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mathematics
calculus

Questions and Answers of Calculus

From 1950 and projected to 2050, the percent of women in the workforce can be modeled by w(x) = 9.42 +8.70 ln x where x is the number of years past 1940. If this model is accurate, at what rate will
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. The table gives the
The following table gives the percent of adult Americans with diabetes for selected years from 2010 and projected to 2050.(a) Find a logarithmic function that models the data, with x equal to the
Find the derivatives of the functions in Problems 1-5. 1. y = 5ex - x 2. y = x2 - 3ex 3. f (x) = ex- xe 4. f (x) = 4ex- ln x 5. g (x) = 500(1 - e-0.1x)
(a) What is the slope of the line tangent to y = xe-x at x = 1? (b) Write the equation of the line tangent to the graph of y =xe-x at x = 1.
(a) What is the slope of the line tangent to y = e-x (1 + e-x) at x = 0? (b) Write the equation of the line tangent to the graph of y = e-x (1 +e-x) at x =0.
The equation for the standard normal probability distribution is(a) At what value of z will the curve be at its highest point? (b) Graph this function with a graphing utility to verify your answer.
(a) Find the mode of the normal distribution* given by(b) What is the mean of this normal distribution?(c) Use a graphing utility to verify your answer.
In Problems 1-4, find any relative maxima and minima. Use a graphing utility to check your results. 1. y = ex / x 2. y = x / ex 3. y = x - ex 4. y = x2 / ex
If $P is invested for n years at 10% compounded continuously, the future value that results after n years is given by the function S = Pe0.1n(a) At what rate is the future value growing at any time
The future value that accrues when $700 is invested at 9%, compounded continuously, is S(t) = 700e0.09t where t is the number of years. (a) At what rate is the money in this account growing when t =
1. Sales decay After the end of an advertising campaign, the sales of a product are given by S = 100,000e-0.5t where S is weekly sales in dollars and t is the number of weeks since the end of the
Suppose that the total cost in dollars of producing x units of a product is given by C(x) = 10,000 + 20xex/600 Find the marginal cost when 600 units are produced.
Suppose that the revenue in dollars from the sale of x units of a product is given by R(x) = 1000xe-x/50 Find the marginal revenue function.
1. The percent concentration y of a certain drug in the bloodstream at any time t (in hours) is given by y = 100(1 - e-0.462t) (a) What function gives the instantaneous rate of change of the
The amount of the radioactive isotope thorium-234 present at time t in years is given by Q(t) = 100e-0.02828t (a) Find the function that describes how rapidly the isotope is decaying. (b) Find the
Suppose the concentration C(t), in mg/ml, of a drug in the bloodstream t minutes after an injection is given by C(t) = 20te-0.04t (a) Find the instantaneous rate of change of the concentration after
1. With U.S. Department of Health and Human Services data from 2000 and projected to 2018, the total public expenditures for health care H can be modeled by H = 624e0.07t where t is the number of
The intensity level of sound, I, is given by I / I0 = 10L/10 where L is the decibel reading and I0 is the standard threshold of audibility. If I / I0 = y, at what rate is y changing with respect to
For selected years from 1900 to 2014, the national debt d, in billions of dollars, can be modeled by d = 1.60e0.083t where t is the number of years past 1900 (a) What function describes how fast the
Medical research has shown that between heartbeats, the pressure in the aorta of a normal adult is a function of time in seconds and can be modeled by the equation P = 95e-0.491t (a) Use the
Suppose that the spread of a disease through the student body at an isolated college campus can be modeled bywhere y is the total number affected at time t (in days). Find the rate of change of y.
The number of people N(t) in a community who are reached by a particular rumor at time t (in days) is given byFind the rate of change of N(t).
By using Social Security Administration data for selected years from 1950 and projected to 2050, the population of Americans ages 20 to 64 can be modeled by the functionwhere t is the number of years
By using U.S. Department of Energy data for selected years from 2010 and projected to 2032, the millions of metric tons of carbon dioxide (CO2) emissions from biomass energy combustion in the United
Obesity (BMI ‰¥ 30) increases the risk of diabetes, heart disease, and many other ailments, but the severely obese (BMI 40) are most at risk and are the most expensive to treat. The table
The following table gives the projected population, in thousands, of Americans over 100 years of age.(a) Create an exponential function that models the projected population y, in thousands, as a
Using data from 2000 and projected to 2050, the receipts (in billions of dollars) for world tourism can be modeled by the function y = 165.55(1.055x) where x is the number of years past 1980. (a)
When U.S. energy use per dollar of GDP is indexed to 1980, that energy use for any year is viewed as a percent of the use per dollar of GDP in 1980. By using U.S. Department of Energy data for
The table gives the average annual wage, in thousands of dollars, for selected years from 2012 and projected to 2050.(a) Find an exponential function that models the data, with x equal to the number
In Problems 1-6, find dy/dx at the given point without first solving for y. 1. x2 = 4y - 17 = 0 at (1, - 4) 2. 3x2 = 10y + 400 = 0 at (10, 70) 3. xy2 = 8 at (2, 2) 4. ey = x at (1, 0) 5. x2 = 3xy - 4
For Problems 1-4, find the slope of the tangent to the curve. 1. x2 + 4x + y2 + 2y - 4 = 0 at (1, - 1) 2. x2 - 4x + 2y2 - 4 = 0 at (2, 2) 3. x2 + 2xy + 3 = 0 at (-1, 2) 4. y + x2 + xy = 13 at (2, 3)
Write the equation of the line tangent to the curve x2 - 2y2 + 4 = 0 at (2, 2).
Write the equation of the line tangent to the curve x2 = y2 + 2x = 3 = 0 at (-1, 2).
Write the equation of the line tangent to the curve 4x2 + 3y2 - 4y - 3 = 0 at (-1, 1).
Write the equation of the line tangent to the curve xy + y2 = 0 at (3, 0).
If ln x = y2, find dy/dx.
If ln (x + y) = y2, find dy/dx.
If y2 ln x = 4, find dy/dx.
If ln (xy - 1) = x + 2, find dy/dx.
Find the slope of the tangent to the curve y2 ln x + x2y = 3 at the point (1, 3).
Write the equation of the line tangent to the curve x ln y + 2xy = 2 at the point (1, 1).
If xey = 6, find dy/dx.
If x + exy = 10, find dy/dx.
If xexy = 4, find dy/dx.
If x - xey = 3, find dy/dx.
If yex - y =3, find dy/dx.
If x2y = ex+y, find dy/dx.
Find the slope of the line tangent to the graph of yex = y2 + x - 2 at (0, 2).
Find the slope of the line tangent to the curve xey = 3x2 + y - 24 at (3, 0).
Write the equation of the line tangent to the curve xey = 2y + 3 at (3, 0).
Write the equation of the line tangent to the curve yex = 2y + 1 at (0, - 1).
At what points does the curve defined by x2 + 4y2 - 4x - 4 = 0 have (a) Horizontal tangents? (b) Vertical tangents?
At what points does the curve defined by x2 + 4y2 - 4 = 0 have (a) Horizontal tangents? (b) Vertical tangents?
In Problem 11, the derivative y was found to bewhen x2 + y2 = 4. (a) Take the implicit derivative of the equation for y` to show that
(a) Find y' implicitly for x3 - y3 = 8.(b) Then, by taking derivatives implicitly, use part (a) to show that(c) Substitute x2/y2 for y' in the expression for y'' and simplify to show that
Find y' for √x + √y = 1 and simplify.
Find y'' for 1 / x - 1 / y = 1.
In Problems 1 and 2, find the maximum and minimum values of y. Use a graphing utility to verify your conclusion. 1. x2 + y2 - 9 = 0 2. 4x2 + y2- 8x = 0
Suppose that a company's sales volume y (in thousands of units) is related to its advertising expenditures x (in thousands of dollars) according to xy - 20x + 10y = 0 Find the rate of change of sales
Suppose that the number of mosquitoes N (in thousands) in a certain swampy area near a community is related to the number of pounds of insecticide x sprayed on the nesting areas according to Nx - 10x
Suppose that a company can produce 12,000 units when the number of hours of skilled labor y and unskilled labor x satisfy 384 = (x + 1)3/4(y + 2)1/3 Find the rate of change of skilled-labor hours
Suppose that production of 10,000 units of a certain agricultural crop is related to the number of hours of labor x and the number of acres of the crop y according to 300x + 30,000y = 11xy - 0.0002x2
If the demand function for q units of a product at $p per unit is given by p(q + 1)2 = 200,000 find the rate of change of quantity with respect to price when p = $80. Interpret this result.
If the demand function for q units of a commodity at $p per unit is given by p2 (2q + 1) = 100,000 find the rate of change of quantity with respect to price when p = $50. Interpret this result.
The number of grams of radium, y, that will remain after t years if 100 grams existed originally can be found by using the equationUse implicit differentiation to find the rate of change of y with
Suppose the proportion P of people affected by a certain disease is described bywhere t is the time in months. Find dP/dt, the rate at which P grows.
The temperature-humidity index (THI) is given by THI = t - 0.55(1 - h)(t - 58) where t is the air temperature in degrees Fahrenheit and h is the relative humidity. If the THI remains constant, find
Find dy/dx for the functions in Problems 1-5. 1. x2 = 2y2 - 4 = 0 2. x = y2 - 4y + 6 = 0 3. x2 = 4x + y2 - 3y + 1 = 0 4. x2 = 5x + y3 - 3y - 3 = 0 5. If x2 = y2 = 4, find y`.
In Problems 1-4, find dy/dt using the given values. 1. y = x3 - 3x for x = 2, dx/dt = 4 2. y = 3x3 + 5x2 - x for x = 4, dx/dt = 3 3. xy = 4 for x = 8, dx/dt = - 2 4. xy = x + 3 for x = 3, dx/dt = - 1
If s = 2πr (r + h), find dr/dt when r = 2, h = 8, dh/dt = 3, and ds/dt = 10π.
A point is moving along the graph of the equation y = - 4x2. At what rate is y changing when x = 5 and is changing at a rate of 2 units sec?
A point is moving along the graph of the equation y = 5x3 - 2x. At what rate is y changing when x = 4 and is changing at a rate of 3 units sec?
The radius of a circle is increasing at a rate of 2 ft min. At what rate is its area changing when the radius is 3 ft? (Recall that for a circle, A = πr2.)
The area of a circle is changing at a rate of 1 in2 / sec. At what rate is its radius changing when the radius is 2 in.?
The volume of a cube is increasing at a rate of 64 in3/sec. At what rate is the length of each edge of the cube changing when the edges are 6 in. long? (Recall that for a cube, V = x3.)
The lengths of the edges of a cube are increasing at a rate of 8 ft min. At what rate is the surface area changing when the edges are 24 ft long? (Recall that for a cube, S = 6x2.)
Suppose that the daily profit (in dollars) from the production and sale of x units of a product is given byAt what rate per day is the profit changing when the number of units produced and sold is
Suppose that the monthly revenue and cost (in dollars) for x units of a product areAt what rate per month is the profit changing if the number of units produced and sold is 200 and is increasing at a
Suppose that the price p (in dollars) of a product is given by the demand functionwhere x represents the quantity demanded. If the daily demand is decreasing at a rate of 20 units per day, at what
The supply function for a product is given by p = 40 + 100 √2x + 9, where x is the number of units supplied and p is the price in dollars. If the price is increasing at a rate of $1 per month, at
Suppose that for a particular product, the number of units x produced per month depends on the number of thousands of dollars y invested, with x = 30y + 20y2. At what rate will production increase if
Boyle's law for enclosed gases states that at a constant temperature, the pressure is related to the volume by the equation P = k / V where k is a constant. If the volume is increasing at a rate of
At what rate is the volume decreasing when the radius is 3 mm? Suppose that a tumor in a person's body has a spherical shape and that treatment is causing the radius of the tumor to decrease at a
At what rate is the surface area of the tumor decreasing when the radius is 3 mm? Suppose that a tumor in a person's body has a spherical shape and that treatment is causing the radius of the tumor
For many species of fish, the allometric relationship between the weight W and the length L is approximately W = kL3, where k is a constant. Find the percent rate of change of the weight as a
The resistance R of a blood vessel to the flow of blood is a function of the radius r of the blood vessel and is given by R = k / r4 where k is a constant. Find the percent rate of change of the
For fiddler crabs, data gathered by Thompson* show that the allometric relationship between the weight C of the claw and the weight W of the body is given by C = 0.11W1.54 Find the percent rate of
For human beings, the surface area S of the body is related to the body's weight W according to S = kW2/3 where k is a constant. Find the percent rate of change of the body's surface area in terms of
A bacterial cell has a spherical shape. If the volume of the cell is increasing at a rate of 4 cubic micrometers per day, at what rate is the radius of the cell increasing when it is 2 micrometers?
Assume that water is being purified by causing it to flow through a conical filter that has a height of 15 inches and a radius of 5 inches. If the depth of the water is decreasing at a rate of 1 inch
Suppose that air is being pumped into a spherical balloon at a rate of 5 in3 min. At what rate is the radius of the balloon increasing when the radius is 5 in.?
Suppose that a boat is being pulled toward a dock by a winch that is 5 ft above the level of the boat deck, as shown in the figure. If the winch is pulling the cable at a rate of 3 ft min, at what
A 30-ft ladder is leaning against a wall. If the bottom is pulled away from the wall at a rate of 1 ft sec, at what rate is the top of the ladder sliding down the wall when the bottom is 18 ft from
A kite is 30 ft high and is moving horizontally at a rate of 10 ft min. If the kite string is taut, at what rate is the string being played out when 50 ft of string is out?
A plane is flying at a constant altitude of 1 mile and a speed of 300 mph. If it is flying toward an observer on the ground, how fast is the plane approaching the observer when it is 5 miles from the
Two boats leave the same port at the same time, with boat A traveling north at 15 knots (nautical miles per hour) and boat B traveling east at 20 knots. How fast is the distance between them changing
Two cars are approaching an intersection on roads that are perpendicular to each other. Car A is north of the intersection and traveling south at 40 mph. Car B is east of the intersection and
Water is flowing into a barrel in the shape of a right circular cylinder at the rate of 200 in3/min. If the radius of the barrel is 18 in., at what rate is the depth of the water changing when the
Suppose that water is being pumped into a rectangular swimming pool of uniform depth at 10/ft3 hr. If the pool is 10 ft wide and 25 ft long, at what rate is the water rising when it is 4 ft deep?

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