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Laurence D. Hoffmann, Gerald Bradley, Applied Calculus for Business, Economics,and the Social and Life Sciences, Tenth Edition
CONSUMER DEMAND In Exercises 57through 60, the demand function p =D(x) and the total cost function C(x) for a particular commodity are given in terms of the level of production x. In each case, find:
(a) The revenue R(x) and profit P(x).
(b) All values of x for which production of the commodity is profitable.
MANUFACTURING COST Suppose the total cost of manufacturing q units of a certain commodity is C(q) thousand dollars, where
C(q) = 0.01q2 + 0.9q + 2
a. Compute the cost of manufacturing 10 units.
b. Compute the cost of manufacturing the 10th unit.
DISTRIBUTION COST Suppose that the number of workerhours required to distribute new
telephone books to x% of the households in a certain rural community is given by the function
a. What is the domain of the function W?
b. For what values of x does W(x) have a practical interpretation in this context?
c. How many workerhours were required to distribute new telephone books to the first 50% of
the households?
d. How many workerhours were required to distribute new telephone books to the entire
community?
e. What percentage of the households in the community had received new telephone books by the time 150 workerhours had been expended?
WORKER EFFICIENCY An efficiency study of the morning shift at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M. will have assembled f (x) = x3 + 6x2 + 15x
television sets x hours later.
a. How many sets will such a worker have assembled by 10:00 A.M.? [Hint: At 10:00 A.M., x = 2.]
b. How many sets will such a worker assemble between 9:00 and 10:00 A.M.?
IMMUNIZATION Suppose that during a nationwide program to immunize the population against a certain form of influenza, public health officials found that the cost of inoculating x% of the population was approximately million dollars.
a. What is the domain of the function C?
b. For what values of x does C(x) have a practical interpretation in this context?
c. What was the cost of inoculating the first 50% of the population?
d. What was the cost of inoculating the second 50% of the population?
e. What percentage of the population had been inoculated by the time 37.5 million dollars had been spent?
BLOOD FLOW Biologists have found that the speed of blood in an artery is a function of the distance of the blood from the artery's central axis. According to Poiseuille's law,* the speed (in centimeters per second) of blood that is r centimeters from the central axis of an artery is given by the function
S(r) = C(R2  r2), where C is a constant and R is the radius of the artery. Suppose that for a certain artery, C = 1.76 105 and R = 1.2 102 centimeters.
a. Compute the speed of the blood at the central axis of this artery.
b. Compute the speed of the blood midway between the artery's wall and central axis.
MANUFACTURING COST A manufacturer can produce digital recorders at a cost of $40 apiece. It is estimated that if the recorders are sold for p dollars apiece, consumers will buy 120  p of them a month. Express the manufacturer's monthly profit as a function of price, graph this function, and use the graph to estimate the optimal selling price.
MANUFACTURING COST A manufacturer can produce tires at a cost of $20 apiece. It is estimated that if the tires are sold for p dollars apiece, consumers will buy 1,560  12p of them each month. Express the manufacturer's monthly profit as a function of price, graph this function, and use the graph to determine the optimal selling price. How many tires will be sold each month at the optimal price?
RENEWABLE RESOURCES The accompanying graph shows how the volume of lumber V in a tree
varies with time t (the age of the tree). Use the graph to estimate the rate at which V is changing with respect to time when t = 30 years. What seems to be happening to the rate of change of V as t increases without bound (that is, in the "long run")?
POPULATION GROWTH 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?
THERMAL INVERSION 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 below 0°C. Use the accompanying graph to estimate the rate at which temperature T is changing with respect to altitude h at an altitude of 1,000 meters and also at 2,000 meters.
PROFIT A manufacturer can produce digital recorders at a cost of $50 apiece. It is estimated that if the recorders are sold for p dollars apiece, consumers will buy q = 120  p recorders each month.
a. Express the manufacturer's profit P as a function of q.
b. What is the average rate of profit obtained as the level of production increases from q = 0 to q = 20?
c. At what rate is profit changing when q = 20 recorders are produced? Is the profit increasing or decreasing at this level of production?
PROFIT A manufacturer determines that when x hundred units of a particular commodity are produced, the profit will be P(x) = 4,000(15  x)(x2) dollars.
a. Find P (x).
b. Determine where P (x) = 0. What is the significance of the level of production xm where this occurs?
MANUFACTURING OUTPUT At a certain factory, it is determined that an output of Q units
is to be expected when L workerhours of labor are employed, where
a. Find the average rate of change of output as the labor employment changes from workerhours to 3,100 workerhours.
b. Use calculus to find the instantaneous rate of change of output with respect to labor level
when L = 3025
COST OF PRODUCTION A business manager determines that the cost of producing x units of a
particular commodity is C thousands of dollars, where C(x) = 0.04x2 + 5.1x + 40
a. Find the average cost as the level of production changes from x = 10 to x = 11 units.
b. Use calculus to find the instantaneous rate of change of cost with respect to production level when x = 10 and compare with the average cost found in part (a). Is the cost increasing or decreasing when 10 units are being produced?
CONSUMER EXPENDITURE The demand for a particular commodity is given by
D(x) = 35x +200; that is, x units will be sold (demanded) at a price of p= D(x) dollars per unit.
a. Consumer expenditure is the total amount of money consumers pay to buy x units. Express
consumer expenditure E as a function of x.
b. Find the average change in consumer expenditure as x changes from x = 4 to x = 5.
c. Use calculus to find the instantaneous rate of change of expenditure with respect to x when x = 4. Is expenditure increasing or decreasing when x = 4?
UNEMPLOYMENT In economics, the graph in Figure 2.2 is called the Phillips curve, after A. W.
Phillips, a New Zealander associated with the London School of Economics. Until Phillips published his ideas in the 1950s, many economists believed that unemployment and inflation were linearly related. Read an article on the Phillips curve (the source cited with Figure 2.2 would be a good place to start), and write a paragraph on the nature of unemployment in the U.S. economy.
BLOOD PRESSURE Refer to the graph of blood pressure as a function of time shown in Figure 2.9.
a. 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.
b. Write a paragraph on the dynamics of the arterial pulse. Pages 131136 in the reference given
with Figure 2.9 is a good place to start, and there is an excellent list of annotated references to related topics on pp. 137138.
ANIMAL BEHAVIOR Experiments indicate that when a flea jumps, its height (in meters) after t seconds is given by the function H(t) = 4.4t  4.9t2
a. 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 does the flea "land" (return to its initial height)? At what rate is H(t) changing at this time? Is it increasing or decreasing?
VELOCITY A toy rocket rises vertically in such a way that t seconds after liftoff, it is
h(t) = 16t2 + 200t feet above ground.
a. How high is the rocket after 6 seconds?
b. What is the average velocity of the rocket over the first 6 seconds of flight (between t = 0 and
t = 6)?
c. What is the (instantaneous) velocity of the rocket at liftoff (t = 0)? What is its velocity after 40 seconds?
CARDIOLOGY A study conducted on a patient undergoing cardiac catheterization indicated that
the diameter of the aorta was approximately Dmillimeters (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 aortic diameter D as p changes from p = 60 to p = 61
b. Use calculus to find the instantaneous rate of change of diameter D with respect to aortic pressure p when p = 60. Is the pressure increasing or decreasing when p = 60?
c. For what value of p is the instantaneous rate of change of D with respect to p equal to 0? What
is the significance of this pressure?
ANNUAL EARNINGS The gross annual earnings of a certain company were A(t) = 0.1t2 + 10t + 20
thousand dollars t years after its formation in 2004.
a. At what rate were the gross annual earnings of the company growing with respect to time in 2008?
b. At what percentage rate were the gross annual earnings growing with respect to time in 2008?
WORKER EFFICIENCY An efficiency study of the morning shift at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M. will have assembled
f (x) = x3 + 6x2 + 15x units x hours later.
a. Derive a formula for the rate at which the worker will be assembling units after x hours.
b. At what rate will the worker be assembling units at 9:00 A.M.?
c. How many units will the worker actually assemble between 9:00 A.M. and 10:00 A.M.?
EDUCATIONAL TESTING It is estimated that x years from now, the average SAT mathematics
score of the incoming students at an eastern liberal arts college will be f (x) = 6x + 582.
a. Derive an expression for the rate at which the average SAT score will be changing with respect
to time x years from now.
b. What is the significance of the fact that the expression in part (a) is a constant? What is the significance of the fact that the constant in part (a) is negative?
PUBLIC TRANSPORTATION After x weeks, the number of people using a new rapid transit system
was approximately N(x) = 6x3 + 500x + 8,000.
a. At what rate was the use of the system changing with respect to time after 8 weeks?
b. By how much did the use of the system change during the eighth week?
PROPERTY TAX Records indicate that x years after 2005, the average property tax on a threebedroom home in a certain community was
T(x) = 20x2 + 40x + 600 dollars.
a. At what rate was the property tax increasing with respect to time in 2005?
b. By how much did the tax change between the years 2005 and 2009?
ADVERTISING A manufacturer of motorcycles estimates that if x thousand dollars is spent on
advertising, then
cycles 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?
POPULATION GROWTH It is projected that x months from now, the population of a certain
town will be P(x) = 2x + 4x3/2 + 5,000.
a. At what rate will the population be changing with respect to time 9 months from now?
b. At what percentage rate will the population be changing with respect to time 9 months from
now?
POPULATION GROWTH It is estimated that t years from now, the population of a certain town
will be P(t) = t2 + 200t + 10,000.
a. Express the percentage rate of change of the population as a function of t, simplify this function
algebraically, and draw its graph.
b. What will happen to the percentage rate of change of the population in the long run (that is,
as t grows very large)?
SPREAD OF AN EPIDEMIC A medical research team determines that t days after an epidemic begins , people will be infected, for 0 t 20. At what rate is the infected population increasing on the ninth day?
SPREAD OF AN EPIDEMIC A disease is spreading in such a way that after t weeks, the
number of people infected is
N(t) = 5,175  t3(t  8) 0 t 8
a. At what rate is the epidemic spreading after 3 weeks?
b. Suppose health officials declare the disease to have reached epidemic proportions when the
percentage rate of change of N is at least 25%.
Over what time period is this epidemic criterion satisfied?
c. Read an article on epidemiology and write a paragraph on how public health policy is related to the spread of an epidemic.
AIR POLLUTION 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) = 0.05t2 + 0.1t + 3.4 parts
per million.
a. At what rate will the carbon monoxide level be changing with respect to time 1 year from now?
b. By how much will the carbon monoxide level change this year?
c. By how much will the carbon monoxide level change over the next 2 years?
NEWSPAPER CIRCULATION It is estimated that t years from now, the circulation of a local
newspaper will be C(t) = 100t2 + 400t + 5,000.
a. Derive an expression for the rate at which the circulation will be changing with respect to time
t years from now.
b. At what rate will the circulation be changing with respect to time 5 years from now? Will the circulation be increasing or decreasing at that time?
c. By how much will the circulation actually change during the sixth year?
SALARY INCREASES Your starting salary will be $45,000, and you will get a raise of $2,000
each year.
a. Express the percentage rate of change of your salary as a function of time and draw the graph.
b. At what percentage rate will your salary be increasing after 1 year?
c. What will happen to the percentage rate of change of your salary in the long run?
GROSS DOMESTIC PRODUCT The gross domestic product of a certain country is growing
at a constant rate. In 1995 the GDP was 125 billion dollars, and in 2003 it was 155 billion dollars.
If this trend continues, at what percentage rate will the GDP be growing in 2010?
ORNITHOLOGY An ornithologist determines that the body temperature of a certain species of bird fluctuates over roughly a 17hour period according to the cubic formula
T(t) = 68.07t3 + 30.98t2 + 12.52t + 37.1 for 0 t 0.713, where T is the temperature in degrees Celsius measured t days from the beginning of a period.
a. Compute and interpret the derivative T (t).
b. At what rate is the temperature changing at the beginning of the period (t = 0) and at the end of the period (t = 0.713)? Is the temperature increasing or decreasing at each of these times?
c. At what time is the temperature not changing (neither increasing nor decreasing)? What is the bird's temperature at this time? Interpret your result.
PHYSICAL CHEMISTRY According to Debye's formula in physical chemistry, the orientation polarization P of a gas satisfies
where , k, and N are positive constants, and T is the temperature of the gas. Find the rate of change of P with respect to T.
COST MANAGEMENT A company uses a truck to deliver its products. To estimate costs, the manager models gas consumption by the function
gal/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 per gallon.
a. Find an expression for the total cost C(x) of the trip.
b. At what rate is the cost C(x) changing with respect to x when the truck is driven at 40 miles per hour? Is the cost increasing or decreasing at that speed?
MOTION OF A PROJECTILE A stone is dropped from a height of 144 feet.
a. When will the stone hit the ground?
b. With what velocity does it hit the ground?
MOTION OF A PROJECTILE You are standing on the top of a building and throw a ball vertically upward. After 2 seconds, the ball passes you on the way down, and 2 seconds after that, it hits the ground below.
a. What is the initial velocity of the ball?
b. How high is the building?
c. What is the velocity of the ball when it passes you on the way down?
d. What is the velocity of the ball as it hits the ground?
SPY STORY Our friend, the spy who escaped from the diamond smugglers in Chapter 1 (Problem 46 of Section 1.4), is on a secret mission in space. An encounter with an enemy agent leaves him with a mild concussion and temporary amnesia. Fortunately, he has a book that gives the formula for the motion of a projectile and the values of g for various heavenly bodies (32 ft/sec2 on earth, 5.5 ft/sec2 on the moon, 12 ft/sec2 on Mars, and 28 ft/sec2 on Venus). To deduce his whereabouts, he throws a
rock vertically upward (from ground level) and notes that it reaches a maximum height of 37.5 ft
and hits the ground 5 seconds after it leaves his hand. Where is he?
POLLUTION CONTROL It has been suggested that one way to reduce worldwide carbon dioxide
(CO2) emissions is to impose a single tax that would apply to all nations. The accompanying graph shows the relationship between different levels of the carbon tax and the percentage of reduction in CO2 emissions.
a. What tax rate would have to be imposed to achieve a worldwide reduction of 50% in CO2
emissions?
b. Use the graph to estimate the rate of change of the percentage reduction in CO2 emissions when
the tax rate is $200 per ton.
c. Read an article on CO2 emissions and write a paragraph on how public policy can be used to
control air pollution.
DEMAND AND REVENUE The manager of a company that produces graphing calculators
determines that when x thousand calculators are produced, they will all be sold when the price is dollars per calculator.
a. At what rate is demand p(x) changing with respect to the level of production x when 3,000 (x = 3) calculators are produced?
b. The revenue derived from the sale of x thousand calculators is R(x) = xp(x) thousand dollars. At
what rate is revenue changing when 3,000 calculators are produced? Is revenue increasing or
decreasing at this level of production?
SALES The manager of the Many Facets jewelry store models total sales by the function where t is the time (years) since the year 2006 and S is measured in thousands of dollars.
a. At what rate were sales changing in the year 2008?
b. What happens to sales in the "long run" (that is, as t )?
PROFIT 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 sell bottles per month at a total cost of C(p) = 0.2p2 + 3p + 200 dollars.
a. Express Bea's profit P(p) as a function of the price p per bottle.
b. At what rate is the profit changing with respect to p when the price is $12 per bottle? Is profit increasing or decreasing at that price?
ADVERTISING 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, where
a. At what rate is the market percentage P(t) changing with respect to time after 5 weeks? Is the percentage increasing or decreasing at this time?
b. What happens to the percentage P(t) in the "long run"; that is, as t + ? What happens to the rate of change of P(t) as t + ?
BACTERIAL POPULATION A bacterial colony is estimated to have a population of million t hours after the introduction of a toxin.
a. At what rate is the population changing 1 hour after the toxin is introduced (t = 1)? Is the
population increasing or decreasing at this time?
b. At what time does the population begin to decline?
POLLUTION CONTROL 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 by
a. At what rate is the percentage of pollution removal P(x) changing when 16 million dollars is spent? Is the percentage increasing or decreasing at this level of expenditure?
b. For what values of x is P(x) increasing? For what values of x is P(x) decreasing?
PHARMACOLOGY An oral painkiller is administered to a patient, and t hours later, the concentration of drug in the patient's bloodstream is given by
a. 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 t?
b. At what rate is the concentration of drug changing after 1 hour? Is the concentration changing at an increasing or decreasing rate at this time?
c. When does the concentration of the drug begin to decline?
d. Over what time period is the concentration changing at a declining rate?
WORKER EFFICIENCY 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 production changing with respect to time at 9:00 A.M.?
POPULATION GROWTH It is estimated that t years from now, the population of a certain
suburban community will be thousand.
a. Derive a formula for the rate at which the population will be changing with respect to time t years from now.
b. At what rate will the population be growing 1 year from now?
c. By how much will the population actually increase during the second year?
d. At what rate will the population be growing 9 years from now?
e. What will happen to the rate of population growth in the long run?
VELOCITY An object moves along a straight line so that after t minutes, its distance from
its starting point is meters.
a. At what velocity is the object moving at the end of 4 minutes?
b. How far does the object actually travel during the fifth minute?
ACCELERATION After t hours of an 8hour trip, a car has gone kilometers.
a. Derive a formula expressing the acceleration of the car as a function of time.
b. At what rate is the velocity of the car changing with respect to time at the end of 6 hours? Is the velocity increasing or decreasing at this time?
c. By how much does the velocity of the car actually change during the seventh hour?
DRUG DOSAGE The human body's reaction to a dose of medicine can be modeled by a function
of the form where K is a positive constant and M is the amount of medicine absorbed in the blood. The derivative can be thought of as a measure of the sensitivity of the body to the medicine.
a. Find the sensitivity S.
b. Find and give an interpretation of the second derivative.
BLOOD CELL PRODUCTION A biological model measures the production of a certain type of white blood cell (granulocytes) by the function
where A and B are positive constants, the exponent m is positive, and x is the number of cells present.
a. Find the rate of production p (x).
b. Find p (x) and determine all values of x for which p (x) = 0 (your answer will involve m).
c. Read an article on blood cell production and write a paragraph on how mathematical methods can be used to model such production. A good place to start is with the article, "Blood Cell Population Model,
Dynamical Diseases, and Chaos" by W. B. Gearhart and M. Martelli, UMAP Module 1990, Arlington, MA: Consortium for Mathematics and Its Applications, Inc., 1991.
ACCELERATION If an object is dropped or thrown vertically, its height (in feet) after t seconds is H(t) = 16t 2 + S0 t + H0, where S0 is the initial speed of the object and H0 its initial height.
a. Derive an expression for the acceleration of the object.
b. How does the acceleration vary with time?
c. What is the significance of the fact that the answer to part (a) is negative?
ANNUAL EARNINGS The gross annual earnings of a certain company are thousand dollars t years after its formation in January 2005.
a. At what rate will the gross annual earnings of the company be growing in January 2010?
b. At what percentage rate will the gross annual earnings be growing in January 2010?
MANUFACTURING COST At a certain factory, the total cost of manufacturing q units is
C(q) = 0.2q2 + q + 900 dollars. It has been determined that approximately q(t) = t2 + 100t units are manufactured during the first t hours of a production run. Compute the rate at which the total manufacturing cost is changing with respect to time 1 hour after production commences.
CONSUMER DEMAND An importer of Brazilian coffee estimates that local consumers will buy approximately pounds of the coffee per week when the price is p dollars per pound. It is also estimated that t weeks from now, the price of Brazilian coffee will be p(t) = 0.02t2 + 0.1t + 6 dollars per pound.
a. At what rate will the demand for coffee be changing with respect to price when the price is $9?
b. At what rate will the demand for coffee be changing with respect to time 10 weeks from now? Will the demand be increasing or decreasing at this time?
CONSUMER DEMAND When a certain commodity is sold for p dollars per unit, consumers will buy units per month. It is estimated that t months from now, the price of the commodity will be p(t) = 0.4t3/2 + 6.8 dollars per unit. At what percentage rate will the monthly demand for the commodity be changing with respect to time 4 months from now?
AIR POLLUTION It is estimated that t years from now, the population of a certain suburban community will be thousand. An environmental study indicates that the average daily level of carbon monoxide in the air will be
parts per million when the population is p thousand.
a. At what rate will the level of carbon monoxide be changing with respect to population when the population is 18 thousand people?
b. At what rate will the carbon monoxide level be changing with respect to time 2 years from now? Will the level be increasing or decreasing at this time?
ANIMAL BEHAVIOR In a research paper, V. A. Tucker and K. SchmidtKoenig demonstrated
that a species of Australian parakeet (the Budgerigar) expends energy (calories per gram of mass per kilometer) according to the formula
where v is the bird's velocity (in km/hr). Find a formula for the rate of change of E with respect to velocity v.
MAMMALIAN GROWTH Observations show that the length L in millimeters (mm) from nose to tip of tail of a Siberian tiger can be estimated using the function L = 0.25w2.6, where w is the weight of the tiger in kilograms (kg). Furthermore, when a tiger is less than 6 months old, its weight (kg) can be estimated in terms of its age A in days by the function w = 3 + 0.21A.
a. At what rate is the length of a Siberian tiger increasing with respect to its weight when it weighs 60 kg?
b. How long is a Siberian tiger when it is 100 days old? At what rate is its length increasing with respect to time at this age?
QUALITY OF LIFE A demographic study models the population p (in thousands) of a community
by the function
where Q is a qualityoflife index that ranges from Q = 0 (extremely poor quality) to Q = 10
(excellent quality). Suppose the index varies with time in such a way that t years from now,
For 0 t 10
a. What value of the qualityoflife index should be expected 4 years from now? What will be the corresponding population at this time?
b. At what rate is the population changing with respect to time 4 years from now? Is the population
increasing or decreasing at this time?
WATER POLLUTION When organic matter is introduced into a body of water, the oxygen content
of the water is temporarily reduced by oxidation. Suppose that t days after untreated sewage is
dumped into a particular lake, the proportion of the usual oxygen content in the water of the lake
that remains is given by the function
a. At what rate is the oxygen proportion P(t) changing after 10 days? Is the proportion increasing or decreasing at this time?
b. Is the oxygen proportion increasing or decreasing after 15 days?
c. If there is no new dumping, what would you expect to eventually happen to the proportion of
oxygen? Use a limit to verify your conjecture.
PRODUCTION The number of units Q of a particular commodity that will be produced when L workerhours of labor are employed is modeled by
Q(L) = 300L1/3
Suppose that the labor level varies with time in such a way that t months from now, L(t) workerhours
will be employed, where
For 0 t 12
a. How many workerhours will be employed in producing the commodity 5 months from now?
How many units will be produced at this time?
b. At what rate will production be changing with respect to time 5 months from now? Will production
be increasing or decreasing at this time?
PRODUCTION The number of units Q of a particular commodity that will be produced with K
thousand dollars of capital expenditure is modeled by
Suppose that capital expenditure varies with time in such a way that t months from now there will be K(t) thousand dollars of capital expenditure, where
a. What will be the capital expenditure 3 months from now? How many units will be produced at this time?
b. At what rate will production be changing with respect to time 5 months from now? Will production
be increasing or decreasing at this time?
DEPRECIATION The value V (in thousands of dollars) of an industrial machine is modeled by
where N is the number of hours the machine is used each day. Suppose further that usage varies
with time in such a way that
where t is the number of months the machine has been in operation.
a. How many hours per day will the machine be used 9 months from now? What will be the value of the machine at this time?
b. At what rate is the value of the machine changing with respect to time 9 months from now? Will the value be increasing or decreasing at this time?
INSECT GROWTH The growth of certain insects varies with temperature. Suppose a particular
species of insect grows in such a way that the volume of an individual is
cm3
when the temperature is T°C, and that its mass is m grams, where
a. Find the rate of change of the insect's volume with respect to temperature.
b. Find the rate of change of the insect's mass with respect to volume.
c. When T = 10°C, what is the insect's volume? At what rate is the insect's mass changing with respect to temperature when T = 10°C?
COMPOUND INTEREST If $10,000 is invested at an annual rate r (expressed as a decimal)
compounded weekly, the total amount (principal P and interest) accumulated after 10 years is
given by the formula
a. Find the rate of change of A with respect to r.
b. Find the percentage rate of change of A with respect to r when r = 0.05 (that is, 5%).
LEARNING When you first begin to study a topic or practice a skill, you may not be very good at it, but in time, you will approach the limits of your ability. One model for describing this behavior involves the function
where T is the time required for a particular person to learn the items on a list of L items and a and b are positive constants.
a. Find the derivative and interpret it in terms of the learning model.
b. Read and discuss in one paragraph an article on how learning curves can be used to study worker productivity.
MARGINAL ANALYSIS In Exercises 1 through 6, C(x) is the total cost of producing x units of a particular commodity and p(x) is the 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 producing the fourth unit.
(c) Find the actual cost of producing the fourth unit.
(d) Use marginal revenue to estimate the revenue derived from the sale of the fourth unit.
(e) Find the actual revenue derived from the sale of the fourth unit.
MARGINAL ANALYSIS A manufacturer's total cost is C(q) = 0.1q3  0.5q2 + 500q + 200
dollars, where q is the number of units produced.
a. Use marginal analysis to estimate the cost of manufacturing the fourth unit.
b. Compute the actual cost of manufacturing the fourth unit.
MARGINAL ANALYSIS A manufacturer's total monthly revenue is R(q) = 240q  0.05q2 dollars
when q units are produced and sold during the month. Currently, the manufacturer is producing
80 units a month and is planning to increase the monthly output by 1 unit.
a. Use marginal analysis to estimate the additional revenue that will be generated by the production and sale of the 81st unit.
b. Use the revenue function to compute the actual additional revenue that will be generated by the
production and sale of the 81st unit.
MARGINAL ANALYSIS Suppose the total cost in dollars of manufacturing q units is
C(q) = 3q2 + q + 500.
a. Use marginal analysis to estimate the cost of manufacturing the 41st unit.
b. Compute the actual cost of manufacturing the 41st unit.
AIR POLLUTION An environmental study of a certain community suggests that t years from now,
the average level of carbon monoxide in the air will be Q(t) = 0.05t2 + 0.1t + 3.4 parts per million. By
approximately how much will the carbon monoxide level change during the coming 6 months?
NEWSPAPER CIRCULATION 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.
MANUFACTURING A manufacturer's total cost is C(q) = 0.1q3  0.5q2 + 500q + 200 dollars when the level of production is q units. The current level of production is 4 units, and the manufacturer is planning to increase this to 4.1 units. Estimate how the total cost will change as a result.
MANUFACTURING A manufacturer's total monthly revenue is R(q) = 240q  0.05q2 dollars
when q units are produced during the month. Currently, the manufacturer is producing 80 units a
month and is planning to decrease the monthly output by 0.65 unit. Estimate how the total
monthly revenue will change as a result.
EFFICIENCY 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 assembled f (x) = x3 + 6x2 + 15x units x hours later. Approximately how many units will the worker assemble between 9:00 and 9:15 A.M.?
PRODUCTION At a certain factory, the daily output is Q(K) = 600K1/2 units, where K denotes the capital investment measured in units of $1,000. The current capital investment is $900,000. Estimate
the effect that an additional capital investment of $800 will have on the daily output.
PRODUCTION At a certain factory, the daily output is Q(L) = 60,000L1/3 units, where L denotes the size of the labor force measured in workerhours. Currently 1,000 workerhours of labor are used each day. Estimate the effect on output that will be produced if the labor force is cut to 940 workerhours.
PROPERTY TAX A projection made in January of 2002 determined that x years later, the average
property tax on a threebedroom 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 2010.
POPULATION GROWTH A 5year projection of population trends suggests that t years from now, the population of a certain community will be P(t) = t3 + 9t2 + 48t + 200 thousand.
a. Find the rate of change of population R(t) = P (t) with respect to time t.
b. At what rate does the population growth rate R(t) change with respect to time?
c. Use increments to estimate how much R(t) changes during the first month of the fourth year. What is the actual change in R(t) during this time period?
PRODUCTION At a certain factory, the daily output is Q = 3,000K1/2L1/3 units, where K denotes the firm's capital investment measured in units of $1,000 and L denotes the size of the labor force measured in workerhours. Suppose that the current capital investment is $400,000 and that
1,331 workerhours of labor are used each day. Use marginal analysis to estimate the effect that
an additional capital investment of $1,000 will have on the daily output if the size of the labor
force is not changed.
PRODUCTION The daily output at a certain factory is Q(L) = 300L2/3 units, where L denotes the
size of the labor force measured in workerhours. Currently, 512 workerhours of labor are used each
day. Estimate the number of additional workerhours of labor that will be needed to increase daily
output by 12.5 units.
MANUFACTURING A manufacturer's total cost is dollars when
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.
GROWTH OF A CELL A certain cell has the shape of a sphere. The formulas S = 4 r2 and are used to compute the surface area and volume of the cell, respectively. Estimate the
effect on S and V produced by a 1% increase in the radius r.
CARDIAC OUTPUT Cardiac output is the volume (cubic centimeters) of blood pumped by a
person's heart each minute. One way of measuring cardiac output C is by Fick's formula
where x is the concentration of carbon dioxide in the blood entering the lungs from the right side
of the heart and a and b are positive constants. If x is measured as x = c with a maximum error of
3%, what is the maximum percentage error that can be incurred by measuring cardiac output with
Fick's formula? (Your answer will be in terms of a, b, and c.)
MEDICINE A tiny spherical balloon is inserted into a clogged artery. If the balloon has an inner
diameter of 0.01 millimeter (mm) and is made from material 0.0005 mm thick, approximately how much material is inserted into the artery? [Hint: Think of the amount of material as a change in volume V, where ]
ARTERIOSCLEROSIS In arteriosclerosis, fatty material called plaque gradually builds up on the
walls of arteries, impeding the flow of blood, which, in turn, can lead to stroke and heart attacks.
Consider a model in which the carotid artery is represented as a circular cylinder with crosssectional
radius R = 0.3 cm and length L. Suppose it is discovered that plaque 0.07 cm thick is distributed
uniformly over the inner wall of the carotid artery of a particular patient. Use increments to estimate the percentage of the total volume of the artery that is blocked by plaque. [Hint: The volume of a cylinder of radius R and length L is V = R2L. Does it matter that we have not specified the length L of the artery?]
BLOOD CIRCULATION In Exercise 57, Section 1.1, we introduced an important law attributed to the French physician, Jean Poiseuille. Another law discovered by Poiseuille says that the volume of the fluid flowing through a small tube in unit time under fixed pressure is given by the formula V = kR4, where k is a positive constant and R is the radius of the tube. This formula is used in medicine to determine how wide a clogged artery must be opened to restore a healthy flow of blood.
a. Suppose the radius of a certain artery is increased by 5%. Approximately what effect does this have on the volume of blood flowing through the artery?
b. Read an article on the cardiovascular system and write a paragraph on the flow of blood.
EXPANSION OF MATERIAL The (linear) thermal expansion coefficient of an object is defined to be
where L(T) is the length of the object when the temperature is T. Suppose a 50meter span of a
bridge is built of steel with = 1.4 105 per degree Celsius. Approximately how much will the length change during a year when the temperature varies from 20°C (winter) to 35°C (summer)?
RADIATION Stefan's law in physics states that a body emits radiant energy according to the formula
R(T) = kT4, where R is the amount of energy emitted from a surface whose temperature is T (in degrees kelvin) and k is a positive constant. Estimate the percentage change in R that results from a 2% increase in T.
MANUFACTURING The output at a certain plant is Q = 0.08x2 + 0.12xy + 0.03y2 units per day, where x is the number of hours of skilled labor used and y the number of hours of unskilled labor used. Currently, 80 hours of skilled labor and 200 hours of unskilled labor are used each day. Use calculus to estimate the change in unskilled labor that should be made to offset a 1hour increase in skilled labor so that output will be maintained at its current level.
MANUFACTURING The output of a certain plant is Q = 0.06x2 + 0.14xy + 0.05y2 units per day, where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor used. Currently, 60 hours of skilled labor and 300 hours of unskilled labor are used each day. Use calculus to estimate the change in unskilled labor that should be made to offset a 1hour increase in skilled labor so that output will be maintained at its current level.
SUPPLY RATE When the price of a certain commodity is p dollars per unit, the manufacturer is willing to supply x hundred units, where 3p2  x2 = 12. How fast is the supply changing when the price is $4 per unit and is increasing at the rate of 87 cents per month?
DEMAND RATE When the price of a certain commodity is p dollars per unit, customers demand x hundred units of the commodity, where
x2 + 3px + p2 = 79
How fast is the demand x changing with respect to time when the price is $5 per unit and is decreasing at the rate of 30 cents per month?
DEMAND RATE When the price of a certain commodity is p dollars per unit, consumers demand x hundred units of the commodity, where
75x2 + 17p2 = 5,300
How fast is the demand x changing with respect to time when the price is $7 and is decreasing at the
rate of 75 cents per month? (That is, .)
REFRIGERATION An ice block used for refrigeration is modeled as a cube of side s. The block
currently has volume 125,000 cm3 and is melting at the rate of 1,000 cm3 per hour.
a. What is the current length s of each side of the cube? At what rate is s currently changing with respect to time t?
b. What is the current rate of change of the surface area S of the block with respect to time? [Note: A cube of side s has volume V = s3 and surface area S = 6s2.]
MEDICINE A tiny spherical balloon is inserted into a clogged artery and is inflated at the rate of
0.002 mm3/min. How fast is the radius of the balloon growing when the radius is R = 0.005 mm?
[Note: A sphere of radius R has volume ].
POLLUTION CONTROL An environmental study for a certain community indicates that there will be Q( p) = p2 + 4p + 900 units of a harmful pollutant in the air when the population is p thousand people. If the population is currently 50,000 and is increasing at the rate of 1,500 per year, at what rate is the level of pollution increasing?
GROWTH OF A TUMOR 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 ?
BOYLE'S LAW Boyle's law states that when gas is compressed at constant temperature, the pressure
P and volume V of a given sample satisfy the equation PV = C, where C is constant. Suppose that at a
certain time the volume is 40 in.3, the pressure is 70 lb/in.2, and the volume is increasing at the rate
of 12 in.3/sec. How fast is the pressure changing at this instant? Is it increasing or decreasing?
METABOLIC RATE 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 warmblooded animal of mass w kilograms (kg) is given by
M = 70w3/4 kilocalories per day
a. Find the rate of change of the metabolic rate of an 80kg cougar that is gaining mass at the rate of 0.8 kg per day.
b. Find the rate of change of the metabolic rate of a 50kg ostrich that is losing mass at the rate of 0.5 kg per day.
SPEED OF A LIZARD 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 11gram lizard increasing if the lizard is growing at the rate of 0.02 grams per day?
PRODUCTION At a certain factory, output is given by Q = 60K1/3L2/3 units, where K is the capital investment (in thousands of dollars) and L is the size of the labor force, measured in workerhours.
If output is kept constant, at what rate is capital investment changing at a time when K = 8, L = 1,000, and L is increasing at the rate of 25 workerhours per week?
[Note: Output functions of the general form Q = AK L1 , where A and are constants with 0 1, are called CobbDouglas production functions. Such functions appear in examples and exercises throughout this text, especially in Chapter 7.]
WATER POLLUTION 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?
A 6foottall 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?
CHEMISTRY In an adiabatic chemical process, there is no net change (gain or loss) of heat. Suppose
a container of oxygen is subjected to such a process. Then if the pressure on the oxygen is P and its volume is V, it can be shown that PV1.4 = C, where C is a constant. At a certain time, V = 5 m3, P = 0.6 kg/m2, and P is increasing at 0.23 kg/m2 per sec. What is the rate of change of V? Is V increasing or decreasing?
MANUFACTURING At a certain factory, output Q is related to inputs x and y by the equation
Q = 2x3 + 3x2y2 + (1 + y)3
If the current levels of input are x = 30 and y = 20, use calculus to estimate the change in input y that should be made to offset a decrease of 0.8 unit in input x so that output will be maintained at its current level.
LUMBER PRODUCTION 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 by
Suppose r, R, and H are increasing at the respective rates of 4 in/yr, 5 in/yr, and 9 in/yr. At what rate is V increasing at the time when r = 2 feet, R = 3 feet, and H = 15 feet?
BLOOD FLOW One of Poiseuille's laws (see Exercise 57, Section 1.1) says that the speed of
blood flowing under constant pressure in a blood vessel at a distance r from the center of the vessel is given by
where K is a positive constant, R is the radius of the vessel, and L is the length of the vessel. Suppose
the radius R and length L of the vessel change with time in such a way that the speed of blood flowing at the center is unaffected; that is, v does not change with time. Show that in this case, the relative rate of change of L with respect to time must be twice the relative rate of change of R.
MANUFACTURING At a certain factory, output Q is related to inputs u and v by the equation
If the current levels of input are u = 10 and v = 25, use calculus to estimate the change in input v that should be made to offset a decrease of 0.7 unit in input u so that output will be maintained at its current level.
POPULATION GROWTH Suppose that a 5year projection of population trends suggests that t years from now, the population of a certain community will be P thousand, where
P(t) = t3 + 9t2 + 48t + 200.
a. At what rate will the population be growing 3 years from now?
b. At what rate will the rate of population growth be changing with respect to time 3 years from now?
RAPID TRANSIT After x weeks, the number of people using a new rapid transit system was
approximately N(x) = 6x3 + 500x + 8,000.
a. At what rate was the use of the system changing with respect to time after 8 weeks?
b. By how much did the use of the system change during the eighth week?
39. PRODUCTION It is estimated that the weekly output at a certain plant is Q(x) = 50x2 + 9,000x
units, where x is the number of workers employed at the plant. Currently there are 30 workers employed at the plant.
a. Use calculus to estimate the change in the weekly output that will result from the addition of 1 worker to the force.
b. Compute the actual change in output that will result from the addition of 1 worker.
POPULATION It is projected that t months from now, the population of a certain town will be
P(t) = 3t + 5t 3/2 + 6,000. At what percentage rate will the population be changing with respect to time
4 months from now?
PRODUCTION At a certain factory, the daily output is Q(L) = 20,000L1/2 units, where L denotes
the size of the labor force measured in workerhours. Currently 900 workerhours of labor are used each day. Use calculus to estimate the effect on output that will be produced if the labor force is cut to 885 workerhours.
GROSS DOMESTIC PRODUCT The gross domestic product of a certain country was N(t) = t2 + 6t + 300 billion dollars t years after 2000. Use calculus to predict the percentage change in the GDP during the second quarter of 2008.
POLLUTION The level of air pollution in a certain city is proportional to the square of the population. Use calculus to estimate the percentage by which the air pollution level will increase if the population increases by 5%.
AIDS EPIDEMIC In its early phase, specifically the period 19841990, the AIDS epidemic could be modeled* by the cubic function
C(t) = 170.36t3 + 1,707.5t2 + 1,998.4t + 4,404.8
for 0 ? t ? 6, where C is the number of reported cases t years after the base year 1984.
a. Compute and interpret the derivative C (t).
b. At what rate was the epidemic spreading in the year 1984?
c. At what percentage rate was the epidemic spreading in 1984? In 1990?
POPULATION DENSITY The formula D = 36 m +1.14 is sometimes used to determine the ideal population density D (individuals per square kilometer) for a large animal of mass m kilograms (kg).
a. What is the ideal population density for humans, assuming that a typical human weighs about 70 kg?
b. The area of the United States is about 9.2 million square kilometers. What would the population of the United States have to be for the population density to be ideal?
c. Consider an island of area 3,000 km2. Two hundred animals of mass m =30 kg are brought to the island, and t years later, the population is given by
P(t) = 0.43t2 + 13.37t + 200
How long does it take for the ideal population density to be reached? At what rate is the population changing when the ideal density is attained?
BACTERIAL GROWTH The population P of a bacterial colony t days after observation begins is modeled by the cubic function
P(t) = 1.035t3 + 103.5t2 + 6,900t + 230,000
a. Compute and interpret the derivative P (t).
b. At what rate is the population changing after 1 day? After 10 days?
c. What is the initial population of the colony? How long does it take for the population to double?
At what rate is the population growing at the time it doubles?
PRODUCTION The output at a certain factory is Q(L) = 600L2/3 units, where L is the size of the labor
force. The manufacturer wishes to increase output by 1%. Use calculus to estimate the percentage
increase in labor that will be required.
PRODUCTION The output Q at a certain factory is related to inputs x and y by the equation
Q = x3 + 2xy2 + 2y3
If the current levels of input are x = 10 and y= 20, use calculus to estimate the change in input y that
should be made to offset an increase of 0.5 in input x so that output will be maintained at its current level.
BLOOD FLOW Physiologists have observed that the flow of blood from an artery into a small capillary is given by the formula
where 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 with respect to pressure C in the capillary if A and D are kept constant? Does the flow increase or decrease with increasing C?
b. What is the percentage rate of change of flow F with respect to A if C and D are kept constant?
POLLUTION CONTROL It is estimated that t years from now, the population of a certain suburban
community will be thousand. An environmental study indicates that the average daily level of carbon monoxide in the air will be units when the population is p thousand. At what percentage rate will the level of carbon monoxide be changing with respect to time 1 year from now?
You measure the radius of a circle to be 12 cm and use the formula A = r2 to calculate the area. If your measurement of the radius is accurate to within 3%, how accurate is your calculation of the
area?
Estimate what will happen to the volume of a cube if the length of each side is decreased by
2%. Express your answer as a percentage.
PRODUCTION 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.
BLOOD FLOW 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 1.2 102 centimeter and makes an error of 5 104 centimeter. Estimate the amount by which the calculated value of the speed of the blood will differ from the true speed if the incorrect value of the radius is used in the formula.
AREA OF A TUMOR You measure the radius of a spherical tumor to be 1.2 cm and use the formula to calculate the surface area. If your measurement of the radius r is accurate to within 3%, how accurate is your calculation of the area?
CARDIOVASCULAR SYSTEM 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 equation
where C1 and C2 are positive constants and T is the (fixed) time length of the systole phase.* Find a relationship between the rates and .
CONSUMER DEMAND When electric toasters are sold for p dollars apiece, local consumers will
buy toasters. It is estimated that t months from now, the unit price of the toasters will be p(t) = 0.04t3/2 + 44 dollars. Compute the rate of change of the monthly demand for toasters with respect to time 25 months from now. Will the demand be increasing or decreasing?
At noon, a truck is at the intersection of two roads and is moving north at 70 km/hr. An hour later, a
car passes through the same intersection, traveling east at 105 km/hr. How fast is the distance between the car and truck changing at 2 P.M.?
POPULATION GROWTH It is projected that t years from now, the population of a certain suburban
community will be thousand. By approximately what percentage will the population grow during the next quarter year?
WORKER EFFICIENCY 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 + 9t2 + 12t 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 production changing with respect to time at 9:00 A.M.?
c. Use calculus to estimate the change in the worker's rate of production between 9:00 and 9:06 A.M.
d. Compute the actual change in the worker's rate of production between 9:00 and 9:06 A.M.
TRAFFIC SAFETY A car is traveling at 88 ft/sec when the driver applies the brakes to avoid hitting a
child. After t seconds, the car is s = 88t  8t2 feet from the point where the brakes were applied. How
long does it take for the car to come to a stop, and how far does it travel before stopping?
CONSTRUCTION MATERIAL Sand is leaking from a bag in such a way that after t seconds, there are
pounds of sand left in the bag.
a. How much sand was originally in the bag?
b. At what rate is sand leaking from the bag after 1 second?
c. How long does it take for all of the sand to leak from the bag? At what rate is the sand leaking from the bag at the time it empties?
INFLATION It is projected that t months from now, the average price per unit for goods in a certain
sector of the economy will be P dollars, where P(t) = 1  t3 + 7t2 + 200t + 300.
a. At what rate will the price per unit be increasing with respect to time 5 months from now?
b. At what rate will the rate of price increase be changing with respect to time 5 months from now?
c. Use calculus to estimate the change in the rate of price increase during the first half of the sixth
month.
d. Compute the actual change in the rate of price increase during the first half of the sixth month.
PRODUCTION COST At a certain factory, approximately q(t) = t2 + 50t units are manufactured
during the first t hours of a production run, and the total cost of manufacturing q units is
C(q) = 0.1q2 + 10q + 400 dollars. Find the rate at which the manufacturing cost is changing with respect to time 2 hours after production commences.
PRODUCTION COST It is estimated that the monthly cost of producing x units of a particular commodity is C(x) = 0.06x + 3x1/2 +20 hundred dollars. Suppose production is decreasing at the
rate of 11 units per month when the monthly production is 2,500 units. At what rate is the cost
changing at this level of production?
Estimate the largest percentage error you can allow in the measurement of the radius of a sphere if you want the error in the calculation of its surface area using the formula S = 4 r2 to be no greater than 8%.
A soccer ball made of leather 1/8 inch thick has an inner diameter of 8.5 inches. Estimate the volume
of its leather shell. [Hint: Think of the volume of the shell as a certain change V in volume.]
A car traveling north at 60 mph and a truck traveling east at 45 mph leave an intersection at the same time. At what rate is the distance between them changing 2 hours later?
A child is flying a kite at a height of 80 feet above her hand. If the kite moves horizontally at a constant speed of 5 ft/sec, at what rate is string being paid out when the kite is 100 feet away from the child?
A person stands at the end of a pier 8 feet above the water and pulls in a rope attached to a buoy. If the rope is hauled in at the rate of 2 ft/min, how fast is the buoy moving in the water when it is 6 feet from the pier?
A 10footlong ladder leans against the side of a wall. The top of the ladder is sliding down the wall at the rate of 3 ft/sec. How fast is the foot of the ladder moving away from the building when the top is 6 feet above the ground?
A lantern falls from the top of a building in such a way that after t seconds, it is h(t) = 150  16t2
feet above ground. A woman 5 feet tall originally standing directly under the lantern sees it start
to fall and walks away at the constant rate of 5 ft/sec. How fast is the length of the woman's
shadow changing when the lantern is 10 feet above the ground?
A baseball diamond is a square, 90 feet on a side. A runner runs from second base to third at
20 ft/sec. How fast is the distance s between the runner and home base changing when he is 15 feet
from third base?
MANUFACTURING COST Suppose the total manufacturing cost C at a certain factory is a
function of the number q of units produced, which in turn is a function of the number t of hours
during which the factory has been operating.
a. What quantity is represented by the derivative ? In what units is this quantity measured?
b. What quantity is represented by the derivative ?In what units is this quantity measured?
c. What quantity is represented by the product ? In what units is this quantity measured?
An object projected from a point P moves along a straight line. It is known that the velocity of the
object is directly proportional to the product of the time the object has been moving and the distance it
has moved from P. It is also known that at the end of 5 seconds, the object is 20 feet from P and is
moving at the rate of 4 ft/sec. Find the acceleration of the object at this time (when t = 5).
Find all the points (x, y) on the graph of the function y = 4x2 with the property that the tangent to the graph at (x, y) passes through the point (2, 0).
Suppose y is a linear function of x; that is, y = mx + b. What will happen to the percentage rate of change of y with respect to x as x increases without bound? Explain.
Find an equation for the tangent line to the curve at the point (x0, y0).
AVERAGE COST The total cost of producing x units of a certain commodity is C(x) thousand
dollars, where
C(x) = x3  20x2 + 179x + 242
a. Find A (x), where A(x) = C(x)/x is the average cost function.
b. For what values of x is A(x) increasing? For what values is it decreasing?
c. For what level of production x is average cost minimized? What is the minimum average cost?
MARGINAL ANALYSIS The total cost of producing x units of a certain commodity is given by Sketch the cost curve and find the marginal cost. Does marginal cost increase or decrease with increasing production?
MARGINAL ANALYSIS Let p = (10  3x)2 for 0 ? x ? 3 be the price at which x hundred units
of a certain commodity will be sold, and let R(x) = xp(x) be the revenue obtained from the sale of the x units. Find the marginal revenue R (x) and sketch the revenue and marginal revenue curves on the same graph. For what level of production is revenue maximized?
PROFIT UNDER A MONOPOLY To produce x units of a particular commodity, a monopolist
has a total cost of C(x) = 2x2 + 3x + 5 and total revenue of R(x) = xp(x), where p(x) = 5  2x is the price at which the x units will be sold. Find the profit function P(x) = R(x)  C(x) and sketch its graph. For what level of production is profit maximized?
MEDICINE The concentration of a drug t hours after being injected into the arm of a patient is
given by
Sketch the graph of the concentration function. When does the maximum concentration occur?
POLLUTION CONTROL Commissioners of a certain city determine that when x million dollars
are spent on controlling pollution, the percentage of pollution removed is given by
a. Sketch the graph of P(x).
b. What expenditure results in the largest percentage of pollution removal?
ADVERTISING A company determines that if x thousand dollars are spent on advertising a
certain product, then S(x) units of the product will be sold, where
S(x) = 2x3 + 27x2 + 132x + 207 0 ? x ? 17
a. Sketch the graph of S(x).
b. How many units will be sold if nothing is spent on advertising?
c. How much should be spent on advertising to maximize sales? What is the maximum sales level?
MORTGAGE REFINANCING When interest rates are low, many homeowners take the opportunity to refinance their mortgages. As rates start to rise, there is often a flurry of activity as latecomers rush in to refinance while they still can do so profitably. Eventually, however, rates reach a level where refinancing begins to wane. Suppose in a certain community, there will be M(r) thousand refinanced mortgages when the 30year fixed mortgage rate is r %, where
1 ? r ? 8
a. For what values of r is M(r) increasing?
b. For what interest rate r is the number of refinanced mortgages maximized? What is this maximum number?
POPULATION DISTRIBUTION A demographic study of a certain city indicates that P(r) hundred people live r miles from the civic center, where
a. What is the population at the city center?
b. For what values of r is P(r) increasing? For what values is it decreasing?
c. At what distance from the civic center is the population largest? What is this largest population?
GROSS DOMESTIC PRODUCT The graph shows the consumption of the baby boom generation, measured as a percentage of total GDP (gross domestic product) during the time period 19701997.
a. At what years do relative maxima occur?
b. At what years do relative minima occur?
c. At roughly what rate was consumption increasing in 1987?
d. At roughly what rate was consumption decreasing in 1972?
DEPRECIATION The value V (in thousands of dollars) of an industrial machine is modeled by
where N is the number of hours the machine is used each day. Suppose further that usage varies with time in such a way that
where t is the number of months the machine has been in operation.
a. Over what time interval is the value of the machine increasing? When is it decreasing?
b. At what time t is the value of the machine the largest? What is this maximum value?
FISHERY MANAGEMENT The manager of a fishery determines that t weeks after 300 fish of a
particular species are released into a pond, the average weight of an individual fish (in pounds)
for the first 10 weeks will be w(t) = 3 + t  0.05t2. He further determines that the proportion of the fish that are still alive after t weeks is given by
a. The expected yield Y(t) of the fish after t weeks is the total weight of the fish that are still alive.
Express Y(t) in terms of w(t) and p(t) and sketch the graph of Y(t) for 0 ? t ? 10.
b. When is the expected yield Y(t) the largest? What is the maximum yield?
FISHERY MANAGEMENT Suppose for the situation described in Exercise 65, it costs the fishery C(t) = 50 + 1.2t hundred dollars to maintain and monitor the pond for t weeks after the fish are released, and that each fish harvested after t weeks can be sold for $2.75 per pound.
a. If all fish that remain alive in the pond after t weeks are harvested, express the profit obtained
by the fishery as a function of t.
b. When should the fish be harvested in order to maximize profit? What is the maximum profit?
MARGINAL ANALYSIS The cost of producing x units of a commodity per week is
C(x) = 0.3x3  5x2 + 28x + 200
a. Find the marginal cost C (x) and sketch its graph along with the graph of C(x) on the same
coordinate plane.
b. Find all values of x where C ?(x) = 0. How are these levels of production related to the graph of
the marginal cost?
MARGINAL ANALYSIS The profit obtained from producing x thousand units of a particular
commodity each year is P(x) dollars, where
a. Find the marginal profit P (x), and determine all values of x such that P (x) = 0.
b. Sketch the graph of marginal profit along with the graph of P(x) on the same coordinate plane.
c. Find P ?(x), and determine all values of x such that P ?(x). How are these levels of production
related to the graph of marginal profit?
SALES A company estimates that if x thousand dollars are spent on marketing a certain product,
then S(x) units of the product will be sold each month, where S(x) = x3 + 33x2 + 60x + 1,000
a. How many units will be sold if no money is spent on marketing?
b. Sketch the graph of S(x). For what value of x does the graph have an inflection point? What is
the significance of this marketing expenditure?
WORKER EFFICIENCY An efficiency study of the morning shift (from 8:00 A.M. to 12:00 noon)
at a factory indicates that an average worker who arrives on the job at 8:00 A.M. will have produced
Q units t hours later, where
a. At what time during the morning is the worker performing most efficiently?
b. At what time during the morning is the worker performing least efficiently?
POPULATION GROWTH A 5year projection of population trends suggests that t years from now, the population of a certain community will be P(t) = t3 + 9t2 + 48t + 50 thousand.
a. At what time during the 5year period will the population be growing most rapidly?
b. At what time during the 5year period will the population be growing least rapidly?
c. At what time is the rate of population growth changing most rapidly?
ADVERTISING The manager of the Footloose sandal company determines that t months after initiating an advertising campaign, S(t) hundred pairs of sandals will be sold, where
a. Find S (t) and S ?(t).
b. At what time will sales be maximized? What is the maximum level of sales?
c. The manager plans to terminate the advertising campaign when the sales rate is minimized. When does this occur? What are the sales level and sales rate at this time?
HOUSING STARTS Suppose that in a certain community, there will be M(r) thousand new
houses built when the 30year fixed mortgage rate is r%, where
a. Find M (r) and M ?(r).
b. Sketch the graph of the construction function M(r).
c. At what rate of interest r is the rate of construction of new houses minimized?
GOVERNMENT SPENDING During a recession, Congress decides to stimulate the economy by providing funds to hire unemployed workers for government projects. Suppose that t months after the stimulus program begins, there are N(t) thousand people unemployed, where
N(t) = t3 + 45t2 + 408t + 3,078
a. What is the maximum number of unemployed workers? When does the maximum level of unemployment occur?
b. In order to avoid overstimulating the economy (and inducing inflation), a decision is made to
terminate the stimulus program as soon as the rate of unemployment begins to decline. When
does this occur? At this time, how many people are unemployed?
SPREAD OF A DISEASE An epidemiologist determines that a particular epidemic spreads in such a way that t weeks after the outbreak, N hundred new cases will be reported, where
a. Find N (r) and N ?(r).
b. At what time is the epidemic at its worst? What is the maximum number of reported new cases?
c. Health officials declare the epidemic to be under control when the rate of reported new cases is
minimized. When does this occur? What number of new cases will be reported at that time?
THE SPREAD OF AN EPIDEMIC Let Q(t) denote the number of people in a city of population N0 who have been infected with a certain disease t days after the beginning of an epidemic. Studies indicate that the rate R(Q) at which an epidemic spreads is jointly proportional to the number of people who have contracted the disease and the number who have not, so R(Q) = kQ(N0  Q). Sketch the graph of the rate function, and interpret your graph. In particular, what is the significance of the highest point on the graph of R(Q)?
SPREAD OF A RUMOR The rate at which a rumor spreads through a community of P people is jointly proportional to the number of people N who have heard the rumor and the number who have not. Show that the rumor is spreading most rapidly when half the people have heard it.
POPULATION GROWTH Studies show that when environmental factors impose an upper bound on the possible size of a population P(t), the population often tends to grow in such a way that the percentage rate of change of P(t) satisfies
where A and B are positive constants. Where does the graph of P(t) have an inflection point? What is the significance of this point? (Your answer will be in terms of A and B.)
TISSUE GROWTH Suppose a particular tissue culture has area A(t) at time t and a potential
maximum area M. Based on properties of cell division, it is reasonable to assume that the area A
grows at a rate jointly proportional to and M  A(t); that is,
where k is a positive constant.
a. Let R(t) = A (t) be the rate of tissue growth. Show that R (t) = 0 when A(t) = M/3.
b. Is the rate of tissue growth greatest or least when A(t) = M/3? [Hint: Use the first derivative
test or the second derivative test.]
c. Based on the given information and what you discovered in part (a), what can you say about
the graph of A(t)?
Water is poured at a constant rate into the vase shown in the accompanying figure. Let h(t) be the
height of the water in the vase at time t (assume the vase is empty when t = 0). Sketch a rough
graph of the function h(t). In particular, what happens when the water level reaches the neck of
the vase?
Let f (x) = x4 + x. Show that even though f ?(0) = 0 the graph of f has neither a relative extremum nor an inflection point where x = 0. Sketch the graph of f(x).
Use calculus to show that the graph of the quadratic function y = ax2 + bx + c is concave upward if a is positive and concave downward if a is negative.
AVERAGE COST The total cost of producing x units of a particular commodity is C thousand
dollars, where C(x) = 3x2 + x + 48, and the average cost is
a. Find all vertical and horizontal asymptotes of the graph of A(x).
b. Note that as x gets larger and larger, the term in A(x) gets smaller and smaller. What does this say about the relationship between the average cost curve y = A(x) and the line y = 3x + 1?
c. Sketch the graph of A(x), incorporating the result of part (b) in your sketch. [Note: The line
y = 3x + 1 is called an oblique (or slant) asymptote of the graph.]
INVENTORY COST A manufacturer estimates that if each shipment of raw materials contains x units, the total cost in dollars of obtaining and storing the year's supply of raw materials will be
a. Find all vertical and horizontal asymptotes of the graph of C(x).
b. Note that as x gets larger and larger, the term in C(x) gets smaller and smaller. What does this say about the relationship between the cost curve y = C(x) and the line y = 2x?
c. Sketch the graph of C(x), incorporating the result of part (b) in your sketch. [Note: The line y = 2x is called an oblique (or slant) asymptote of the graph.]
DISTRIBUTION COST The number of workerhours W required to distribute new telephone books to x% of the households in a certain community is modeled by the function
a. Sketch the graph of W(x).
b. Suppose only 1,500 workerhours are available for distributing telephone books. What percentage of households do not receive new books?
PRODUCTION A business manager determines that t months after production begins on a new product, the number of units produced will be P million per month, where
a. Find P (t) and P ?(t).
b. Sketch the graph of P(t).
c. What happens to production in the long run.
SALES A company estimates that if x thousand dollars are spent on the marketing of a certain
product, then Q(x) thousand units of the product will be sold, where
a. Sketch the graph of the sales function Q(x).
b. For what marketing expenditure x are sales maximized? What is the maximum sales level?
c. For what value of x is the sales rate minimized?
CONCENTRATION OF DRUG A patient is given an injection of a particular drug at noon, and samples of blood are taken at regular intervals to determine the concentration of drug in the patient's system. It is found that the concentration increases at an increasing rate with respect to time until 1 P.M., and for the next 3 hours, continues to increase but at a decreasing rate until the peak concentration is reached at 4 P.M. The concentration then decreases at a decreasing rate until 5 P.M.,
after which it decreases at an increasing rate toward zero. Sketch a possible graph for the concentration of drug C(t) as a function of time.
BACTERIAL POPULATION The population of a bacterial colony increases at an increasing rate
for 1 hour, after which it continues to increase but at a rate that gradually decreases toward zero. Sketch a possible graph for the population P(t) as a function of time t.
EPIDEMIOLOGY Epidemiologists studying a contagious disease observe that the number of newly infected people increases at an increasing rate during the first 3 years of the epidemic. At that time, a new drug is introduced, and the number of infected people declines at a decreasing rate. Two years after its introduction, the drug begins to lose effectiveness. The number of new cases continues to decline for 1 more year but at an increasing rate before rising again at an increasing rate. Draw a
possible graph for the number of new cases N(t) as a function of time.
ADOPTION OF TECHNOLOGY Draw a possible graph for the percentage of households
adopting a new type of consumer electronic technology if the percentage grows at an
increasing rate for the first 2 years, after which the rate of increase declines, with the market
penetration of the technology eventually approaching 90%.
EXPERIMENTAL PSYCHOLOGY To study the rate at which animals learn, a psychology
student performed an experiment in which a rat was sent repeatedly through a laboratory maze.
Suppose the time required for the rat to traverse the maze on the nth trial was approximately
a. Graph the function f (n).
b. What portion of the graph is relevant to the practical situation under consideration?
c. What happens to the graph as n increases without bound? Interpret your answer in practical terms.
AVERAGE TEMPERATURE A researcher models the temperature T (in degrees Celsius) during the time period from 6 A.M. to 6 P.M. in a certain city by the function
for 0 ? t ? 12
where t is the number of hours after 6 A.M.
a. Sketch the graph of T(t).
b. At what time is the temperature the greatest? What is the highest temperature of the day?
IMMUNIZATION During a nationwide program to immunize the population against a new strain
of influenza, public health officials determined that the cost of inoculating x% of the susceptible
population would be approximately million dollars.
a. Sketch the graph of the cost function C(x).
b. Suppose 40 million dollars are available for providing immunization. What percentage of the
susceptible population will not be inoculated?
POLITICAL POLLING A poll commissioned by a politician estimates that t days after she comes
out in favor of a controversial bill, the percentage of her constituency (those who support her at the
time she declares her position on the bill) that still supports her is given by
The vote is to be taken 10 days after she announces her position.
a. Sketch the graph of S(t) for 0 ? t ? 10.
b. When is her support at its lowest level? What is her minimum support level?
c. The derivative S (t) may be thought of as an approval rate. Is her approval rate positive or negative
when the vote is taken? Is the approval rate increasing or decreasing at this time? Interpret your results.
ADVERTISING A manufacturer of motorcycles estimates that if x thousand dollars are spent on
advertising, then for x _ 0, cycles will be sold.
a. Sketch the graph of the sales function M(x).
b. What level of advertising expenditure results in maximum sales? What is the maximum sales
level?
COST MANAGEMENT A company uses a truck to deliver its products. To estimate costs, the manager models gas consumption by the function
gal/mile, assuming that the truck is driven at a constant speed of x miles per hour for x ? 5. The driver is paid $18 per hour to drive the truck 400 miles, and gasoline costs $4.25 per gallon. Highway regulations require 30 ? x ? 65.
a. Find an expression for the total cost C(x) of the trip. Sketch the graph of C(x) for the legal speed
interval 30 ? x ? 65.
b. What legal speed will minimize the total cost of the trip? What is the minimal total cost?
Find constants A, B, and C so that the function f (x) = Ax3 + Bx2 + C will have a relative extremum at (2, 11) and an inflection point at (1, 5). Sketch the graph of f.
MAXIMUM PROFIT AND MINIMUM AVERAGE COST In Exercises 17 through 22, you are given the price p(q) at which q units of a particular commoditycan be sold and the total cost C(q) of producing the q units. In each case:
(a) Find the revenue function R(q), the profit function P(q), the marginal revenue R (q), and marginal
cost C (q). Sketch the graphs of P(q), R (q), and C (q) on the same coordinate axes and determine the level of production q where P(q) is maximized.
(b) Find the average cost A(q) = C(q)/q and sketch the graphs of A(q), and the marginal cost C (q)
on the same axes. Determine the level of production q at which A(q) is minimized.
ELASTICITY OF DEMAND In Exercises 23 through 28, compute the elasticity of demand for the given demand function D(p) and determine whether the demand is elastic, inelastic, or of unit elasticity at the indicated price p.
D(p) = 1.3p + 10; p = 4
At what point does the tangent to the curve y = 2x3  3x2 + 6x have the smallest slope? What is the slope of the tangent at this point?
AVERAGE PROFIT A manufacturer estimates that when q units of a certain commodity are produced, the profit obtained is P(q) thousand dollars, where
P(q) = 2q2 + 68q  128
a. Find the average profit and the marginal profit functions.
b. At what level of production is average profit equal to marginal profit?
c. Show that average profit is maximized at the level of production found in part (b).
d. On the same set of axes, graph the relevant portions of the average and marginal profit functions.
MARGINAL ANALYSIS A manufacturer estimates that if x units of a particular commodity
are produced, the total cost will be C(x) dollars, where
C(x) = x3  24x2 + 350x + 338
a. At what level of production will the marginal cost C (x) be minimized?
b. At what level of production will the average cost be minimized?
GROUP MEMBERSHIP A national consumers' association determines that x years after its founding
in 1993, it will have P(x) members, where
P(x) = 100(2x3  45x2 + 264x)
a. At what time between 1995 and 2008 was the membership largest? Smallest?
b. What were the largest and smallest membership levels between 1995 and 2008?
BROADCASTING An allnews radio station has made a survey of the listening habits of local
residents between the hours of 5:00 P.M. and midnight. The survey indicates that the percentage
of the local adult population that is tuned in to the station x hours after 5:00 P.M. is
a. At what time between 5:00 P.M. and midnight are the most people listening to the station?
What percentage of the population is listening at this time?
b. At what time between 5:00 P.M. and midnight are the fewest people listening? What percentage
of the population is listening at this time?
LEARNING In a learning model, two responses (A and B) are possible for each of the series of
observations. If there is a probability p of getting response A in any individual observation, the probability of getting response A exactly n times in a series of m observations is
F( p) = pn(1  p)mn. The maximum likelihood estimate is the value of p that maximizes F( p) for
0 ? p ? 1. For what value of p does this occur?

