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mathematics
mathematical applications for the management
Mathematical Applications For The Management, Life And Social Sciences 12th Edition Ronald J. Harshbarger, James J. Reynolds - Solutions
In Problem, find y'.y = √ln (3x + 1)
In Problem, find y'.y = log5 x
In Problem, find y'.y = log6 (x4 - 4x3 + 1)
In Problem, find y'.y = log2 (1 - x - x2)
Can increasing the tax per unit sold actually lead to a decrease in tax revenues?
In Problem, find the relative maxima and relative minima, and sketch the graph with a graphing calculator to check your results.y = ln x - x
Suppose that the price p (in dollars) of a product is given by the demand functionwhere x represents the quantity demanded and x < 300. If the daily demand is decreasing at a rate of 20 units per day, at what rate is the price changing when the price per unit is $30? 18,000 – 60x 400 – x
Find the derivatives of the functions in Problem.h(x) = 750e0.04x
In Problem, find dy/dt using the given values.xy = x + 3 for x = 3, dx/dt = -1
In Problem 11, the derivative y9 was found to bey' = -x/ywhen x2 + y2 = 4.(a) Take the implicit derivative of the equation for y9 to show thaty" = -y 9+ xy'/y2(b) Substitute -x/y for y' in the expression for y" in part (a) and simplify to show that y" = - (x2 + y2)/y3(c) Does y" = -4/y3? Why
(a) Find y' implicitly for x3 - y3 = 8.(b) By taking derivatives implicitly, use part (a) to show that
The Richter scale reading, R, used for measuring the magnitude of an earthquake with intensity I is determined bywhere I0 is a standard minimum threshold of intensity. If I0 = 1, what is the rate of change of the Richter scale reading with respect to intensity? In (1/I,) R = In 10
Find y" for 1/x - 1/y = 1.
With U.S. Department of Health and Human Services data from 2002 and projected to 2024, the total public expenditures for health care H can be modeled byH = 1500e0.053twhere t is the number of years past 2000 and H is in billions of dollars. If this model is accurate, at what rate will health care
Using U.S. Bureau of Labor Statistics data for selected years from 1994 and projected to 2024, the billions of dollars spent for personal consumption in the United States can be modeled byP = 6010e0.0252twhere t is the number of years after 1990. If this model is accurate, find and interpret the
In Problem, find dy,dx at the given point without first solving for y.x2 + 5xy + 4 = 0 at (1, -1)
Using Social Security Administration data for selected years from 2012 and projected to 2050, the U.S. Consumer Price Index (CPI) can be modeled by the functionC(t) = 92.7 e0.0271 t where t is the number of years past 2010. With the reference year as 2012, a 2015 CPI = 106.15 means that goods
In the late 1980s, J. Pepi developed the most recent model for measuring the effects of high temperature and humidity. Pepi’s model is called the Summer Simmer Index and is given byS = 1.98T - 1.09(1 - H)(T - 58) - 56.8where T is the air temperature (in degrees Fahrenheit) and H is the relative
Using U.S. Energy Information Administration data for selected years from 2010 and projected to 2040, the U.S. real disposable income per capita (in thousands of dollars) can be modeled byI(t) = 32.11(1.014)twhere t is the number of years after 2010.(a) Write the function that models the rate of
Find the relative maxima, relative minima, points of inflection, and asymptotes, if they exist, for each of the functions in Problem. Graph each function.f (x) = x3 + 6x2 + 9x + 3
Find the relative maxima, relative minima, points of inflection, and asymptotes, if they exist, for each of the functions in Problem. Graph each function.y = 4x3 - x4 - 10
In Problem, determine whether each function is concave up or concave down at the indicated points.g(x) = 2x4 - 5x3 + 11x - 4 at(a) x = 3(b) x = 12
Find the relative maxima, relative minima, points of inflection, and asymptotes, if they exist, for each of the functions in Problem. Graph each function.y = x2 - 3x + 6/x - 2
Use the graph of y = f (x) in Problem 1 to identify at which of the indicated points the derivative f'(x)(a) Changes from positive to negative(b) Changes from negative to positive(c) Does not change sign. y 6+(1,5) (-1,2) (4, 1) + 4 -2 6 -2+ 2.
The profit per acre from a grove of orange trees is given by x(200 - x) dollars, where x is the number of orange trees per acre. How many trees per acre will maximize the profit?
In Problem, determine whether each function is concave up or concave down at the indicated points.g(x) = 7 - 8x3 + 4x5 at(a) x = 4(b) x = 1/2
In Problem, find the absolute maxima and minima for f(x) on the interval [a, b].f(x) = x3 - 3x2 - 24x + 7, [-3, 6]
In Problem, use the function y = 3x5 - 5x3 + 2.Over what intervals is the graph of this function concave up?
Use the graph of y = f (x) in Problem 2 to identify at which of the indicated points the derivative f'(x)(a) Changes from positive to negative(b) Changes from negative to positive(c) Does not change sign. y (5,6) 6. (2, 4) (-1, 2) + + -2 4 -2T
In Problem, find the absolute maxima and minima for f (x) on the interval [a, b].f(x) = 2x3 - 3x2 - 12x - 5, [-10, 10]
In Problem, use the function y = 3x5 - 5x3 + 2.Find the points of inflection of this function.
In Problem, find the absolute maxima and minima for f (x) on the interval [a, b].f(x) = 9 + 6x2 - x3, [-3, 10]
In Problem, use the function y = 3x5 - 5x3 + 2.Find the relative maxima and minima of this function.
Find the absolute maximum and minimum for f(x) = 2x3 - 15x2 + 3 on the interval [-2, 8].
The figure shows a typical curve that gives the volume of sales S as a function of time t after an ad campaign.(a) What is the horizontal asymptote?(b) What is limt∞ S(t)?(c) What is the horizontal asymptote for S'(t)?(d) What is limt∞ S'(t)? Sales volume t
Suppose that the demand x (in units) for a product is x = 10,000 - 100p, where p dollars is the market price per unit. Then the consumer expenditure for the product isE = px = 10,000p - 100p2For what market price will expenditure be greatest?
Find all horizontal and vertical asymptotes of the function 200х - - 500 f(x) = х+ 300
In Problem, use the indicated x-values on the graph of y = f (x) to find the following.Find the x-coordinates of three points of inflection. b def a
Use the following graph of y = f(x) and the indicated points to complete the chart. Enter +, -, or 0 according to whether f, f', and f" are positive, negative, or zero at each point.
In Problem, use the indicated x-values on the graph of y = f (x) to find the following.Find the x-coordinate of a horizontal point of inflection. b def a
Use the following figure to complete parts (a)–(c).(a) limx-∞ f(x) = ?(b) What is the vertical asymptote?(c) Find the horizontal asymptote. y 4 2 -8 -6 -2 4 -2 y = f(x)
In Problem(a) Find all critical values, including those at which f (x) is undefined.(b) Find the relative maxima and minima, if any exist.(c) Find the horizontal points of inflection, if any exist.(d) Sketch the graph.y = x2/3(x - 4)2
In Problem,(a) Find the critical values of the function and(b) Make a sign diagram and determine the relative maxima and minima.y = 15x3 - x5 + 7
In Problem, find any horizontal and vertical asymptotes for each function.y = 6x3/4x2 + 9
If f(6) = 10, f'(6) = 0, and f"(6) 5= -3, what can we conclude about the point on the graph of y = f (x) where x = 6? Explain.
The aged dependency ratio is defined as the number of individuals age 65 or older per 100 individuals ages 20–64. The aging of the baby boomer generation along with medical advancements and lifestyle changes for all individuals have caused this ratio to rise, shaping society’s plans for the
The revenue function for a product is R(x) = 164x dollars and the cost function for the product is C(x) = 0.01x2 + 20x + 300 dollarswhere x is the number of units produced and sold.(a) How many units of the product should be sold to obtain maximum profit?(b) What is the maximum possible profit?
The cost of producing x units of a product is given byC(x) = 100 + 20x + 0.01x2 dollarsHow many units should be produced to minimize average cost?
A firm sells 100 TVs per month at $300 each, but market research indicates that it can sell 1 more TV per month for each $2 reduction of the price. At what price will the revenue be maximized?
For the revenue function given by R(x) = 2800x + 8x2 - x3(a) find the maximum average revenue.(b) show that R̅(x) attains its maximum at an x-value where R̅(x) = M̅R̅.
An open-top box is made by cutting squares from the corners of a piece of tin and folding up the sides. If the piece of tin was originally 20 cm on a side, how long should the sides of the removed squares be to maximize the resulting volume?
For each function in Problem, find any horizontal and vertical asymptotes, and use information from the first derivative to sketch the graph.y = (x + 2/x - 3)2
A company estimates that it will need 784,000 items during the coming year. It costs $420 to manufacture each item, $2500 to prepare for each production run, and $5 per year for each item stored. How many units should be in each production run so that the total costs of production and storage are
A rectangular field with one side along a river is to be fenced. Suppose that no fence is needed along the river, the fence on the side opposite the river costs $40 per foot, and the fence on the other sides costs $10 per foot. If the field must contain 45,000 square feet, what dimensions will
For each function in Problem, find any horizontal and vertical asymptotes, and use information from the first derivative to sketch the graph.f (x) = 16x/x2 + 1
In Problem, use the graphs to find the following items.(a) Vertical asymptotes(b) Horizontal asymptotes(c) limx∞ f (x)(d) limx-∞ f (x) y y = f(x) -2 4 2.
From a tract of land, a developer plans to fence a rectangular region and then divide it into two identical rectangular lots by putting a fence down the middle. Suppose that the fence for the outside boundary costs $20 per foot and the fence for the middle costs $8 per foot. If each lot ontains
For each function in Problem, find any horizontal and vertical asymptotes, and use information from the first derivative to sketch the graph.f (x) = 4x2/x4 + 1
In Problem, use the graphs to find the following items.(a) Vertical asymptotes(b) Horizontal asymptotes(c) limx∞ f (x)(d) limx-∞ f (x) y y f(x) -2 -2 4. 2. 4. 2.
Find the relative maxima, relative minima, and points of inflection and sketch the graphs of the functions in Problem.y = x4 - 8x3 + 16x2
For each function in Problem(a) Find y' = f'(x).(b) Find the critical values.(c) Find the critical points.(d) Find intervals of x-values where the function is increasing and where it is decreasing.(e) Flassify the critical points as relative maxima, relative minima, or horizontal points of
A 4-pen kennel of 640 square feet is to be constructed as shown. The cost is $20 per foot for the sides and $5 per foot for the ends and dividers. What are the dimensions of the kennel that will minimize the cost?
The base of a rectangular box is to be twice as long as it is wide. The volume of the box is 256 cubic inches. The material for the top costs $0.50 per square inch, and the material for the sides and bottom costs $0.25 per square inch. Find the dimensions that will make the cost a minimum.
In Problem, a function and its first and second derivatives are given. Use these to find relative maxima, relative minima, and points of inflection; sketch the graph of each function.
In Problem, a function and its first and second derivatives are given. Use these to find any horizontal and vertical asymptotes, critical points, relative maxima, relative minima, and points of inflection. Then sketch the graph of each function. 1 y = 3 Vx + - 1 y' = 6 – 2x4/3 3x
In Problem, a function and its first and second derivatives are given. Use these to find any horizontal and vertical asymptotes, critical points, relative maxima, relative minima, and points of inflection. Then sketch the graph of each function.
In Problem, a function and its first and second derivatives are given. Use these to find any horizontal and vertical asymptotes, critical points, relative maxima, relative minima, and points of inflection. Then sketch the graph of each function.
Suppose that a company needs 60,000 items during a year and that preparation for each production run costs $800. Suppose further that it costs $8 to produce each item and $1.50 to store an item for one year. Use the inventory cost model to find the number of items in each production run that will
A company needs 150,000 items per year. It costs the company $720 to prepare a production run of these items and $14 to produce each item. If it also costs the company $1.50 per year for each item stored, find the number of items that should be produced in each run so that total costs of production
A company needs 450,000 items per year. Production costs are $1000 to prepare for a production run and $20 for each item produced. Inventory costs are $4 per item per year. Find the number of items that should be produced in each run so that the total costs of production and storage are minimized.
For each function in Problem, find the relative maxima, relative minima, and horizontal points of inflection; then sketch the graph. Check your graph with a graphing utility.y = 1/6x6 - x4 + 7
In Problem, both a function and its derivative are given. Use them to find critical values, critical points, intervals on which the function is increasing and decreasing, relative maxima, relative minima, and horizontal points of inflection; sketch the graph of each function.y = (x2 - 2x)2
The figure is a typical graph of worker productivity as a function of time on the job.(a) If P represents the productivity and t represents the time, write a mathematical symbol that represents the rate of change of productivity with respect to time.(b) Which of A, B, and C is the critical point
For each function in Problem, complete the following steps.(a) Use a graphing calculator to graph the function in the standard viewing window.(b) Analytically determine the location of any asymptotes and extrema.(c) Graph the function in a viewing window that shows all features of the graph. State
In Problem, both a function and its derivative are given. Use them to find critical values, critical points, intervals on which the function is increasing and decreasing, relative maxima, relative minima, and horizontal points of inflection; sketch the graph of each function. x/3 – 2 f(x) = x –
The figure shows the graph of a marginal profit function for a company. At what level of sales will profit be maximized? Explain. MP 50 25 + 200 400 600 -25 Units Dollars per unit
An entrepreneur starts new companies and sells them when their growth is maximized. Suppose that the annual profit for a new company is given bywhere P is in thousands of dollars and x is the number of years after the company is formed. If she wants to sell the company before profits begin to
MMR II Extreme Bike Shop sells 54 of its most popular mountain bikes per month at a price of $3080 each. Market research indicates that MMR II could sell one more of these bikes for each $20 price reduction. At what selling price will MMR II maximize the revenue from these bikes?
The table gives the percent of men 65 years or older in the workforce for selected years from 1920 and projected to 2030.(a) With x as the number of years after 1900, find the cubic function that models these data. Report the model with three significant digits.(b) Use the reported model to
The figure shows a barograph readout of the barometric pressure as recorded by Georgia Southern University’s meteorological equipment. The figure shows a tremendous drop in barometric pressure on March 13, 1993.(a) If B(t) is barometric pressure expressed as a function of time, as shown in the
The figure shows the graph of a quadratic revenue function and a linear cost function.(a) At which of the four x-values shown is the distance between the revenue and the cost greatest?(b) At which of the four x-values shown is the profit largest?(c) At which of the four x-values shown is the slope
The figure shows the graph of revenue function y = R(x) and cost function y = C(x).(a) At which of the four x-values shown is the profit largest?(b) At which of the four x-values shown is the slope of the tangent to the revenue curve equal to the slope of the tangent to the cost curve?(c) What is
The figure shows the Dow Jones Industrial Average for all of 2001, the year of the terrorist attacks on New York City and Washington, D.C.(a) Approximate when during 2001 the Dow reached its absolute maximum for that year.(b) When do you think the Dow reached its absolute minimum for this period?
The numbers of millions of Social Security beneficiaries for selected years from 1950 and projected to 2030 are given in the following table.(a) Find the cubic function that models these data, with x equal to the number of years past 1950. Report the model with 3 significant digits.(b) Find the
In Problem, two graphs are given. One is the graph of f, and the other is the graph of f'. Decide which is which and explain your reasoning. 3 3. -8-6-4 +++ 2 4 6 8 -8-6-4 -1 4 6 8 -2+ -3+ 4.
Analysis of daily output of a factory shows that, on average, the number of units per hour y produced after t hours of production isy = 70t + 0.5t2 - t3, 0 ≤ t ≤ 8(a) Find the critical values of this function.(b) Which critical values make sense in this particular problem?(c) For which values
For the years from 2002 and projected to 2024, the U.S. per capita out-of-pocket cost for health care C (in dollars) can be modeled bythe functionC(t) = 0.908t3 - 25.3t2 + 549t + 4540where t is the number of years past 2000.(a) When does the rate of change of health care costs per capita reach its
Because the derivative of a function represents both the slope of the tangent to the curve and the instantaneous rate of change of the function, it is possible to use information about one to gain information about the other. In Problem, use the graph of the function y = f (x) given in Figure(a) At
In Problem, find the derivative of each function.h(x) = 7/x7 - 3/x3 + 8√x
At the indicated point, for each function in Problems, find(a) The slope of the tangent line.(b) The instantaneous rate of change of the function.y = √x3 + 1 at (2, 3)
In Problem, use the numerical derivative feature of a graphing calculator to approximate the given second derivatives.f "(3) for f (x) = 1/√x2 + 7
In Problem, do the following for each function f (x).(a) Find f'(x) and f"(x).(b) Graph f (x), f'(x), and f"(x) with a graphing utility.(c) Identify x-values where f"(x) = 0, f"(x) > 0, and f"(x) < 0.(d) Identify x-values where f '(x) has a maximum point or a minimum point, where f'(x) is
Find dC/dx for C = 5x4 - 2x2 + 1/x3 + 1.
Find y' if y = ( x + 1/1 - x2)3.
Find y' if y = x√x2 - 4.
In Problem, find the second derivatives.y = x4 - 1/x
In Problems, find the fifth derivatives. d²y If dx? d'y find x + 1 dx
In Problem, total revenue is in dollars and x is the number of units.Suppose the total revenue for a certain product is given by R(x) = 36x - 0.01x2.(a) Graph the marginal revenue function for this product.(b) At what value of x will total revenue be maximized?(c) What is the maximum revenue?
In Problem, find the indicated derivative.Find y(4) if y = x6 - 15x3.
In Problem, find the indicated derivatives and simplify. (x + 1)(x – 2) f'(x) for f(x) x + 1
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