
In Problems 1-16, find each indefinite integral. Check by differentiating.(a)4(b)6 (c)8 (d)10 (e) (f) (g)
In Problems 51-58, find the particular anti derivative of each derivative that satisfies thegiven condition.a.R'(x)=600 - 0.6x; R(0) = 0,b.c. d.
Find the equation of the curve that passes through (1, 3) if its slope is given byFor each x.
In Problems 61-66, find each indefinite integral.a.b. c.
In Problems 67-72, find the particular anti derivative of each derivative that satisfies the given condition.a.b. c.
Assuming that x
In 2007, U.S. consumption of renewable energy was 6.8 quadrillion Btu (or 6.8 × 1015 Btu). Since the 1960s, consumption has been growing at a rate (in quadrillion Btu's per year) given by f'{t)= 0.004r + 0.062 where this years after 1960. Find f(t)and estimate U.S. consumption of renewable energy
The graph of the marginal revenue function from the sale of x sports watches is given in the figure.(A) Using the graph shown, describe the shape of the graph of the revenue function R(x)as x increases from 0 to 1,000.(B) Find the equation of the marginal revenue function (the linear function shown
The rate of change of the monthly sales of a newly released football game is given by S'(t) = 500t1/4 S(0) = 0 where t is the number of months since the game was released and S{t)is the number of games sold each month. Find S(i). When will monthly sales reach 20,000 games?
Repeat Problem 88 if S'(t)=500f1/4 + 300 and all other information remains the same. Use a graphing calculator to approximate the solution of the equation S(t)20,000 to two decimal places.
If the rate of labor use in Problem 91 is g(x) = 2,000x,-1/3 and if the first 8 control units require 12,000 labor-hours, how many labor-hours, L(x), will be required for the first x control units? The first 27 control units?
The area A of a healing wound changes at a rate given approximately bywhere t is time in days and A(1) =2 square centimeters. What will the area of the wound be in 10 days?
A college language class was chosen for an experiment in learning. Using a list of 50 words, the experiment involved measuring the rate of vocabulary memorization at different times during a continuous 5-hour study session. It was found that the average rate of learning for the entire class was
In Problems 1-36, find each indefinite integral and check the result by differentiating.1.2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
In Problems 37-42, the indefinite integral can be found in more than one way. First use the substitution method to find the indefinite integral. Then find it without using substitution. Check that your answers are equivalent.a.b. c.
ln Problems 43-48, suppose that the indicated "solutions" were given to you by a student whom you are tutoring. (A) How would you have the student check each solution? (B) Is the solution right or wrong? If the solution is wrong, explain what is wrong and how it can be corrected. (C) Show a correct
In Problems 49-60, find each indefinite integral and check the result by differentiating.a.b. c. d. e. f.
In problem 61-66 find each indefinite integral.
In Problems 61-66, find the anti derivative of each derivative.a.b.
Price-supply equation. The marginal price at a supply level of x bottles of shampoo per week is given byFind the price-supply equation if the distributor of the shampoo is willing to supply 75 bottles a week at a price of $5.00 per bottle. How many bottles would the supplier be willing to supply at
Revenue function. The weekly marginal revenue from the sale of x pairs of tennis shoes is given byWhere R(x)is revenue in dollars. Find the revenue function. Find the revenue from the sale of 1,000 pairs of shoes.
Marketing. Repeat Problem 71 if the monthly rate of increase in sales is found to be approximated by sS' (t) 20 - 20e -0.05t 0 ≤ t ≤ 24
Oil production. Assume that the rate in Problem 73 is found to be(A) When is the rate of production greatest?(B) How many barrels of oil Q(t)will the field produce in the first t years if Q(0)=0? How many barrels will be produced in the first 5 years?(C) How long (to the nearest tenth of a year)
The rate of healing for a skin wound (in square centimeters per day) is approximated by A'(t) - 0.9e-0.1t. If the initial wound has an area of 9 square centimeters, what will its area A(t)be after t days? After 5 days?
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