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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 Exercises 1 through 20, find the indicated indefinite integral. |(₁³² + 7/7) dt - 31² +
For the consumers’ demand functions D(q) in Exercises 1 through 6:(a) Find the total amount of money consumers are willing to spend to obtain q0 units of the commodity.(b) Sketch the demand curve and interpret the consumer willingness to spend in part (a) as an area.D(q) = 50e−0.04q dollars per
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. 2 f3d1 dt
In Exercises 1 through 20, find the indicated indefinite integral. fax + (x + 1)(2x² + √x) dx
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. ftox [(x - 1)5 + 3(x - 1)² + 5] dx
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. X -0.3 dx
In Exercises 5 through 18, sketch the given region R and then find its area.R is the region bounded by the curves y = x2, y = −x2, and the line x = 1.
Suppose q hundred units of a certain commodity are demanded by consumers when the price is p = 25 − q2 dollars per unit. What is the consumers’ surplus for the commodity when the level of production is q0 = 4 (400 units)?
In Exercises 7 through 10, p = D(q) is the price (dollars per unit) at which q units of a particular commodity will be demanded by the market (that is, all q units will be sold at this price), and q0 is a specified level of production. In each case, find the price p0 = D(q0) at which q0 units will
In Exercises 5 through 18, sketch the given region R and then find its area.R is the region bounded by the x axis and the curve y = −x2 + 4x − 3.
In Exercises 7 through 14, find the volume of the solid formed by rotating the region R about the x axis.R is the region under the line y = 3x + 1 from x = 0 to x = 1.
Money is transferred continuously into Rosa’s account at the constant rate of $5,000 per year. The account earns interest at the annual rate of 5% compounded continuously. How much will be in Rosa’s account at the end of 3 years?
In Exercises 7 through 10, p = D(q) is the price (dollars per unit) at which q units of a particular commodity will be demanded by the market (that is, all q units will be sold at this price), and q0 is a specified level of production. In each case, find the price p0 = D(q0) at which q0 units will
In Exercises 5 through 18, sketch the given region R and then find its area.R is the region bounded by the curves y = ex, y = e−x, and the line x = ln 2.
In Exercises 7 through 14, find the volume of the solid formed by rotating the region R about the x axis.R is the region under the curve from y = √x from x = 1 to x = 4.
In Exercises 9 through 14, estimate the area under the graph of y = f(x) over the interval 0 ≤ x ≤ 4 by computing a Riemann sum of f over the interval with 8 subintervals, using left endpoints. Then find the actual area using the fundamental theorem of calculus.f(x) = x
The magnitude formula for the Richter scale iswhere E is the energy released by the earthquake (in joules), and E0 = 104.4 joules is the energy released by a small reference earthquake used as a standard of measurement.a. The 1906 San Francisco earthquake released approximately 5.96 × 1016 joules
The gross domestic product (GDP) of a certain country was 100 billion dollars in 1995 and 165 billion dollars in 2005. Assuming that the GDP is growing exponentially, what will it be in the year 2015?
Suppose the percentage of alcohol in the blood t hours after consumption is given bya. At what rate is the blood alcohol level changing at time t?b. How much time passes before the blood alcohol level begins to decrease?c. Suppose the legal limit for blood alcohol is 0.04%. How much time must pass
How much should you invest now at an annual interest rate of 6.25% so that your balance 10 years from now will be $2,000 if interest is compounded:a. Monthlyb. Continuously
A manufacturer determines that the supply function for x units of a particular commodity is S(x) = ln(x + 2) and the corresponding demand function is D(x) = 10 − ln(x + 1).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 = 100
Corwin Lewis, the manager of a fishery, determines that t days after 1,000 fish of a particular species are released into a pond, the average weight of an individual fish will be w(t) pounds and the proportion of the fish still alive after t days will be p(t), wherea. The expected yield E(t) from
The following data were compiled by a researcher during the first 10 minutes of an experiment designed to study the growth of bacteria:Assuming that the number of bacteria grows exponentially, how many bacteria will be present after 30 minutes? Number Number of minutes of bacteria 0 5,000 10 8,000
Based on the estimate that there are 10 billion acres of arable land on the earth and that each acre can produce enough food to feed 4 people, some demographers believe that the earth can support a population of no more than 40 billion people. The population of the earth was approximately 3 billion
In Exercises 53 through 56, find the second derivative of the given function.f(x) = e2x + 2e−x
In 1626, Peter Minuit traded trinkets worth $24 to a tribe of Native Americans for land on Manhattan Island. Assume that in 1990 the same land was worth $25.2 billion. If the sellers in this transaction had invested their $24 at 7% annual interest compounded continuously during the entire 364-year
In Exercises 53 through 56, find the second derivative of the given function.f(x) = ln(2x) + x2
An economist has compiled these data on the gross domestic product (GDP) of a certain country:Use these data to predict the GDP in the year 2015 if the GDP is growing:a. Linearly, so that GDP = at + b.b. Exponentially, so that GDP = Aekt. Year GDP (in billions) 1995 100 2005 180
An investment firm estimates that the value of its portfolio after t years is A million dollars, wherea. What is the value of the account when t = 0?b. How long does it take for the account to double its initial value?c. How long does it take before the account is worth a billion dollars? A(t) =
The following data were compiled by a researcher during an experiment designed to study the decay of a radioactive substance:Assuming that the sample of radioactive substance decays exponentially, how much is left after 20 hours? Number of hours Grams of substance 0 1,000 5 700
The manager of a fishery determines that t weeks after 3,000 fish of a particular species are hatched, the average weight of an individual fish will be w(t) = 0.8te−0.05t pounds, for 0 ≤ t ≤ 20. Moreover, the proportion of the fish that will still be alive after t weeks is estimated to bea.
It is estimated that if x thousand dollars are spent on advertising, approximately Q(x) = 50 − 40e−0.1x thousand units of a certain commodity will be sold.a. Sketch the sales curve for x ≥ 0.b. How many units will be sold if no money is spent on advertising?c. How many units will be sold if
A drug is injected into a patient’s bloodstream and t seconds later, the concentration of the drug is C grams per cubic centimeter (g/cm3), wherea. What is the drug concentration after 10 seconds?b. How long does it take for the drug concentration to reach 0.12 g/cm3? C(t) = 0.1(1+ 3e-0.031)
An epidemic spreads throughout a community so that t weeks after its outbreak, the number of residents who have been infected is given by a function of the formwhere A is the total number of susceptible residents. Show that the epidemic is spreading most rapidly when half of the susceptible
The concentration of a drug in a patient’s kidneys at time t (seconds) is C grams per cubic centimeter (g/cm3), wherea. What is the drug concentration after 20 seconds? After 60 seconds?b. How long does it take for the drug concentration to reach 0.75 g/cm3? C(t) = 0.4(2 0.13e-0.021) -
A manufacturer will supply S(x) = 300e0.03x − 310 units of a particular commodity when the price is x dollars per unit.a. How many units will be supplied when the unit price is $10?b. How many more units will be supplied when the unit price is $100 than when it is $80?
The gross domestic product (GDP) of a certain country was $500 billion at the beginning of the year 2005 and increases at the rate of 2.7% per year.a. Express the GDP of this country as a function of the number of years t after 2005.b. What does this formula predict the GDP of the country will be
In Exercises 53 through 56, find the second derivative of the given function.f(t) = t2 ln t
How quickly will $2,000 grow to $5,000 when invested at an annual interest rate of 8% if interest is compounded:a. Quarterlyb. Continuously
Psychologists believe that when a person is asked to recall a set of facts, the number of facts recalled after t minutes is given by a function of the form Q(t) = A(1 − e−kt), where k is a positive constant and A is the total number of relevant facts in the person’s memory.a. Sketch the
In Exercises 53 through 56, find the second derivative of the given function.g(t) = t2e−t
An employer determines that the daily output of a worker on the job for t weeks is Q(t) = 120 − Ae−kt units. Initially, the worker can produce 30 units per day, and after 8 weeks, can produce 80 units per day. How many units can the worker produce per day after 4 weeks?
In Exercises 57 through 64, use logarithmic differentiation to find the derivative f'(x). f(x) = = (x + 2)5 / 3 x 3x - 5
Determine the monthly car payment for a new car costing $15,675, if there is a down payment of $4,000 and the car is financed over a 5-year period at an annual rate of 6% compounded monthly.
In a learning model proposed by C. L. Hull, the habit strength H of an individual is related to the number r of reinforcements by the equationH(r) = M(1 − e−kr)a. Sketch the graph of H(r). What happens to H(r) as r→ + ∞?b. Show that if the number of reinforcements is doubled from r to 2r,
In Exercises 57 through 64, use logarithmic differentiation to find the derivative f'(x).f(x) = (2x + 3)2(x − 5x2)1/2
An archaeologist has found a fossil in which the ratio ofthe ratio found in the atmosphere. Approximately how old is the fossil? 14C to ¹2C is 12, 1 3
Suppose that for the first 6 years of a moose’s life, its shoulder height H(t) and tip-to-tip antler length A(t) increase with time t (years) according to the formulas H(t) = 125e0.08t and A(t) = 50e0.16t, where H and A are both measured in centimeters (cm).a. On the same graph, sketch y = H(t)
A home loan is made for $150,000 at 9% annual interest, compounded monthly, for 30 years. What is the monthly mortgage payment on this loan?
In Exercises 57 through 64, use logarithmic differentiation to find the derivative f'(x). f(x) = 4/2x + 1 V 1 - 3x
In Exercises 57 through 64, use logarithmic differentiation to find the derivative f'(x).f(x) = x2e−x(3x + 5)3
The world’s population grows at the rate of approximately 2% per year. If it is assumed that the population growth is exponential, then the population t years from now will be given by a function of the form P(t) = P0e0.02t, where P0 is the current population. Assuming that this model of
Suppose a family figures it can handle monthly mortgage payments of no more than $1,200. What is the largest amount of money they can borrow, assuming the lender is willing to amortize over 30 years at 8% annual interest compounded monthly?
You have a debt of $10,000, which is scheduled to be repaid at the end of 10 years. If you want to repay your debt now, how much should your creditor demand if the prevailing annual interest rate is:a. 7% compounded monthlyb. 6% compounded continuously
The concentration of a certain drug in a patient’s bloodstream t hours after being administered orally is assumed to be given by the function C(t) = Ate−kt, where C is measured in micrograms of drug per milliliter of blood. Monitoring devices indicate that a maximum concentration of 5 occurs 20
You are selling your car for $6,000. A potential buyer says, “I will pay you $1,000 now for the car and pay off the rest at 12% interest with monthly payments for 3 years. Let’s see . . . 12% of the $5,000 is $600 and $5,600 divided by 36 months is $155.56, but I’ll pay you $160 per month for
A bank compounds interest continuously. What annual interest rate does it offer if $1,000 grows to $2,054.44 in 12 years?
In Exercises 57 through 64, use logarithmic differentiation to find the derivative f'(x). 1 + x₂(x −9)(I + x) = (x)ƒ
In a classic paper,* E. Heinz modeled the concentration y(t) of a drug injected into the body intramuscularly by the functionwhere t is the number of hours after the injection and a, b, and c are positive constants, with b > a.a. When does the maximum concentration occur? What happens to the
According to the Ebbinghaus model, the fraction F(t) of subject matter you will remember from this course t months after the final exam can be estimated by the formulawhere B is the fraction of the material you will never forget and k is a constant that depends on the quality of your memory.
Tests of an artifact discovered at the Debert site in Nova Scotia show that 28% of the original 14C is still present. Approximately how old is the artifact?
Ravi Patel, the marketing editor at a major publishing house, estimates that if x thousand complimentary copies of a new text are distributed to instructors, the first-year sales will be approximatelythousand copies. How many complimentary copies should Ravi send out to instructors if his goal is
A bank pays 5% interest compounded quarterly, and a savings institution pays 4.9% interest compounded continuously. Over a 1-year period, which account pays more interest? What about a 5-year period?
Suppose an environmental change affects the organism in Exercise 49 in such a way that an individual is only half as likely as before to survive to age x years. If the number of female births f(x) remains the same, how is the ideal age for reproduction affected by this change?Data from Exercises
Suppose during a 10-year period of rapid inflation, it is estimated that prices inflate at an annual rate of 5% per year. If an item costs $3 at the beginning of the period, what would you expect to pay for the same item 10 years later?
A psychologist measures a child’s capability to learn and remember by the functionwhere t is the child’s age in years, for 0 ≤ t ≤ 5.Answer these questions about this model.a. At what age does a child have the greatest learning capability?b. At what age is a child’s learning capability
It is estimated that t years from now, the population of a certain country will bea. Sketch the graph of P(t).b. What is the current population?c. What will the population be 50 years from now?d. What will happen to the population in the long run? P(t) 20 2 + 3e-0.06t million.
Pilar owns a parcel of land whose value t years from now will be If the prevailing interest rate remains constant at 6% per year compounded continuously, when should Pilar sell the land to maximize its present value? V(t) = 8,000e dollars.
A company organizes a training program in which it is determined that after t weeks the average trainee produces P(t) = 50(1 − e−0.15t) units while a typical new worker without special training producesa. How many units does the average trainee produce during the third week of the training
In Exercises 21 through 36, solve the given equation for x. 1 5 = 3 ln x -In 2 -In x
In Exercises 15 through 30, find the derivative dy/dx. In some of these problems, you may need to use implicit differentiation or logarithmic differentiation. y = e e-2x (2-x²³)³/² V1 + x²
In Exercises 1 through 38, differentiate the given function. f(x)= = et + ex et - ex
Suppose your family owns a rare book whose value t years from now will beIf the prevailing interest rate remains constant at 6% per year compounded continuously, when will it be most advantageous for your family to sell the book and invest the proceeds? V(t) = 200e √21 dollars.
In Exercises 39 through 42, find the largest and smallest values of the given function over the prescribed closed, bounded interval. f(x) e-x/2 1² 2 for -5 ≤ x ≤ −1
In Exercises 21 through 36, solve the given equation for x. In x=(In 16 + 2 In 2) 1370
In Exercises 41 through 46, find the rate of change of the given function f (x) with respect to x for the prescribed value x = c.f(x) = x3 − 3x + 5; x = 2
If you invest $2,000 at 5% compounded continuously, how much will your account be worth in 3 years? How long does it take before your account is worth $3,000?
Public health records indicate that t weeks after the outbreak of a certain strain of influenza, approximatelythousand people had contracted the disease.a. Sketch the graph of f (t).b. How many people had the disease initially?c. How many had caught the disease by the end of 3 weeks?d. If the trend
In Exercises 29 through 32, use a graphing calculator to sketch the graph of the given exponential function.y = ex + 2
Find if ln a = 2 and ln b = 3. 1 Vab³ In-
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 39 through 42, find the largest and smallest values of the given function over the prescribed closed, bounded interval. g(t) = In(Vt) 2 for 1 t ≤2
In Exercises 1 through 38, differentiate the given function.f(x) = ln(e−x + x)
A traffic accident was witnessed by 10% of the residents of a small town, and 25% of the residents had heard about the accident 2 hours later. Suppose the number N(t) of residents who had heard about the accident t hours after it occurred is given by a function of the form where B is the population
In Exercises 39 through 46, find the largest and smallest values of the given function over the prescribed closed, bounded interval. g(x) 2x + 1 for 0 ≤ x ≤ 1
In Exercises 29 through 32, use a graphing calculator to sketch the graph of the given exponential function.y = 4 − e−x
Findif ln b = 6 and ln c = −2. Vb a In (VD)* C с 1 -In a
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(s) = es + ln s
The aerobic rating of a person x years old is modeled by the functiona. At what age is a person’s aerobic rating largest?b. At what age is a person’s aerobic rating decreasing most rapidly? A(x) = 110(In x - 2) X for x 10
It is known that fluorocarbons have the effect of depleting ozone in the upper atmosphere. Suppose it is found that the amount of original ozone Q0 that remains after t years is given bya. At what percentage rate is the ozone level decreasing at time t?b. How many years will it take for 10% of the
In Exercises 29 through 32, use a graphing calculator to sketch the graph of the given exponential function.y = 2x/2
In Exercises 21 through 36, solve the given equation for x.ln x = 2(ln 3 − ln 5)
In Exercises 39 through 42, find the largest and smallest values of the given function over the prescribed closed, bounded interval.h(t) = (e−t + et)5 for −1 ≤ t ≤ 1
In Exercises 39 through 46, find the largest and smallest values of the given function over the prescribed closed, bounded interval.f(x) = (3x − 1)e−x for 0 ≤ x ≤ 2
In Exercises 39 through 42, find the effective interest rate re for the given investment.Nominal annual rate of 5%, compounded continuously
In Exercises 39 through 46, find the largest and smallest values of the given function over the prescribed closed, bounded interval. f(x) = In(x + 1) x + 1 for 0≤x≤ 2
According to Hoorweg’s law, when a nerve is stimulated by discharges from an electrical condenser of capacitance C, the electric energy required to elicit a minimal response (a muscle contraction) is given bywhere a, b, and R are positive constants.a. For what value of C is the energy E(C)
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 functiona. What market price corresponds to the production of x = 0 units?b. How much revenue is obtained when 200 units of the commodity are produced?c.
In Exercises 39 through 42, find the effective interest rate re for the given investment.Nominal annual rate of 7.3%, compounded continuously
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