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mathematics
calculus with applications
Calculus For Business, Economics And The Social And Life Sciences 11th Brief Edition Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price - Solutions
It is projected that t years from now, the circulation of a local newspaper will be C(t) = 100t2 + 400t + 5,000. Estimate the amount by which the circulation will increase during the next 6 months.
In Exercises 1 through 28, differentiate the given function. || + 2 3
In Exercises 24 through 27, find all points on the graph of the given function where the tangent line is horizontal.f(x) = (x − 1)(x2 − 8x + 7)
In Exercises 13 through 24, compute the derivative of the given function and find the equation of the line that is tangent to its graph for the specified value x = c.f(x) = x3 − 1; c = 1
A projection made in January of 2005 determined that x years later, the average property tax on a three-bedroom home in a certain community will be T(x) = 60x3/2 + 40x + 1,200 dollars. Estimate the percentage change by which the property tax will increase during the first half of the year 2013.
An object moves along a line in such a way that its position at time t is given by s(t) = 2t3 − 3t2 + 2 for t ≥ 0.a. Find the velocity v(t) and acceleration a(t) of the object.b. When is the object stationary? When is it advancing? Retreating?c. What is the total distance traveled by the object
In Exercises 1 through 12, use the chain rule to compute the derivative dy/dx and simplify your answer.y = 1 − 3u2; u = 3 − 2x
Find the second derivative of the function f(t) = t(2t + 1)2.
In Exercises 1 through 6, C(x) is the total cost of producing x units of a particular commodity and p(x) is the unit price at which all x units will be sold. Assume p(x) and C(x) are in dollars.(a) Find the marginal cost and the marginal revenue.(b) Use marginal cost to estimate the cost of
In Exercises 1 and 2, use the definition of the derivative to find f'(x). f(x) = 1 x-2
In Exercises 1 through 28, differentiate the given function.y = 3
In Exercises 1 through 12, compute the derivative of the given function and find the slope of the line that is tangent to its graph for the specified value of the independent variable.f(x) = −3; x = 1
In Exercises 1 through 8, find dy/dx in two ways:(a) By implicit differentiation(b) By differentiating an explicit formula for y.In each case, show that the two answers are the same.5x − 7y = 3
In Exercises 1 through 18, differentiate the given function.f(x) = (x − 5)(1 − 2x)
In Exercises 1 through 12, use the chain rule to compute the derivative dy/dx and simplify your answer.y = u2 + 1; u = 3x − 2
In each case, find the derivative dy/dx. 5 a. y = 3x² − 4√√x +- - c. y 1 - 2x d. y = (3-4x + 3x²)³/2² 7 b. y = (3x³x + 1)(4 - x²) 5x² - 3x + 2
In Exercises 1 through 28, differentiate the given function.y = −2
In Exercises 1 through 12, compute the derivative of the given function and find the slope of the line that is tangent to its graph for the specified value of the independent variable.f(x) = 4; x = 0
In Exercises 1 through 18, differentiate the given function.f(x) = (2x + 1)(3x − 2)
A manufacturer’s total cost iswhen q units are produced. The current level of production is 4 units. Estimate the amount by which the manufacturer should decrease production to reduce the total cost by $130. C(q) 1 =-=-9³ 6 = q+642q + 400 dollars
In Exercises 13 through 24, compute the derivative of the given function and find the equation of the line that is tangent to its graph for the specified value x = c. f(x) = 1 Vx =;c = 1
In Exercises 19 through 23, find an equation for the tangent line to the given curve at the point where x = x0. y= x+7 5 - 2x² Xj = 0 Хо
In each of these cases, find the percentage rate of change of the function f(t) with respect to t at the given value of t.a. f(t) = t2(3 − 2t)3 at t = 1b. f(t) || 1 - t + 1 at t = 0
In Exercises 1 through 6, C(x) is the total cost of producing x units of a particular commodity and p(x) is the unit price at which all x units will be sold. Assume p(x) and C(x) are in dollars.(a) Find the marginal cost and the marginal revenue.(b) Use marginal cost to estimate the cost of
In Exercises 1 through 6, C(x) is the total cost of producing x units of a particular commodity and p(x) is the unit price at which all x units will be sold. Assume p(x) and C(x) are in dollars.(a) Find the marginal cost and the marginal revenue.(b) Use marginal cost to estimate the cost of
In Exercises 1 through 28, differentiate the given function. y 1 t + 1² 1
Use the chain rule to find dy/dx.a. y = 5u2 + u − 1; u = 3x + 1b. y = 1/u2; u = 2x + 3
In Exercises 19 through 23, find an equation for the tangent line to the given curve at the point where x = x0. y || X 2x + 3 Хо = - 1
In Exercises 13 through 24, compute the derivative of the given function and find the equation of the line that is tangent to its graph for the specified value x = c.f(x) = 2√x; c = 4
In Exercises 9 through 22, find dy/dx by implicit differentiation.(3xy2 + 1)4 = 2x − 3y
In Exercises 1 through 12, use the chain rule to compute the derivative dy/dx and simplify your answer. y = 1 - И U u = 3. 3x² + 5
In Exercises 1 through 6, C(x) is the total cost of producing x units of a particular commodity and p(x) is the unit price at which all x units will be sold. Assume p(x) and C(x) are in dollars.(a) Find the marginal cost and the marginal revenue.(b) Use marginal cost to estimate the cost of
In Exercises 9 through 22, find dy/dx by implicit differentiation.y2 + 3xy − 4x2 = 9
In Exercises 1 through 8, find dy/dx in two ways:(a) By implicit differentiation(b) By differentiating an explicit formula for y.In each case, show that the two answers are the same.xy + 2y = 3
In Exercises 1 through 12, use the chain rule to compute the derivative dy/dx and simplify your answer. I + x = n U n - اية y:
In Exercises 13 through 24, compute the derivative of the given function and find the equation of the line that is tangent to its graph for the specified value x = c.f(x) = 7 − 2x; c = 5
In Exercises 3 through 13, find the derivative of the given function. y= 2-x² 3x² + 1 2
In Exercises 1 through 12, compute the derivative of the given function and find the slope of the line that is tangent to its graph for the specified value of the independent variable.f(x) = −x3; x = 1
In Exercises 1 through 28, differentiate the given function.y = x7/3
In Exercises 1 through 6, C(x) is the total cost of producing x units of a particular commodity and p(x) is the unit price at which all x units will be sold. Assume p(x) and C(x) are in dollars.(a) Find the marginal cost and the marginal revenue.(b) Use marginal cost to estimate the cost of
In Exercises 1 through 12, compute the derivative of the given function and find the slope of the line that is tangent to its graph for the specified value of the independent variable.f(x) = x2 − 1; x = −1
In Exercises 1 through 8, find dy/dx in two ways:(a) By implicit differentiation(b) By differentiating an explicit formula for y.In each case, show that the two answers are the same.x + 1/y = 5
In Exercises 1 through 18, differentiate the given function. f(x) = 1 - 2x2 + 1( - ) + x. 3 X کړ)
In Exercises 3 through 13, find the derivative of the given function.y = (x3 + 2x − 7)(3 + x − x2)
Records indicate that x years after the year 2010, the average property tax on a four-bedroom home in a suburb of a major city was T(x) = 3x2 + 40x + 1,800 dollars.a. At what rate is the property tax increasing with respect to time in 2013?b. At what percentage rate is the property tax increasing
In Exercises 1 through 18, differentiate the given function.f(x) = −3(5x3 − 2x + 5)(√x + 2x)
Find the rate of change of the functionwith respect to x when x = 1. f(x) = x + 1 1 – 5x
In Exercises 1 through 8, find dy/dx in two ways:(a) By implicit differentiation(b) By differentiating an explicit formula for y.In each case, show that the two answers are the same.xy = 4
In Exercises 1 through 28, differentiate the given function.y = x−4
In Exercises 3 through 13, find the derivative of the given function. f(x) = x³ - 1 3x³ +2√x 3 X + 1 - 2x x³
In Exercises 1 through 12, compute the derivative of the given function and find the slope of the line that is tangent to its graph for the specified value of the independent variable.f(x) = 2x2 − 3x − 5; x = 0
In Exercises 1 through 12, use the chain rule to compute the derivative dy/dx and simplify your answer.y = 2u2 − u + 5; u = 1 − x2
In Exercises 1 through 28, differentiate the given function.y = −2x + 7
In Exercises 1 through 8, find dy/dx in two ways:(a) By implicit differentiation(b) By differentiating an explicit formula for y.In each case, show that the two answers are the same.x2 + y3 = 12
In Exercises 1 through 12, compute the derivative of the given function and find the slope of the line that is tangent to its graph for the specified value of the independent variable.f(x) = 2 − 7x; x = −1
Find an equation for the tangent line to the curve y = x2 − 2x + 1 at the point where x = −1.
In Exercises 1 through 18, differentiate the given function.y = 400(15 − x2)(3x − 2)
In Exercises 1 through 12, use the chain rule to compute the derivative dy/dx and simplify your answer.y = √u; u = x2 + 2x − 3
In Exercises 1 through 28, differentiate the given function.y = 5x − 3
In Exercises 1 through 12, compute the derivative of the given function and find the slope of the line that is tangent to its graph for the specified value of the independent variable.f(x) = 5x − 3; x = 2
In Exercises 3 through 13, find the derivative of the given function.f(x) = 6x4 − 7x3 + 2x + √2
In Exercises 1 through 8, find dy/dx in two ways:(a) By implicit differentiation(b) By differentiating an explicit formula for y.In each case, show that the two answers are the same.x3 − y2 = 5
In Exercises 1 and 2, use the definition of the derivative to find f'(x).f(x) = x2 − 3x + 1
In Exercises 1 through 6, C(x) is the total cost of producing x units of a particular commodity and p(x) is the unit price at which all x units will be sold. Assume p(x) and C(x) are in dollars.(a) Find the marginal cost and the marginal revenue.(b) Use marginal cost to estimate the cost of
In Exercises 1 through 18, differentiate the given function.y = 10(3u + 1)(1 − 5u)
Bea Johnson, the owner of the Bea Nice boutique, estimates that when a particular kind of perfume is priced at p dollars per bottle, she will sellbottles per month at a total cost of C(p) = 0.2p2 + 3p + 200 dollarsa. Express Bea’s profit P(p) as a function of the price p per bottle.b. At what
In Exercises 47 through 52, find all values of x = c so that the tangent line to the graph of f(x) at (c, f (c)) will be horizontal. f(x) = 2x + 5 (1 - 2x)³
A tumor is modeled as being roughly spherical, with radius R. If the radius of the tumor is currently R = 0.54 cm and is increasing at the rate of 0.13 cm per month, what is the corresponding rate of change of the volume V =4/3πR3?
In Exercises 47 through 50, find the relative rate of change of f (x) with respect to x for the prescribed value x = c.f(x) = (4 − x)x−1; c = 3
The accompanying graph shows how a population P of fruit flies (Drosophila) changes with time t (days) during an experiment. Use the graph to estimate the rate at which the population is growing after 20 days and also after 36 days. At what time is the population growing at the greatest rate?Growth
A company manufactures a “thin” DVD burner kit that can be plugged into personal computers. The marketing manager determines that t weeks after an advertising campaign begins, P(t) percent of the potential market is aware of the burners, wherea. At what rate is the market percentage P(t)
A circular oil slick spreads in such a way that its radius is increasing at the rate of 20 ft/hr. How fast is the area of the slick changing when the radius is 200 feet?
An object moves along a straight line in such a way that t seconds after it begins to move, it ismeters from its starting point.a. What is the object’s (instantaneous) velocity at time t?b. What is the object’s initial velocity (at time t = 0)?c. How far from the starting point is the object at
Air temperature usually decreases with increasing altitude. However, during the winter, thanks to a phenomenon called thermal inversion, the temperature of air warmed by the sun in mountains above a fog may rise above the freezing point, while the temperature at lower elevations remains near or
In Exercises 47 through 52, find all values of x = c so that the tangent line to the graph of f(x) at (c, f (c)) will be horizontal. f(x)=√x² - 4x + 5
A study indicates that spending money on pollution control is effective up to a point but eventually becomes wasteful. Suppose it is known that when x million dollars is spent on controlling pollution, the percentage of pollution removed is given bya. At what rate is the percentage of pollution
The gross annual earnings of a certain company were A(t) = 0.1t2 + 10t + 20 thousand dollars t years after its formation in 2008.a. At what rate were the gross annual earnings of the company growing with respect to time in 2012?b. At what percentage rate were the gross annual earnings growing with
Physiologists have observed that the flow of blood from an artery into a small capillary is given by the formulawhere D is the diameter of the capillary, A is the pressure in the artery, C is the pressure in the capillary, and k is a positive constant.a. By how much is the flow of blood F changing
A company uses a truck to deliver its products. To estimate costs, the manager models gas consumption by the functiongal/mile, assuming that the truck is driven at a constant speed of x miles per hour, for x ≥ 5. The driver is paid $20 per hour to drive the truck 250 miles, and gasoline costs $4
Currently, a company is selling 1,000 units of a certain commodity at a price of $5 per unit. The manager of the company estimates that the price is currently increasing at the rate of 5 cents per week, while demand is decreasing at the rate of 4 units per week.a. If x is the level of production at
The basal metabolic rate is the rate of heat produced by an animal per unit time. Observations indicate that the basal metabolic rate of a warm-blooded animal of mass w kilograms (kg) is given by M = 70w3/4 kilocalories per daya. Find the rate of change of the metabolic rate of an 80-kg cougar that
A 5-year projection of population trends suggests that t years after 2010, the population of a certain community will be P thousand, where P(t) = −6t2 + 12t + 151a. At what average rate will the population be growing between 2010 and 2012?b. At what instantaneous rate will the population be
Experiments indicate that when a flea jumps, its height (in meters) after t seconds is given by the function H(t) = 4.4t − 4.9t2a. Find H´(t). At what rate is H(t) changing after 1 second? Is it increasing or decreasing?b. At what time is H´(t) = 0? What is the significance of this time?c. When
Refer to the graph of blood pressure as a function of time shown in Figure 2.9.Estimate the average rate of change in blood pressure over the time periods [0.7, 0.75] and [0.75, 0.8]. Interpret your results.Figure 2.9. Blood pressure 120 80 40 0 P mm of mercury (Hg) 0.25 + 0.50 Sharp point Not
An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 8:00 A.M. will have produced Q(t) = −t3 + 8t2 + 15t units t hours later.a. Compute the worker’s rate of production R(t) = Q'(t).b. At what rate is the worker’s rate of
The output at a certain factory is Q = 600K1/2L1/3 units, where K denotes the capital investment and L is the size of the labor force. Estimate the percentage increase in output that will result from a 2% increase in the size of the labor force if capital investment is not changed.
You measure the radius r of a spherical tumor to be 1.2 cm with an error no greater than 3%. Use calculus to estimate the error incurred by using this approximate value of r in the formula S = 4πr2 to compute the surface area of the tumor.
A manufacturer of motorcycles estimates that if x thousand dollars is spent on advertising, thencycles will be sold. At what rate will sales be changing when $9,000 is spent on advertising? Are sales increasing or decreasing for this level of advertising expenditure? M(x) = 2,300 + 125 X 517 1² 3
The manager of a company estimates that it will cost $10,000 to produce 400 units of her product 1 year from now and that all those units can then be sold at a price of $30 per unit. She also estimates that in 1 year, the price will be increasing at the rate of 75 cents per unit per month, while
The speed of blood flowing along the central axis of a certain artery is S(R) = 1.8 × 105R2 centimeters per second, where R is the radius of the artery. A medical researcher measures the radius of the artery to be 1.2 × 10−2 centimeter and makes an error of 5 × 10−4 centimeter. Estimate the
In Exercises 47 through 52, find all values of x = c so that the tangent line to the graph of f(x) at (c, f (c)) will be horizontal.f(x) = (x − 1)2(2x + 3)3
An oral painkiller is administered to a patient, and t hours later, the concentration of drug in the patient’s bloodstream is given bya. At what rate R(t) is the concentration of drug in the patient’s bloodstream changing t hours after being administered? At what rate is R(t) changing at time
The human body’s reaction to a dose of medicine can be modeled* by a function of the formwhere K is a positive constant and M is the amount of medicine absorbed in the blood. The derivative S = dF/dM can be thought of as a measure of the sensitivity of the body to the medicine.a. Find the
One model of the cardiovascular system relates V(t), the stroke volume of blood in the aorta at a time t during systole (the contraction phase), to the pressure P(t) in the aorta during systole by the equationwhere C1 and C2 are positive constants and T is the (fixed) time length of the systole
A study conducted on a patient undergoing cardiac catheterization indicated that the diameter of the aorta was approximately D millimeters (mm) when the aortic pressure was p (mm of mercury), where D(p) = −0.0009p2 + 0.13p+ 17.81 for 50 ≤ p ≤ 120.a. Find the average rate of change of the
To estimate the amount of wood in the trunk of a tree, it is reasonable to assume that the trunk is a cutoff cone (see the figure).If the upper radius of the trunk is r, the lower radius is R, and the height is H, the volume of the wood is given bySuppose r, R, and H are increasing at the
Lupe Garcia is the efficiency expert for a large manufacturing company. She finds that the average worker who arrives on the job when the morning shift begins at 8:00 A.M. will have assembled f(x) = −x3 + 6x2 + 15x units x hours later.a. Lupe wants to find the rate at which the average worker is
Herpetologists have proposed using the formula s = 1.1w0.2 to estimate the maximum sprinting speed s (meters per second) of a lizard of mass w (grams). At what rate is the maximum sprinting speed of an 11-gram lizard increasing if the lizard is growing at the rate of 0.02 grams per day?
In Exercises 55 through 60, find the second derivative of the given function. f(t) 2 5t + 1
A 6-foot-tall man walks at the rate of 4 ft/sec away from the base of a street light 12 feet above the ground. At what rate is the length of his shadow changing when he is 20 feet away from the base of the light? 12 20
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