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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
In the Think About It essay at the end of Chapter 3, we examined several models associated with the AIDS epidemic. Using a data analysis program, we obtain the functionas a model for the number of cases of AIDS reported t years after the base year of 1990.a. According to this model, in what year
In Exercises 39 through 46, find the largest and smallest values of the given function over the prescribed closed, bounded interval.g(t) = t3/2e−2t for 0 ≤ t ≤ 1
How quickly will money double if it is invested at an annual interest rate of 6% compounded continuously?
A manufacturer estimates that when x units of a particular commodity are produced, the market price p (dollars per unit) is given by the demand function p = 300e−0.02xa. What market price corresponds to the production of x = 100 units?b. How much revenue is obtained when 100 units of the
Esmeralda needs $5,000 for a trip to Peru when she graduates from college in 4 years. How much must she invest now at an annual interest rate of 5% compounded continuously to achieve her goal?
Bob and Alice want to remodel their bathroom in 3 years. They estimate the job will cost $25,000. How much must they invest now at an annual interest rate of 7% compounded quarterly to achieve their goal? What if the compounding were continuous?
An epidemic spreads through a community so that t weeks after its outbreak, the number of people who have been infected is given by a function of the formwhere B is the number of residents in the community who are susceptible to the disease. If 1/5 of the susceptible residents were infected
In Exercises 43 through 46, find an equation for the tangent line to the given curve at the specified point.y = x ln x2 where x = 1
In Exercises 39 through 46, find the largest and smallest values of the given function over the prescribed closed, bounded interval.f(x) = e−2x − e−4x for 0 ≤ x ≤ 1.
In terms of effective interest rate, order the following nominal rate investments from lowest to highest:a. 7.9% compounded semiannuallyb. 7.8% compounded quarterlyc. 7.7% compounded monthlyd. 7.65% compounded continuously
How quickly will money double if it is invested at an annual interest rate of 7% compounded continuously?
In Exercises 43 through 46, find an equation for the tangent line to the given curve at the specified point.y = (x2 − x)e−x where x = 0
In Exercises 39 through 46, find the largest and smallest values of the given function over the prescribed closed, bounded interval.h(s) = 2s ln s − s2 for 0.5 ≤ s ≤ 2
Money deposited in a certain bank doubles every 13 years. The bank compounds interest continuously. What annual interest rate does the bank offer?
A medical researcher determines that t hours from the time a toxin is introduced to a bacterial colony, the population will bea. What is the population at the time the toxin is introduced?b. When does the maximum bacterial population occur? What is the maximum population?c. What eventually happens
In terms of effective interest rate, order the following nominal rate investments from lowest to highest:a. 4.87% compounded quarterlyb. 4.85% compounded monthlyc. 4.81% compounded daily (365 days)d. 4.79% compounded continuously
In Exercises 43 through 46, find an equation for the tangent line to the given curve at the specified point.y = x3e2−x where x = 2
In Exercises 47 through 52, find an equation for the tangent line to y = f (x) at the specified point. e2x f(x) = where x = 1
How long will it take for a quantity of money A0 to triple in value if it is invested at an annual interest rate r compounded continuously?
According to a certain logistic model, the world’s population (in billions) t years after 1960 will be approximatelya. If this model is correct, at what rate was the world’s population increasing with respect to time in the year 2000? At what percentage rate was the population increasing at
In Exercises 43 through 46, find an equation for the tangent line to the given curve at the specified point.y = (x + ln x)3 where x = 1
If an account that earns interest compounded continuously takes 12 years to double in value, how long will it take to triple in value?
The Morenos invest $10,000 in an account that grows to $12,000 in 5 years. What is the annual interest rate r if interest is compoundeda. Quarterly b. Continuously
In Exercises 47 through 52, find an equation for the tangent line to y = f (x) at the specified point. f(x) = In x X where x = 1
An efficiency expert hired by a manufacturing firm has compiled these data relating workers’ output to their experience:Suppose output Q is related to experience t by a function of the form Q(t) = 500 − Ae−kt. Find the function of this form that fits the data. What output is expected from a
Find f(9) if f(x) = ekx and f(3) = 2.
In Exercises 47 through 52, find an equation for the tangent line to y = f (x) at the specified point.f(x) = (x + 1)e−2x; where x = 0
Jingfei is the marketing manager for a major firm. She estimates that t days after termination of an advertising campaign for a new product, S(t) units will be sold, wherea. How many units are being sold at the time the advertising ends?b. How many units should Jingfei expect her company to have
A certain bank offers an interest rate of 6% per year compounded annually. A competing bank compounds its interest continuously. What (nominal) interest rate should the competing bank offer so that the effective interest rates of the two banks will be equal?
A sum of money is invested at a certain fixed interest rate, and the interest is compounded quarterly. After 15 years, the money has doubled. How will the balance at the end of 30 years compare with the initial investment?
Find f(8) if f(x) = A · 2kx, f(0) = 20, and f(2) = 40.
A statistical study indicates that the fraction of the electric toasters manufactured by a certain company that are still in working condition after t years of use is approximately f(t) = e−0.2t.a. What fraction of the toasters can be expected to work for at least 3 years?b. What fraction of the
Lyle buys a rare stamp for $500. If the annual rate of inflation is 4%, how much should he ask when he sells it in 5 years to break even?
In Exercises 47 through 52, find an equation for the tangent line to y = f (x) at the specified point.f(x) = x2 ln √x; where x = 1
A manufacturer determines that the supply function for x units of a particular commodity is S(x) = e0.02x and the corresponding demand function is D(x) = 3e−0.03x.a. Find the demand price p = D(x) when the level of production is x = 10 units.b. Find the supply price p = S(x) when x = 12 units.c.
A radioactive substance decays exponentially. If 500 grams of the substance were present initially and 400 grams are present 50 years later, how many grams will be present after 200 years?
A sum of money is invested at a certain fixed interest rate, and the interest is compounded continuously. After 10 years, the money has doubled. How will the balance at the end of 20 years compare with the initial investment?
Parkinson’s law* states that in any administrative department not actively at war, the staff will grow by about 6% per year, regardless of need.a. Parkinson applied his law to the size of the British Colonial Office. He noted that the Colonial Office had 1,139 staff members in 1947. How many
In Exercises 47 through 52, find an equation for the tangent line to y = f (x) at the specified point.f(x) = x − ln x; where x = e
Which investment has the greater effective interest rate: 8.25% per year compounded quarterly or 8.20% per year compounded continuously?
The population density x miles from the center of a certain city is D(x) = 12e−0.07x thousand people per square mile.a. What is the population density at the center of the city?b. What is the population density 10 miles from the center of the city?
A function of the form C(t) = Ate−kt, where A and k are positive constants, is called a surge function and is sometimes used to model the concentration of drug in a patient’s bloodstream t hours after the drug is administered. Assume t 0.a. Find C'(t), and determine the time interval where the
In Exercise 60, Section 2.3, you were given a function to model the production of blood cells. Such models are useful in the study of leukemia and other so-called dynamical diseases in which certain physiological systems begin to behave erratically. An alternative model* for blood cell production
It is estimated that t years from now the population of a certain country will be P million people, wherea. Sketch the graph of P(t).b. What is the current population?c. What will be the population in 20 years?d. What happens to the population in the long run? P(t) 30 1 + 2e -0.05t
A manufacturer of digital cameras estimates that when cameras are sold for x dollars apiece, consumers will buy 8,000e−0.02x cameras each week. He also determines that profit is maximized when the selling price x is 1.4 times the cost of producing each unit. What price maximizes weekly profit?
A community grows in such a way that t years from now, its population is P(t) thousand, wherea. What is the population when t = 0.b. How long does it take for the population to double its initial value?c. What is the average rate of growth of the population over the first 10 years? P(t) = 51 +100
The size of a bacterial culture grows in such a way that after t minutes, there are P(t) = A · 20.001t bacteria present, for some constant A. After 10 minutes, there are 10,000 bacteria.a. What is A?b. How many bacteria are initially present (t = 0)? After 20 minutes? After 1 hour?c. At what
A medical student studying the growth of bacteria in a certain culture has compiled these data:Use these data to find an exponential function of the form Q(t) = Q0ekt expressing the number of bacteria in the culture as a function of time. How many bacteria are present after 1 hour? Number of
Radioactive iodine 133I has a half-life of 20.9 hours. If injected into the bloodstream, the iodine accumulates in the thyroid gland.a. After 24 hours, a medical technician scans a patient’s thyroid gland to determine whether thyroid function is normal. If the thyroid has absorbed all the iodine,
The concentration of drug in a patient’s bloodstream t hours after an injection is given by C(t) = Ae−0.87t milligrams per milliliter (mg/ml) for constant A. The concentration is 4 mg/ml after 1 hour.a. What is A?b. What is the initial concentration (t = 0)? The concentration after 2 hours?c.
When a chain, a telephone line, or a TV cable is strung between supports, the curve it forms is called a catenary. A typical catenary curve isSketch this catenary curve. y = 0.125(ex + e-4x)
A Gompertz curve is the graph of a function of the general form N(t) = CABt where A, B, and C are constants. Such curves are used by psychologists and others to describe such things as learning and growth within an organization.†a. Suppose the personnel director of a large corporation determines
An actuary measures the probability that a person in a certain population will die at age x by the formula where λ is a constant such that 0 a. Find the maximum value of P(x) in terms of λ.b. Sketch the graph of P(x).P P(x) = Ả xe - λx
The concentration of drug in a patient’s bloodstream t hours after an injection is given by C(t) = 3 · 2−0.75t milligrams per milliliter (mg/ml).a. What is the concentration when t = 0? After 1 hour?b. What is the average rate of change of concentration during the second hour?
A forged painting allegedly painted by Rembrandt in 1640 is found to have 99.7% of its original 14C. When was it actually painted? What percentage of the original 14C should remain if it were legitimate?
In Exercises 65 through 68, the demand function q = D(p) for a particular commodity is given in terms of a price p per unit at which all q units can be sold. In each case:(a) Find the elasticity of demand and determine the values of p for which the demand is elastic, inelastic, and of unit
In Exercises 65 through 68, the demand function q = D(p) for a particular commodity is given in terms of a price p per unit at which all q units can be sold. In each case:(a) Find the elasticity of demand and determine the values of p for which the demand is elastic, inelastic, and of unit
An environmental study of a certain suburban community suggests that t years from now, the average level of carbon monoxide in the air will be Q(t) = 4e0.03t parts per million.a. At what rate will the carbon monoxide level be changing with respect to time 2 years from now?b. At what percentage rate
In 1389, Pierre d’Arcis, the bishop of Troyes, wrote a memo to the pope, accusing a colleague of passing off “a certain cloth, cunningly painted,” as the burial shroud of Jesus Christ. Despite this early testimony of forgery, the image on the cloth is so compelling that many people regard it
In Exercises 57 through 64, use logarithmic differentiation to find the derivative f'(x).f(x) = log2(√x)
The number of bacteria in a certain culture grows exponentially. If 5,000 bacteria were initially present and 8,000 were present 10 minutes later, how long will it take for the number of bacteria to double?
The size of a bacterial population P(t) grows at the rate of 3.1% per day. If the initial population is 10,000, what is the population after 10 days?
It is estimated that t years after 2005, the population of a certain country will be P(t) million people where P(t) = 2 · 50.018ta. What was the population in 2005?b. What will the population be in 2015?
In Exercises 57 through 64, use logarithmic differentiation to find the derivative f'(x).f(x) = 5x2
It is projected that t years from now, the population of a certain country will be P(t) = 50e0.02t million.a. What is the current population?b. What will the population be 30 years from now?
In Exercises 57 through 64, use logarithmic differentiation to find the derivative f'(x). f(x) e -3x √2x-5 (6 – 5x)*
The value of a certain industrial machine decreases exponentially. If the machine was originally worth $50,000 and was worth $20,000 five years later, how much will it be worth when it is 10 years old?
The Dead Sea Scrolls were written on parchment in about 100 B.C. What percentage of the original 14C in the parchment remained when the scrolls were discovered in 1947?
The double declining balance formula in accounting iswhere V(t) is the value after t years of an article that originally cost V0 dollars and L is a constant, called the “useful life” of the article.a. A refrigerator costs $875 and has a useful life of 8 years. What is its value after 5 years?
In Exercises 31 through 38, determine where the given function is increasing and decreasing and where its graph is concave upward and concave downward. Sketch the graph, showing as many key features as possible (high and low points, points of inflection, asymptotes, intercepts, cusps, vertical
In Exercises 1 through 38, differentiate the given function. g(u) = In(u + √² + 1)
Suppose $1,000 is invested at an annual interest rate of 7%. Compute the future value of the investment after 10 years if the interest is compounded:a. Annuallyb. Quarterlyc. Monthlyd. Continuously
In Exercises 1 through 38, differentiate the given function. L(x) = In x² + 2x - 3 x² + 2x + 1
Suppose you win a parcel of land whose market value t years from now is estimated to bedollars. If the prevailing interest rate remains constant at 7% compounded continuously, when will it be most advantageous to sell the land? (Use a graphing utility and ZOOM and TRACE to make the required
Suppose $5,000 is invested at an annual interest rate of 10%. Compute the future value of the investment after 10 years if the interest is compounded:a. Annuallyb. Semiannuallyc. Daily (using 365 days per year)d. Continuously
A manufacturer can produce digital recorders at a cost of $125 apiece and estimates that if they are sold for x dollars apiece, consumers will buy approximately 1,000e−0.02x each week.a. Express the profit P as a function of x. Sketch the graph of P(x).b. At what price should the manufacturer
In Exercises 31 through 38, determine where the given function is increasing and decreasing and where its graph is concave upward and concave downward. Sketch the graph, showing as many key features as possible (high and low points, points of inflection, asymptotes, intercepts, cusps, vertical
In Exercises 1 through 38, differentiate the given function. f(x) 2.t X
In Exercises 21 through 36, solve the given equation for x. 5 1 + 2e * = 3 =
In Exercises 21 through 36, solve the given equation for x. 25e⁰.1x 0.1x +3 + 3 10
In Exercises 21 through 36, solve the given equation for x.3x = e2
In Exercises 31 through 38, determine where the given function is increasing and decreasing and where its graph is concave upward and concave downward. Sketch the graph, showing as many key features as possible (high and low points, points of inflection, asymptotes, intercepts, cusps, vertical
In Exercises 33 and 34, find the values of the constants C and b so that the curve y = Cbx contains the indicated points.(2, 12) and (3, 24)
In Exercises 33 and 34, find the values of the constants C and b so that the curve y = Cbx contains the indicated points.(2, 3) and (3, 9)
Aaron owns a stamp collection that is currently worth $1,200 and whose value increases linearly at the rate of $200 per year. If the prevailing interest rate remains constant at 8% per year compounded continuously, when will it be most advantageous for Aaron to sell the collection and invest the
In Exercises 21 through 36, solve the given equation for x.ak = ekx
In Exercises 31 through 38, determine where the given function is increasing and decreasing and where its graph is concave upward and concave downward. Sketch the graph, showing as many key features as possible (high and low points, points of inflection, asymptotes, intercepts, cusps, vertical
Find the present value of $10,000 over a term of 5 years at an annual interest rate of 7% if interest is compounded:a. Annuallyb. Quarterlyc. Daily (use 365 days)d. Continuously
In Exercises 31 through 38, determine where the given function is increasing and decreasing and where its graph is concave upward and concave downward. Sketch the graph, showing as many key features as possible (high and low points, points of inflection, asymptotes, intercepts, cusps, vertical
In Exercises 1 through 38, differentiate the given function. f(x)= log₂ x Vx
In Exercises 1 through 38, differentiate the given function.f(x) = x23x2
A product is introduced and t months later, its unit price is p(t) hundred dollars, wherea. For what values of t is the price increasing? When is it decreasing?b. When is the price decreasing most rapidly?c. What happens to the price in the long run (as t → + ∞)? P In(t+1) t + 1 +5
Find the present value of $25,000 over a term of 10 years at an annual interest rate of 5% if interest is compounded:a. Semiannuallyb. Monthlyc. Continuously
In Exercises 5 through 12, evaluate the given expressions. a. 82/3 + 163/4 b. 27 +36 3/2 121
In Exercises 5 through 20, determine where the given function is increasing and decreasing and where its graph is concave upward and concave downward. Sketch the graph of the function. Show as many key features as possible (high and low points, points of inflection, vertical and horizontal
In Exercises 1 through 38, differentiate the given function.f(x) = x log10 x
If log2 x = 5, what is ln x?
Let V(t) be the value of an asset t years from now, and assume that the prevailing annual interest rate remains fixed at r (expressed as a decimal) compounded continuously.a. Show that the present value of the asset P(t) = V(t)e−rt has a critical number where V'(t) = V(t)r. (Using economic
It is determined that q units of a commodity can be sold when the price is p hundred dollars per unit, wherea. Verify that the demand function q( p) decreases as p increases for p 0.b. For what price p is revenue R = pq maximized? What is the maximum revenue? q(p) = 1,000(p + 2)e P
In Exercises 31 through 38, determine where the given function is increasing and decreasing and where its graph is concave upward and concave downward. Sketch the graph, showing as many key features as possible (high and low points, points of inflection, asymptotes, intercepts, cusps, vertical
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