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mathematics
calculus

Questions and Answers of Calculus

In Problems 1-4, assume that x and y are differentiable functions of t. In each case, find dx/dt given that x = 5, y = 12, and dy/dt = 2. 1. x2 + y2 = 169 2. y2x2 = 119 3. y22xy + 24 4. x2(y - 6) =
If x2 + y2 = z2, find dy/dt when x = 3, y = 4, dx/dt = 10, and dz/dt =2.
(a) Find the elasticity of the demand function p + 4q = 80 at (10, 40). (b) How will a price increase affect total revenue?
Suppose the weekly demand function for a product is q = 5000 / 1 + e2p - 1 where p is the price in thousands of dollars and q is the number of units demanded. What is the elasticity of demand when
The demand functions for specialty steel products are given, where p is in dollars and q is the number of units. For both problems: (a) Find the elasticity of demand as a function of the quantity
South West Electronics Corporation (SWEC) designs high-tech business and residential security systems. The company's marketing analyst has been assigned to analyze market demand for SWEC's
The owner and manager of Pleasantville Deli is considering the expansion of current menu offerings to include a new line of take-out sandwiches. The deli serves primarily the Pleasantville business
If the demand function for a fixed period of time is given by p = 38 - 2q and the supply function before taxation is p = 8 + 3q, what tax per item will maximize the total tax revenue? p is the price
If the demand and supply functions for a product are p = 800 - 2q and p = 100 + 0.5q, respectively, find the tax per unit t that will maximize the tax revenue T. p is the price per unit in dollars
If the demand and supply functions for a product are p = 2100 - 3q and p = 300 + 1.5q, respectively, find the tax per unit t that will maximize the tax revenue T. p is the price per unit in dollars
If the weekly demand function is p = 200 - 2q2 and the supply function before taxation is p = 20 + 3q, what tax per item will maximize the total tax revenue? p is the price per unit in dollars and q
(a) Find the elasticity of the demand function 2p + 3q = 150 at the price p = 15. (b) How will a price increase affect total revenue?
If the monthly demand function is p = 7230 - 5q2 and the supply function before taxation is p3030q2, what tax per item will maximize the total revenue? p is the price per unit in dollars and q is
Suppose the weekly demand for a product is given by p + 2q = 840 and the weekly supply before taxation is given by p = 0.02q2 + 0.55q + 7.4. Find the tax per item that produces maximum tax revenue.
If the daily demand for a product is given by the function p + q = 1000 and the daily supply before taxation is p = q2/30 + 2.5q + 920, find the tax per item that maximizes tax revenue. Find the tax
If the demand and supply functions for a product are P = 2100 - 10q - 0.5q2 and p = 300 + 5q + 0.5q2, respectively, find the tax per unit t that will maximize the tax revenue T. p is the price per
If the demand and supply functions for a product are p = 5000 - 20q - 0.7q2 and p = 500 + 10q + 0.3q2, respectively, find the tax per unit t that will maximize the tax revenue T.
(a) Find the elasticity of the demand function p2 + 2p + q = 49 at p = 6. (b) How will a price increase affect total revenue?
(a) Find the elasticity of the demand function pq = 81 at p = 3. (b) How will a price increase affect total revenue?
Suppose that the demand for a product is given by pq + p = 5000. (a) Find the elasticity when p = $50 and q = 99. (b) Tell what type of elasticity this is: unitary, elastic, or inelastic. (c) How
Suppose that the demand for a product is given by 2p2q = 10,000 + 9000p2. (a) Find the elasticity when p = $50 and q = 4502. (b) Tell what type of elasticity this is: unitary, elastic, or
Suppose that the demand for a product is given by pq + p + 100q = 50,000. (a) Find the elasticity when p = $401. (b) Tell what type of elasticity this is. (c) How would a price increase affect
Suppose that the demand for a product is given by (p + 1) √q + 1 = 1000 (a) Find the elasticity when p = $39. (b) Tell what type of elasticity this is. (c) How would a price increase affect revenue?
Suppose the demand function for a product is given bywhere p is in hundreds of dollars and q is the number of tons. (a) What is the elasticity of demand when the quantity demanded is 2 tons and the
In Problems 1-5, find the derivative of each function. 1. y = 10e3x2-x 2. y = 3ln (4x + 11) 3. p = ln (q / q2 - 1) 4. y = xex2 5. f (x) = 5e2x - 40e-0.1x + 11
In Problems 1-5, find the indicated derivative. 1. If y ln x = 5y2 + 11, find dy/dx. 2. Find dy/dx for exy = y. 3. Find dy/dx for y2 = 4x - 1. 4. Find dy/dx for x2 + 3y2 - 2x3y + 2 = 0. 5. Find
Find the slope of the tangent to the curve x2 + 4x - 3y2 + 6 = 0 at (3, 3).
Find the points where tangents to the graph of the equation in Problem 21 are horizontal. In problem 21 Find the slope of the tangent to the curve x2 + 4x - 3y2 + 6 = 0 at (3, 3).
Suppose 3x2 - 2y3 = 10y, where x and y are differentiable functions of t. If dx/dt = 2, find dy/dt when x = 10 and y = 5.
A right triangle with legs of lengths x and y has its area given by A = 1 / 2 xy If the rate of change of x is 2 units per minute and the rate of change of y is 5 units per minute, find the rate of
By using U.S. Census Bureau data for selected years from 1980 and projected to 2040, the number of White non-Hispanic individuals in the U.S. civilian non-institutional population age 16 years and
By using Social Security Administration data for selected years from 1950 and projected to 2050, the additional years of life expectancy at age 65 can be modeled by L(y) = - 14.6 + 7.10 ln y where y
If $1000 is invested for n years at 12% compounded continuously, the future value of the investment is given by S = 1000e0.12n (a) Find the function that gives the rate of change of this
Disposable income is the amount available for spending and saving after taxes have been paid and is one gauge for the state of the economy. By using U.S. Energy Administration data for selected years
A breeder reactor converts stable uranium-238 into the isotope plutonium-239. The decay of this isotope is given by A(t) = A0e-0.00002876t where A(t) is the amount of isotope at time t, in years,
The average cost of producing x units of a product is C̅ 600ex/600 dollars per unit. What is the marginal cost when 600 units are produced?
The impact of inflation on a $60,000 pension can be measured by the purchasing power P of $60,000 after t years. For an inflation rate of 5% per year, compounded annually, P is given by P =
A spherical droplet of water evaporates at a rate of 1 mm3 min. Find the rate of change of the radius when the droplet has a radius of 2.5 mm.
A sign is being lowered over the side of a building at the rate of 2 ft min. A worker handling a guide line is 7 ft away from a spot directly below the sign. How fast is the worker taking in the
Suppose that in a study of water birds, the relationship between the area A of wetlands (in square miles) and the number of different species S of birds found in the area was determined to be S =
Suppose the demand and supply functions for a product are respectively, where p is in dollars and q is the number of units. Find the tax per unit t that will maximize the tax revenue T, and find the
Suppose the supply and demand functions for a product are respectively, where p is in dollars and q is the number of units.Find the tax t that maximizes the tax revenue T, and find the maximum tax
A demand function is given by pq = 27 where p is in dollars and q is the number of units. (a) Find the elasticity of demand at (9, 3). (b) How will a price increase affect total revenue?
Suppose the demand for a product is given by p2(2q + 1) = 10,000 where p is in dollars and q is the number of units. (a) Find the elasticity of demand when p = $20. (b) How will a price increase
Elasticity Suppose the weekly demand function for a product is given by p = 100e-0.1q where p is the price in dollars and q is the number of tons demanded. (a) What is the elasticity of demand when
A product has the demand function p = 100 - 0.5q where p is in dollars and q is the number of units. (a) Find the elasticity (q) as a function of q, and graph the function f (q) = η (q) (b) Find
1. If f'(x) = 4x3, what is f (x)? 2. If f'(x) = 5x4, what is f (x)? 3. If f'(x) = x6, what is f (x)? 4. If g'(x) = x4, what is g(x)?
In Problems 1-4, use algebra to rewrite the integrands; then integrate and simplify. 1. ∫(4x2 - 1)2x3 dx 2. ∫(x3 + 1)2x dx 3. ∫ x + 1 / x3 dx 4. ∫ x - 3 / √x dx
In Problems 1 and 2, find the antiderivatives and graph the resulting family members that correspond to C = 0, C = 4, C = - 4, C = 8, and C = - 8. 1. ∫ (2x+3) dx 2. ∫ (4 -x) dx
If ∫ f(x) dx = 2x9 - 7x5 + C, find f(x).
If ∫ g(x) dx = 11x10 - 4x3 + C, find g(x).
In each of Problems 1-4, a family of functions is given and graphs of some members of the family are shown. Write the indefinite integral that gives the family.1.2. 3. 4.
If the marginal revenue (in dollars per unit) for a month for a commodity is M̅R̅= - 0.4x + 30, find the total revenue function.
If the marginal revenue (in dollars per unit) for a month for a commodity is M̅R̅ = 0.05x + 25, find the total revenue function.
If the marginal revenue (in dollars per unit) for a month is given by M̅R̅= 0.3x + 450, what is the total revenue from the production and sale of 50 units?
Revenue If the marginal revenue (in dollars per unit) for a month is given by M̅R̅ = -0.006x + 36, find the total revenue from the sale of 75 units.
Suppose that when a sense organ receives a stimulus at time t, the total number of action potentials is P(t). If the rate at which action potentials are produced is t3 + 4t2 + 6, and if there are 0
Suppose that a particle has been shot into the air in such a way that the rate at which its height is changing is v = 320 - 32t, in feet per second, and suppose that it is 1600 feet high when t = 10
A factory is dumping pollutants into a river at a rate given by dx/dt = t3/4/600 tons per week, where t is the time in weeks since the dumping began and x is the number of tons of pollutants. (a)
The rate of growth of the population of a city is predicted to be dp / dt = 1000t1.08 where p is the population at time t and t is measured in years from the present. Suppose that the current
The DeWitt Company has found that the rate of change of its average cost for a product iswhere x is the number of units and cost is in dollars. The average cost of producing 20 units is $40.00. (a)
Evaluate the integrals in Problems 1-5. Check your answers by differentiating. 1. ∫x7dx 2. ∫x5 dx 3. ∫8x3dx 4. ∫16x9 dx 5. ∫(33 + x13) dx
An oil tanker hits a reef and begins to leak. The efforts of the workers repairing the leak cause the rate at which the oil is leaking to decrease. The oil was leaking at a rate of 31 barrels per
National health expenditures per capita E (in dollars) have risen dramatically since 2000. By using data from the Centers for Medicare and Medicaid Services from 2006 and projected to 2021, the rate
By using Social Security Administration data for selected years from 2012 and projected to 2050, the rate of change of the total U.S. taxable payroll can be modeled by dP/dt = 27.0t + 112 billions of
When the air temperature is 20° F, the rate of change of the wind chill temperature t (°F) is given by dt / dw = - 4.352w-0.84 where w is wind speed in miles per hour. (a) Will both the rate of
Hail is produced in severe thunderstorms when an updraft draws moist surface air into subfreezing air above 10,000 feet. The speed of the updraft s, in mph, affects the diameter of hail (in inches).
With U.S. Census Bureau data (actual and projected) for selected years from 1960 to 2050, the rate of change of U.S. population P can be modeled by dP / dt = - 0.0002187t2 + 0.0276t + 1.98 million
The consumer price index (CPI) measures how prices have changed for consumers. With 1995 as a reference of 100, a year with CPI = 150 indicates that consumer costs in that year were 1.5 times the
In Problems 1 and 2, find du. 1. u = 2x5 +9 2. u = 3x4 - 4x3
In each of Problems 1 and 2, one of parts (a) and (b) can be integrated with the Power Rule and the other cannot. Integrate the part that can be done with the Power Rule, and explain why the Power
If ∫ f(x) dx = (7x - 13)10 + C, find f(x).
If ∫ g(x) dx = (5x2 + 2)6 + C, find g(x).
In Problems 1 and 2, (a) evaluate each integral and (b) graph the members of the solution family for C = - 5, C = 0, and C = 5. 1. ∫ x(x2 - 1)3 dx 2. ∫ (3x - 11)1/3 dx
Each of Problems 1 and 2 has the form f (x) dx.(a) Evaluate each integral to obtain a family of functions.(b) Find and graph the family member that passes through the point (0, 2). Call that function
In each of Problems 1 and 2, a family of functions is given, together with the graphs of some functions in the family. Write the indefinite integral that gives the family.1. F(x) = (x2 - 1)4/3 + C2.
In parts (a)-(c) of Problems 1 and 2, three integrals are given. Integrate those that can be done by the methods studied so far. Additionally, as part (d), give your own example of an integral that
Suppose that the marginal revenue for a product is given bywhere x is the number of units and revenue is in dollars. Find the total revenue.
The marginal revenue for a new calculator is given bywhere x represents hundreds of calculators and revenue is in dollars. Find the total revenue function for these calculators.
The total physical output of a number of machines or workers is called physical productivity and is a function of the number of machines or workers. If P = f (x) is the productivity, dP/dx is the
The rate of production of a new line of products is given bywhere x is the number of items and t is the number of weeks the product has been in production. (a) Assuming that x = 0 when t = 0, find
The rate of change in data entry speed of the average student is ds/dx = 5(x + 1)-1/2, where x is the number of lessons the student has had and s is in entries per minute. (a) Find the data entry
Evaluate the integrals in Problems 1-4. Check your results by differentiation. 1. ∫ (x2 + 3)3 2x dx 2. ∫ (3x3 + 1)4 9x2 dx 3. ∫ (5x3 + 11)4 15x2 dx 4. ∫ (8x4 +5)3 (32x3) dx
Because a new employee must learn an assigned task, production will increase with time. Suppose that for the average new employee, the rate of performance is given bywhere N is the number of units
An excellent film with a very small advertising budget must depend largely on word-of-mouth advertising. In this case, the rate at which weekly attendance might grow can be given bywhere t is the
An inferior product with a large advertising budget does well when it is introduced, but sales decline as people discontinue use of the product. Suppose that the rate of weekly sales revenue is given
Because of job outsourcing, a Pennsylvania town predicts that its public school population will decrease at the ratewhere x is the number of years and N is the total school population. If the present
A new fast-food firm predicts that the number of franchises for its products will grow at the ratewhere t is the number of years, 0 ‰¤ t ‰¤ 10. If there is one franchise (n =
The rate of change of the percent of obese Americans who are severely obese can be modeled by the function percentage points per year, where x is the number of years after 1980.(a) If 8.3% of obese
Suppose the rate of change of the number of Social Security beneficiaries (in millions per year) can be modeled bywhere t is the number of years past 1950. (a) Use integration and the data point for
Suppose the rate of change of the percent p of total U.S. workers who are female can be modeled bypercentage points per year, where t is the number of years past 1950. (a) Use integration and the
Energy use per dollar of GDP in the United States indexed to 1980 means that energy use for any year is viewed as a percent of the use per dollar of GDP in 1980. The table shows the energy use per
Evaluate the integrals in Problems 1-4. 1. ∫ 3e3x dx 2. ∫ 4e4x dx 3. ∫ e-x dx 4. ∫ e2x dx
In Problems 1 and 2, graphs of two functions labeled g(x) and h(x) are given. Decide which is the graph of ˆ« f (x) and which is one member of the family ˆ« f (x) dx. Check your
In Problems 1 and 2, a function f (x) and its graph are given. Find the family F(x) = f (x) dx and graph the member that satisfies F(0) = 0.1.2.
In parts (a)-(d) of Problems 1 and 2, integrate those that can be done by the methods studied so far.1.a. âˆ« xex3 dxb.c.d.2.a.b.c.d. âˆ« 6xe-x2/8 dx
Suppose that the marginal revenue from the sale of x units of a product is MR = R'(x) = 6e0.01x. What is the revenue in dollars from the sale of 100 units of the product?
Suppose that the rate at which the concentration of a drug in the blood changes with respect to time t is given by
Radioactive substances decay at a rate that is proportional to the amount present. Thus, if k is a constant and the amount present is x, the decay rate is dx/dt = kx (t in hours) This means that the
The rate of vocabulary memorization of the average student in a foreign language course is given by dv / dt = 40 / t + 1 where t is the number of continuous hours of study, 0 < t ≤ 4, and v is the

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