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mathematics
calculus

Questions and Answers of Calculus

The numbers of millions of Social Security beneficiaries for selected years and projected into the future are given in the table.(a) Find the cubic function that models these data, with x equal to
For women age 16 and older, the table gives the percent of this group that participates in the U.S. workforce for selected years from 1950 and projected to 2050.(a) With x as the number of years past
The figure shows the daily Dow Jones Industrial Average (DJIA) and its 30-day moving average from late July to early November. Use the figure to complete the following.(a) Approximate the absolute
The graph shows the number of workers, W=f (t), still in the workforce per Social Security beneficiary (historically and projected into the future) as a function of time t, in calendar years with
(a) If the total revenue function for a hammer is R=36x-0.01x2, then sale of how many hammers, x, will maximize the total revenue in dollars? Find the maximum revenue. (b) Find the maximum revenue
(a) If the total revenue function for a blender is R(x) = 25x-0.05x2, sale of how many units, x, will provide the maximum total revenue in dollars? Find the maximum revenue.(b) Find the maximum
If the total revenue function for a computer is R(x) = 2000x-20x2-x3, find the level of sales, x, that maximizes revenue and find the maximum revenue in dollars.
A firm has total revenues given by R(x) = 2800x-8x2-x3 dollars for x units of a product. Find the maximum revenue from sales of that product.
An agency charges $100 per person for a trip to a concert if 70 people travel in a group. But for each person above the 70, the charge will be reduced by $1.00. How many people will maximize the
The manufacturer of GRIPPER tires modeled its return to sales from television advertising expenditures in two regions, as follows: Region 1: S1 =30 + 20x1 - 0.4x12 Region 2: S2 =20 + 36x2 -
For Problems 9 and 10, consider that when medicine is administered, reaction (measured in change of blood pressure or temperature) can be modeled bywhere c is a positive constant and m is the amount
An inferior product with a large advertising budget sells well when it is introduced, but sales fall as people discontinue use of the product. Suppose that the weekly sales S are given byWhere S is
A newly released film has its weekly revenue given byWhere R is in millions of dollars and t is in weeks.(a) After how many weeks will the weekly revenue be maximized?(b) What is the maximum weekly
Suppose that the percent p (as a decimal) of people who could correctly identify two of eight defendants in a drug case t days after their trial began is given byFind the number of days before the
Suppose that in an election year the proportion p of voters who recognize a certain candidate's name t months after the campaign started is given byAfter how many months is the proportion maximized?
Two equal rectangular lots are to be enclosed by fencing the perimeter of a rectangular lot and then putting a fence across its middle. If each lot is to contain 1200 square feet, what is the minimum
The running yard for a dog kennel must contain at least 900 square feet. If a 20-foot side of the kennel is used as part of one side of a rectangular yard with 900 square feet, what dimensions
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 $20 per foot, and the fence on the
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
A rectangular area is to be enclosed and divided into thirds. The family has $800 to spend for the fencing material. The outside fence costs $10 per running foot installed, and the dividers cost $20
A ball thrown into the air from a building 100 ft high travels along a path described byWhere y is its height in feet and x is the horizontal distance from the building in feet. What is the maximum
A kennel of 640 square feet is to be constructed as shown. The cost is $4 per running foot for the sides and $1 per running foot for the ends and dividers. What are the dimensions of the kennel that
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 $0.10 per square inch and the material for the sides and
According to B. F. Visser, the velocity v of air in the trachea during a cough is related to the radius r of the trachea according to v=ar2 (r0 - r) where a is a constant and r0 is the radius of the
Suppose that a company needs 1,500,000 items during a year and that preparation for each production run costs $600. Suppose also that it costs $15 to produce each item and $2 per year to store an
Suppose that a company needs 60,000 items during a year and that preparation for each production run costs $400. Suppose further that it costs $4 to produce each item and $0.75 to store an item for
A company needs 150,000 items per year. It costs the company $360 to prepare a production run of these items and $7 to produce each item. If it also costs the company $0.75 per year for each item
A company needs 450,000 items per year. Production costs are $500 to prepare for a production run and $10 for each item produced. Inventory costs are $2 per item per year. Find the number of items
A rectangular box with a square base is to be formed from a square piece of metal with 12-inch sides. If a square piece with side x is cut from each corner of the metal and the sides are folded up to
(a) A square piece of cardboard 36 centimeters on a side is to be formed into a rectangular box by cutting squares with length x from each corner and folding up the sides. What is the maximum volume
The owner of an orange grove must decide when to pick one variety of oranges. She can sell them for $24 a bushel if she sells them now, with each tree yielding an average of 5 bushels. The yield
(a) A box with an open top and a square base is to be constructed to contain 4000 cubic inches. Find the dimensions that will require the minimum amount of material to construct the box.(b) A box
A printer has a contract to print 100,000 posters for a political candidate. He can run the posters by using any number of plates from 1 to 30 on his press. If he uses x metal plates, they will
A vacationer on an island 8 miles offshore from a point that is 48 miles from town must travel to town occasionally. (See the figure.) The vacationer has a boat capable of traveling 30 mph and can go
By using U.S. Department of Energy data for selected years from 2011 and projected to 2040, the U.S. onshore oil reserves in the lower 48 states (in billions of barrels) can be modeled by theFunction
The velocity v of an autocatalytic reaction can be represented by the equation v = x (a - x) where a is the amount of material originally present and x is the amount that has been decomposed at any
Analysis of daily output of a factory during an 8-hour shift shows that the hourly number of units y produced after t hours of production is y=70t + ½ t2 - t3, 0 ≤ t ≤8 (a) After how many hours
A time study showed that, on average, the productivity of a worker after t hours on the job can be modeled by P = 27t + 6t2 - t3 , 0 ≤ t ≤ 8 where P is the number of units produced per hour.
Suppose that the monthly cost in dollars of mining a certain ore is related to the number of pieces of equipment used, according toWhere x is the number of pieces of equipment used. Using how many
For Problems 9 and 10, consider that when medicine is administered, reaction (measured in change of blood pressure or temperature) can be modeled byWhere c is a positive constant and m is the amount
In Problems 1-4, a function and its graph are given. Use the graph to find each of the following, if they exist. Then confirm your results analytically.(a) Vertical asymptotes(d) Horizontal
For each function in below Problems find any horizontal and vertical asymptotes, and use information from the first derivative to sketch the graph.1. f(x) = 2x+2 / x-32. f(x) = 5x - 15 / x+23. y = x2
In below Problems 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
In below Problems, a function and its graph are given.(a) Use the graph to estimate the locations of any horizontal or vertical asymptotes.(b) Use the function to determine precisely the locations of
For each function in below Problems 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
The percent p of particulate pollution that can be removed from the smokestacks of an industrial plant by spending C dollars is given by p = 100cC / 7300 + C (a) Find any C-values at which the rate
The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by p = 100C / 8100 + C (a) Find any C-values at which the rate of
A recently released film has its weekly revenue given by R (t) = 50t / t2 + 36, t ≥ 0 where R(t) is in millions of dollars and t is in weeks. (a) Graph R(t). (b) When will revenue be maximized? (c)
If the total daily cost, in dollars, of producing plastic rafts for swimming pools is given by C(x) = 500 + 8x + 0.05x2 where x is the number of rafts produced per day, then the average cost per raft
If x is the wind speed in miles per hour and is greater than or equal to 5, then the wind chill (in degrees Fahrenheit) for an air temperature of 00F can be approximated by the function f (x) =
An entrepreneur starts new companies and sells them when their growth is maximized. Suppose that the annual profit for a new company is given by P(x) = 22 - ½ x - 18 / x + 1 where P is in thousands
The figure is a typical graph of worker productivity per hour P as a function of time t on the job.(a) What is the horizontal asymptote?(b) What is Limx †’ ˆž P (t)?(c) What is
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 Limx → ∞ S(t)? (c) What is the
For selected years from 1950 and projected to 2050, the table shows the percent of total U.S. workers who were female.Assume these data can be modeled with the function p(t) = 78.6t + 2090 / 1.38t +
Obesity (BMI ‰¥ 30) is a serious problem in the United States and expected to get worse. Being overweight increases the risk of diabetes, heart disease, and many other ailments, but the
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
In below Problems, find any horizontal and vertical asymptotes for each function. 1. y = 2x / x-3 2. y = 3x-1 / x +5 3. y = x+1/ x2 -4 4. y = 4x / 9-x2 5. y = 3x3 - 6 / x2 +4
In Problems 1-4, find all critical points and determine whether they are relative maxima, relative minima, or horizontal points of inflection. 1. y= -x2 2. p = q2 - 4q - 5 3. f(x) = 1 - 3x + 3x2 -
Is the graph of y = x4 - 3x3 + 2x - 1 concave up or concave down at x =2?
Find intervals on which the graph of y = x4 - 2x3 - 12x2 +6 is concave up and intervals on which it is concave down, and find points of inflection.
Find the relative maxima, relative minima, and points of inflection of the graph of y = x3 - 3x2 - 9x + 10
In below Problems, find any relative maxima, relative minima, and points of inflection, and sketch each graph. 1. y = x3 - 12x 2. y =2 + 5x3 - 3x5
Given R = 280x- x2, find the absolute maximum and minimum for R when (a) 0 ≤ x ≤ 200 and (b) 0 ≤ x ≤ 100
Given y = 6400x - 18x2 - x3 / 3, Find the absolute maximum and minimum for y when (a) 0 ≤ x ≤ 50 and (b) 0 ≤ x ≤ 100
In below Problems, find any horizontal asymptotes and any vertical asymptotes.1.2.
In below Problems :(a) Find any horizontal and vertical asymptotes.(b) Find any relative maxima and minima.(c) Sketch each graph.1.2. 3.
In below Problems, a function and its graph are given.(a) Use the graph to determine (estimate) x-values where f'(x) > 0, where f'(x) (b) Use the graph to determine x-values where f''(x) >
In Problems below, f '(x) and its graph are given.(a) Use the graph of f '(x) to determine (estimate) where the graph of f (x) is increasing, where it is decreasing, and where it has relative
In below Problems, f (x) and its graph are given.(a) Use the graph to determine (estimate) where the graph of f (x) is concave up, where it is concave down, and where it has points of inflection.(b)
In below Problems cost, revenue, and profit are in dollars and x is the number of units. 1. Suppose the total cost function for a product is C(x) = 3x2 + 15x + 75 How many units will minimize the
Suppose the productivity P of an individual worker (in number of items produced per hour) is a function of the number of hours of training t according toFind the number of hours of training at which
The figure shows a typical graph of output y (in thousands of dollars) as a function of capital Investment I (also in thousands of dollars)(a) Is the point of diminishing returns closest to the point
If in Problem 40 the mountain bikes cost the shop $680 each, at what selling price will MMR II's profit be a maximum?
Suppose that for a product in a competitive market, the demand function is p =1200 - 2x and the supply function is p =200 +2x, where x is the number of units and p is in dollars. A firm's average
The monthly demand function for x units of a product sold at $p per unit by a monopoly is p=800 - x, and its average cost is C̅=200+x. (a) Determine the quantity that will maximize profit. (b) Find
Suppose that in a monopolistic market, the demand function for a commodity iswhere x is the number of units and p is in dollars. If a company's average cost function for this commodity is find the
The reaction R to an injection of a drug is related to the dose x (in milligrams) according toFind the dose that yields the maximum reaction.
The number of parts produced per hour by a worker is given by n= 4 + 3t2 - t3where t is the number of hours on the job without a break. If the worker starts at 8 a.m., when will she be at maximum
Population estimates show that the equation P=300+10t-t2 represents the size of the graduating class of a high school, where t represents the number of years after 2015, 0 ≤ t ≤ 10. What will be
Suppose that an observatory is to be built between cities A and B, which are 30 miles apart. For the best viewing, the observatory should be located where the night brightness from these cities is
A playpen manufacturer wants to make a rectangular enclosure with maximum play area. To remain competitive, he wants the perimeter of the base to be only 16 feet. What dimensions should the playpen
In below Problems (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
A page is to contain 56 square inches of print and have a ¾ @inch margin at the bottom and 1-inch margins at the top and on both sides. Find the dimensions that minimize the size of the page (and
The reaction R to an injection of a drug is related to the dose x, in milligrams, according toThe sensitivity to the drug is defined by dR/ dx. Find the dose that maximizes sensitivity.
For the years from 2000 and projected to 2018, the U.S. per capita out-of-pocket cost for health care C (in dollars) can be modeled by the function C(t) = 0.118t3 - 2.51t2 + 40.2t +677 Where t is the
A company needs to produce 288,000 items per year. Production costs are $1500 to prepare for a production run and $30 for each item produced. Inventory costs are $1.50 per year for each item stored.
Suppose the total cost of producing x units of a product is given by C(x) = 4500 + 120x + 0.05x2 (a) Find any asymptotes of the average cost function C̅(x) = C(x)/x. (b) Graph the average cost
Suppose a company's percent share of the market (actual and projected) for a new product t quarters after its introduction is given by(a) Find the company's market share when the product is
Find the derivatives of the functions in Problems 1-5. 1. f (x) = 4 ln x 2. y = 3 ln x 3. y = ln 8x 4. y = ln 5x 5. y = ln x4
In each of Problems 1-3, find the derivative of the function in part (a). Then find the derivative of the function in part (b) or show that the function in part (b) is the same function as that in
In Problems 1-4, find y`. 1. y = x - ln x 2. y = x2 ln (2x + 3) 3. y = ln x / x 4. y = 1 + ln x / x2
In Problems 1-3, find the relative maxima and relative minima, and sketch the graph with a graphing utility to check your results. 1. y = x ln x 2. y = x2 ln x 3. y = x2 - 8 ln x
Suppose that the total cost (in dollars) for a product is given by C(x) = 1500 + 200 ln (2x + 1) where x is the number of units produced. (a) Find the marginal cost function. (b) Find the marginal
If money is invested at the constant rate r, the time to increase the investment by a factor x is t = ln x / r (a) At what rate dt / dx is the time changing at x = 2? (b) What happens to dt / dx as x
The total revenue, in dollars, from the sale of x units of a product is given by R(x) = 2500x / ln (10x + 10) (a) Find the marginal revenue function. (b) Find the marginal revenue when 100 units are
Suppose that the supply of x units of a product at price p dollars per unit is given by p = 10 + 50 ln (3x + 1) (a) Find the rate of change of supply price with respect to the number of units
The demand function for a product is given by p = 4000 ln (x =10), where p is the price per unit in dollars when x units are demanded. (a) Find the rate of change of price with respect to the number
The pH of a solution is given by pH = - log [H+] where [H+] is the concentration of hydrogen ions (in gram atoms per liter). What is the rate of change of pH with respect to [H+]?
Concentration (in mg/ml) in the bloodstream of a certain drug is related to the time t (in minutes) after an injection and can be calculated using y in the equation y = A ln (t) - Bt + C where A, B,
The loudness of sound (L, measured in decibels) perceived by the human ear depends on intensity levels (I) according to L = 10 log (I / I0) where I0 is the standard threshold of audibility. If x =

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