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mathematics
calculus

Questions and Answers of Calculus

1. Find y' if (x + 1/1 - x2)3. 2. Find y' if y = x √x2 - 4.
Find dy / dx if y = x/3√3x - 1.
Find the second derivatives. 1. y = √x - x2 2. y = x4 - 1/x
Find the fifth derivatives.1. y = (2x + 1)42. y = (1 - x)6 / 243. If dy/dx = √x2 - 4, find d3y/dx3.
Assume that a company's monthly total revenue and total cost (both in dollars) are given byR(x) = 140x - 0.01x2 and C (x) = 60x + 70,000where x is the number of units. (Let P(x) denote the profit
If RÌ…(x) = R(x) /x and CÌ…(x) = C(x)/x are, respectively, the company's average revenue per unit and average cost per unit, find
Evaluate and explain the meanings of
For the period from 1990 to 2030, find and interpret the annual average rate of change of(a) Elderly men in the workforce(b) Elderly women in the workforce.The graph shows the percent of elderly men
(a) Find the annual average rate of change of the percent of elderly men in the workforce from 1970 to 1980 and from 2030 to 2040.(b) Find the annual average rate of change of the percent of elderly
Suppose that the demand for x units of a product is given by x = (100 p) - 1, where p is the price per unit of the product. Find and interpret the rate of change of demand with respect to price if
Thunderstorms severe enough to produce hail develop when an upper-level low (a pool of cold air high in the atmosphere) moves through a region where there is warm, moist air at the surface. These
In a 100-unit apartment building, when the price charged per apartment rental is (830 + 30x) dollars, then the number of apartments rented is 100 - x and the total revenue for the building is R(x) =
Suppose the productivity of a worker (in units per hour) after x hours of training and time on the job is given by(a) Find and interpret P(20). (b) Find and interpret P'(20).
1. The demand q for a product at price p is given by q = 10,000 - 50 √0.02p2 + 500 Find the rate of change of demand with respect to price. 2. The number of units x of a product that is supplied at
Suppose an object moves so that its distance to a sensor, in feet, is given by s(t) = 16 + 140t + 8√t where t is the time in seconds. Find the acceleration at time t = 4 seconds.
Suppose a company's profit (in dollars) is given by P(x) = 70x - 0.1x2 - 5500 where x is the number of units. Find and interpret P'(300) and P"(300).
In Problems below use the indicated points on the graph of yf (x) to identify points at whichf (x) has(a) a relative maximum,(b) a relative minimum, and(c) a horizontal point of inflection.1.2. 3. 4.
For each function and graph in below Problems:(a) Estimate the coordinates of the relative maxima, relative minima, or horizontal points ofinflection by observing the graph.(b) Use yf (x) to find the
For each function in below Problems: (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
For each function and graph in Problems below(a) Use the graph to identify x-values for which y' Ëƒ 0, y'Ë‚ 0, y'=0, and y' does not exist.(b) Use the derivative to check your
For each function in below Problems find the relative maxima, relative minima, horizontal points of inflection, and sketch the graph check your graph with a graphing utility. 1. y = 1/3x3 - x2 + x
In below Problems, 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,
In below Problems, use the derivative to locate critical points and determine a viewing window that shows all features of the graph. Use a graphing utility to sketch a complete graph. 1. f(x) = x3 -
In each of Problems below a graph of f (x) is given. Use the graph to determine the critical values of f (x), where f (x) is increasing, where it is decreasing, and where it has relative maxima,
In Problems 47 and 48, 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.1.2.
Suppose that the daily sales (in dollars) t days after the end of an advertising campaign are given by S = 1000 + 400/ t+ 1 , t ≥ 0 Does S increase for all t ≥ 0, decrease for all t ≥ 0, or
In Problems below use the sign diagram for f '(x) to determine(a) the critical values of f (x),(b) intervals on which f (x) increases,(c) intervals on which f (x) decreases,(d) x-values at which
Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that its monthly sales volume y (in thousands of dollars) is related to the price p (in dollars) of an oil change byIs y
A time study showed that, on average, the productivity of a worker after t hours on the job Can be modeled by p(t) = 27t + 6t2 - t3, 0 ≤ t ≤ 8 Where P is the number of units produced per
Analysis of daily output of a factory shows that, on average, the number of units per hour yProduced after t hours of production is(a) Find the critical values of this function. (b) Which critical
Suppose that the average cost, in dollars, of producing a shipment of a certain product isWhere x is the number of machines used in the production process. (a) Find the critical values of this
Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given byWhere x is the number of machines used.(a) Find the critical values
Suppose the weekly marginal revenue function for selling x units of a product is given by the graph in the figure.(a) At each of x =150, x = 250, and x = 350, what is happening to revenue? (b) Over
Suppose that the rate of change f' (x) of the average annual earnings of new car salespersons is shown in the figure.(a) If a, b, and c represent certain years, what is happening to f (x), the
The weekly revenue of a certain recently released film is given byWhere R is in millions of dollars and t is in weeks. (a) Find the critical values. (b) For how many weeks will weekly revenue
Suppose that the concentration C of a medication in the bloodstream t hours after an injection is given by(a) Determine the number of hours before C attains its maximum. (b) Find the maximum
Suppose that the proportion P of voters who recognize a candidate's name t months after the start of the campaign is given by(a) How many months after the start of the campaign is recognition at its
The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by(a) For what t-values is x increasing? (b) Find the t-value at which x is maximum. (c)
In 2013, worldwide cell phone subscribership's surpassed the world's total population of 6.8 billion. Using data from 2009 and projected to 2020, the billions of subscribership's can be modeled by
The economic dependency ratio is defined as the number of persons in the total population who are not in the workforce per 100 in the workforce. Since 1960, Baby Boomers in the workforce and a
The following table shows the millions of individuals, ages 15 to 59, in China's labor pool for selected years from 1975 and projected to 2050.(a) Find the cubic function L(t) that best models the
The table shows the total energy supply from crude oil products, in quadrillion BTUs, for selected years from 2010 and projected to 2040.(a) Find the cubic function that is the best model for the
The table shows the total employment in U.S. manufacturing, in millions, for selected years from 2010 and projected to 2040.(a) Find the cubic function that is the best model for the data. Use t as
In Problems below,(a) find the critical values of the function, and(b) make a sign diagram and determine the relative maxima and minima.1. y = 2x3 - 12x2 + 62. y = x3 - 3x2 + 6x +13. y = 2x5 + 5x4 -
In Problems 1 and 2, determine whether each function is concave up or concave down at the indicated points. 1. f (x) = x3 - 3x2+1 at (a) x = -2 (b) x =3 2. f (x) = x3 + 6x - 4 at (a) x = -5 (b)
Find the relative maxima, relative minima, and points of inflection, and sketch the graphs of the functions, in below Problems1. y = x2 - 4x + 22. y = x3 - x24. y = x3 - 3x2 +6 5. y = x4 -16x2
In below Problems 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.1.3.4.
In Problems below , a function and its graph are given.1.2. (a) From the graph, estimate where f ''(x) Ëƒ 0, where f ''(x) Ë‚ 0, and where f ''(x)=0. (b) Use (a) to decide where f
In below Problems , f (x) and its graph are given. Use the graph of f (x) to determine the following.1.2.(a) Where is the graph of f (x) concave up and where is it concave down?(b) Where does f (x)
In Problems below use the graph shown in the figure and identify points from A through I that satisfy the given conditions.1.0 (b) f'(x) 0 (d) f'(x) >0 and f"(x) = 0 (e) f'(x) = 0 an" alt = "In
In Problems below, a graph is given. Tell where f (x) is concave up, where it is concave down, and where it has points of inflection on the interval - 2 Ë‚ x Ë‚ 2, if the given graph is the
In below Problems, use the indicated x-values on the graph of y =f (x) to find the following.1. Find intervals over which the graph is concave down. 2. Find intervals over which the graph is concave
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
The figure shows the growth of a population as a function of time.(a) If P represents the population and t represents the time, write a mathematical symbol that Represents the rate of change (growth
The figure shows the daily sales volume S as a function of time t since an ad campaign began.(a) Which of A, B, and C is the point of inflection for the graph? (b) On which side of C is d2 S/dt2
The figure shows the oxygen level P (for purity) in a lake t months after an oil spill.(a) Which of A, B, and C is the point of inflection for the graph? (b) On which side of C is d2 P/ dt2
Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function P(t): P(t) = 27t +12t2 - t3 (a)Find the number of hours before
Acording to Poiseuille's law, the speed S of blood through an artery of radius r at a distance x from the artery wall is given by S= k [ r2 - (r-x)2] where k is a constant. Find the distance x that
Suppose that a company's daily sales volume attributed to an advertising campaign is given by(a) Find how long it will be before sales volume is maximized.(b) Find how long it will be before the rate
Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill is given by(a) Find how long it will be before the oxygen level reaches its minimum.(b) Find how long it
Energy use per capita indexed to 1995 means that per capita energy use for any year is viewed as a percent of per capita use in 1995. Using U.S. Department of Energy data for selected years from 1985
The figure gives the percent of the U.S. population that was foreign born for selected years from 1910 and projected to 2020.(a) Find the cubic function that is the best fit for the data. Use x = 0
The table gives the size of the U.S. civilian labor force (in millions) for selected years from 1950 and projected to 2050.(a) Find a cubic function that models these data, with x equal to the number
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 =0 representing 1900, find the cubic function that models these
The table gives the annual cost per person age 65 and older for home health care from 2006 and projected through 2021. While these annual costs per person may seem modest, the majorities of older
In below Problem, a function and its graph are given. Use the second derivative to determine intervals on which the function is concave up, to determine intervals on which it is concave down, and to
In Problems 1-4, find the absolute maxima and minima for f (x) on the interval [a, b].1. f(x) = x3 - 2x2 -4x + 2, [ -1, 3]2. f(x) = x3 - 3x + 3, [ -3, 1.5]3. f(x) = x3 + x2 - x + 1, [ -2, 0]4. f(x) =
A company handles an apartment building with 70 units. Experience has shown that if the rent for each of the units is $1080 per month, all the units will be filled, but 1 unit will become vacant for
A cable TV company has 4000 customers paying $110 each month. If each $1 reduction in price attracts 50 new customers, find the price that yields maximum revenue. Find the maximum revenue.
If club members charge $5 admission to a classic car show, 1000 people will attend, and for each $1 increase in price, 100 fewer people will attend. What price will give the maximum revenue for the
The function R̅(x)=R̅(x) x defines the average revenue for selling x units. ForR(x)=2000x+20x2+x3(a) find the maximum average revenue.(b) show that R̅(x) attains its maximum at an x-value where
For the revenue function given by R(x) = 280x+ 8x2 +x3(a) find the maximum average revenue.(b) Show that R̅(x) attains its maximum at an x-value where R̅(x) = M̅R̅.
If the total cost function for a lamp is C(x)=250+33x+0.1x2 dollars, producing how many units, x, will result in a minimum average cost per unit? Find the minimum average cost.
If the total cost function for a product is C(x) = 300+10x+0.03x2 dollars, producing how many units, x, will result in a minimum average cost per unit? Find the minimum average cost.
If the total cost function for a product is C(x) = 810+0.1x2 dollars, producing how many units, x, will result in a minimum average cost per unit? Find the minimum average cost.
If the total cost function for a product is C(x)= 250+6x+0.1x2 dollars, producing how many units, x, will minimize the average cost? Find the minimum average cost.
If the total cost function for a product is C(x) = 100(0.02x+4)3 dollars, where x represents the number of hundreds of units produced, producing how many units will minimize average cost? Find the
If the total cost function for a product is C(x)= (x+5)3 dollars, where x represents the number of hundreds of units produced, producing how many units will minimize average cost? Find the minimum
For the cost function C(x)=25+13x+x2, show that average costs are minimized at the x-value where C̅(x)=M̅C̅
For the cost function C(x)=300+10x+0.03x2, show that average costs are minimized at the x-value whereC̅(x)=M̅C̅
The graphs in Problems 23 show total cost functions. For each problem:(a) Explain how to use the total cost graph to determine the level of production at which average cost is minimized.(b) Determine
The graphs in Problems 24 show total cost functions. For each problem:(a) Explain how to use the total cost graph to determine the level of production at which average cost is minimized.(b) Determine
If the profit function for a product is P(x) = 5600x + 85x2-x3-200,000 dollars, selling how any items, x, will produce a maximum profit? Find the maximum profit.
If the profit function for a commodity is P=6400x - 18x2-1/3x3-40,000 dollars, selling how many units, x, will result in a maximum profit? Find the maximum profit.
A manufacturer estimates that its product can be produced at a total cost of C (x) =45,000+100x+x3 dollars. If the manufacturer's total revenue from the sale of x units is R(x)=4600x dollars,
A product can be produced at a total cost C(x) = 800+100x2+x3 dollars, where x is the number produced. If the total revenue is given by R(x) = 60,000x-50x2 dollars, determine the level of production,
A firm can produce only 1000 units per month. The monthly total cost is given by C(x) =300+200x dollars, where x is the number produced. If the total revenue is given by R(x) =250x-1/100x2 dollars,
A firm can produce 100 units per week. If its total cost function is C=500+1500x dollars and its total revenue function is R=1600x-x2 dollars, how many units, x, should it produce to maximize its
A company handles an apartment building with 50 units. Experience has shown that if the rent for each of the units is $720 per month, all of the units will be filled, but 1 unit will become vacant
A travel agency will plan a tour for groups of size 25 or larger. If the group contains exactly 25 people, the cost is $500 per person. However, each person's cost is reduced by $10 for each
A firm has monthly average costs, in dollars, given bywhere x is the number of units produced per month. The firm can sell its product in a competitive market for $1600 per unit. If production is
A small business has weekly average costs, in dollars, ofWhere x is the number of units produced each week. The competitive market price for this business's product is $46 per unit. If production is
The weekly demand function for x units of a product sold by only one firm is p=600-1/2 x dollars, and the average cost of production and sale is C̅300+2x dollars. (a) Find the quantity that will
The monthly demand function for x units of a product sold by a monopoly is p=8000-x dollars, and its average cost is C̅=4000+5x dollars. (a) Determine the quantity that will maximize profit. (b)
The monthly demand function for a product sold by a monopoly is p=1960-1/3 x2 dollars, and the average cost is C̅1000+2x+x2 dollars. Production is limited to 1000 units and x is in hundreds of
The monthly demand function for x units of a product sold by a monopoly is p=5900-1/2 x2 dollars, and its average cost is C̅=3020+2x dollars. If production is limited to 100 units, find the number
An industry with a monopoly on a product has its average weekly costs, in dollars, given byThe weekly demand for x units of the product is given by p=120-0.015x dollars. Find the price the industry
A large corporation with monopolistic control in the marketplace has its average daily costs, in dollars, given byThe daily demand for x units of its product is given by p=60,000-50x dollars. Find
Coastal Soda Sales has been granted exclusive market rights to the upcoming Beaufort Seafood Festival. This means that during the festival Coastal will have a monopoly, and it is anxious to take
A retiree from a large Atlanta financial services firm decides to keep busy and supplement her retirement income by opening a small upscale folk art company near Charleston, South Carolina. The

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