a. b. c.
d. e.
f.Graph the function defined by y5C1x2 for 0,x#5.
Ca21 9b Ca15
8b
C112 Ca9
10b Ca3
4b
C1x2 a21.
a.0, f1x25loga x;
a21.
a.0, f1x25ax;
log215m2222log21m132 52 log14p112 1log p5log 3
log312x15255 logk 6456
a112p
5b253
a11m
3b5515
e3x21514 e2522x55
122k59 212m57
3z22511 6p517
3 log4 r222 log4 r 4 log3 y22 log3 x
log3 2y32log3 8y2 log5 3k1log5 7k3
log100 1000 log4 8
log32 16 log3 81
g. What is the independent variable?
h. What is the dependent variable?
87.Pollution The cost to remove xpercent of a pollutant is
in thousands of dollars. Find the cost of removing the follow- ing percents of the pollutant.
a. 80% b. 50% c. 90%
d. Graph the function.
e. Can all of the pollutant be removed?
Interest Find the amount of interest earned by each deposit.
88.$6902 if interest is 6% compounded semiannually for 8 years 89.$2781.36 if interest is 4.8% compounded quarterly for 6 years Interest Find the compound amount if $12,104 is invested at 6.2% compounded continuously for each period.
90.2 years 91. 4 years
Interest Find the compound amounts for the following deposits if interest is compounded continuously.
92.$1500 at 6% for 9 years 93.$12,000 at 5% for 8 years
94.How long will it take for $1000 deposited at 6% compounded semiannually to double? To triple?
95.How long will it take for $2100 deposited at 4% compounded quarterly to double? To triple?
Effective Rate Find the effective rate to the nearest hundredth for each nominal interest rate.
96.7% compounded quarterly 97.6% compounded monthly 98.5% compounded continuously
Present Value Find the present value of each amount.
99.$2000 if interest is 6% compounded annually for 5 years 100.$10,000 if interest is 8% compounded semiannually for
6 years
101.Interest To help pay for college expenses, Julie Davis borrowed
$10,000 at 7% interest compounded semiannually for 8 years.
How much will she owe at the end of the 8-year period?
102.Inflation How long will it take for $1 to triple at an annual inflation rate of 8% compounded continuously?
103.Interest Find the interest rate needed for $6000 to grow to
$8000 in 3 years with continuous compounding.
104.Present Value Frank Steed wants to open a camera shop.
How much must he deposit now at 6% interest compounded monthly to have $25,000 at the end of 3 years?
105.Revenue A concert promoter finds she can sell 1000 tickets at
$50 each. She will not sell the tickets for less than $50, but she finds that for every $1 increase in the ticket price above
$50, she will sell 10 fewer tickets.
y5 7x 1002x,
APPLICATIONS
CHAPTER 2 Review 115 a.Express n, the number of tickets sold, as a function of p, the
price.
b.Express R, the revenue, as a function of p, the price.
c.Find the domain of the function found in part b.
d.Express R, the revenue, as a function of n, the number sold.
e.Find the domain of the function found in part d.
f. Find the price that produces the maximum revenue.
g.Find the number of tickets sold that produces the maximum revenue.
h.Find the maximum revenue.
i. Sketch the graph of the function found in part b.
j. Describe what the graph found in part i tells you about how the revenue varies with price.
106.Cost Suppose the cost in dollars to produce xposters is given by
a.Sketch a graph of
b.Find a formula for the cost to produce an additional poster when xposters are already produced.
c.Find a formula for the average cost per poster.
d.Find a formula for the change in the average cost per poster when one additional poster is pro- duced. (This quantity is approximately equal to the mar- ginal average cost, which will be discussed in the chapter on the derivative.)
107.Cost Suppose the cost in dollars to produce xhundreds of nails is given by
a.Sketch a graph of
b.Find a formula for the cost to produce an additional hundred nails when x hundred are already produced. (This quantity is approximately equal to the mar- ginal cost.)
c.Find a formula for the average cost per hundred nails.
d.Find a formula for the change in the average cost per nail when one additional batch of 100 nails is produced. (This quantity is approximately equal to the marginal average cost, which will be discussed in the chapter on the derivative.)
108.Consumer Price Index The U.S. consumer price index (CPI, or cost of living index) has risen over the years, as shown in the table in the next column, using an index in which the aver- age over the years 1982 to 1984 is set to 100. Source:Bureau of Labor Statistics.
a.Lettingtbe the years since 1960, write an exponential func- tion in the form that fits the data at 1960 and 2005.
b.If your calculator has an exponential regression feature, find the best fitting exponential function for the data.
y5at
A1x112 2A1x2, A1x2,
C1x112 2C1x2, C1x2.
C1x25x214x17.
A1x1122A1x2, A1x2,
C1x112 2C1x2, C1x2.
C1x2 55x13 x11.
c.Use a graphing calculator to plot the answers to parts a and b on the same axes as the data. Are the answers to parts a and b close to each other?
d.If your calculator has a quadratic and cubic regression fea- ture, find the best-fitting quadratic and cubic functions for the data.
e.Use a graphing calculator to plot the answers to parts b and d on the same window as the data. Discuss the extent to which any one of these functions models the data better than the others.
L i fe S c i e n c e s
109.Fever A certain viral infection causes a fever that typically lasts 6 days. A model of the fever (in on day x,
is
According to the model, on what day should the maximum fever occur? What is the maximum fever?
110.Sunscreen An article in a medical journal says that a sun- screen with a sun protection factor (SPF) of 2 provides 50% protection against ultraviolet B (UVB) radiation, an SPF of 4 provides 75% protection, and an SPF of 8 pro- vides 87.5% protection (which the article rounds to 87%).
Source:Family Practice.
a.87.5% protection means that 87.5% of the UVB radiation is screened out. Write as a fraction the amount of radiation that is let in, and then describe how this fraction, in general, relates to the SPF rating.
b.Plot UVB percent protection against x, where c.Based on your graph from part b, give an equation relating
UVB protection to SPF rating.
d.An SPF of 8 has double the chemical concentration of an SPF 4. Find the increase in the percent protection.
e.An SPF of 30 has double the chemical concentration of an SPF 15. Find the increase in the percent protection.
f. Based on your answers from parts d and e, what happens to the increase in the percent protection as the SPF continues to double?
1/SPF. 1y2 x5
F1x25 22 3x2114
3x196.
1#x#6,
°F)
Year CPI
1960 29.6
1970 38.8
1980 82.4
1990 130.7 1995 152.4 2000 172.2 2005 195.3
111. HIV in Infants The following table lists the reported number of cases of infants born in the United States with HIV in recent years because their mother was infected.* Source:
Centers for Disease Control and Prevention.
114. Population Growth A population of 15,000 small deer in a specific region has grown exponentially to 17,000 in 4 years.
a.Write an exponential equation to express the population growth yin terms of time tin years.
b.At this rate, how long will it take for the population to reach 45,000?
115. Population Growth In 1960 in an article in Sciencemaga- zine, H. Van Forester, P. M. Mora, and W. Amiot predicted that world population would be infinite in the year 2026. Their pro- jection was based on the rational function defined by
where gives population in year t. This function has pro- vided a relatively good fit to the population until very recently. Source:Science.
a.Estimate world population in 2010 using this function, and compare it with the estimate of 6.909 billion. Source:
United Nations.
b.What does the function predict for world population in 2020? 2025?
c.Discuss why this function is not realistic, despite its good fit to past data.
116. Intensity of Light The intensity of light (in appropriate units) passing through water decreases exponentially with the depth it penetrates beneath the surface according to the function where xis the depth in meters. A certain water plant requires light of an intensity of 1 unit. What is the greatest depth of water in which it will grow?
117. Drug Concentration The concentration of a certain drug in the bloodstream at time t(in minutes) is given by
Use a graphing calculator to find the maximum concentration and the time when it occurs.
118. Glucose Concentration When glucose is infused into a person’s bloodstream at a constant rate of cgrams per minute, the glu- cose is converted and removed from the bloodstream at a rate proportional to the amount present. The amount of glucose in grams in the bloodstream at time t(in minutes) is given by
where ais a positive constant. Assume and
a.At what time is the amount of glucose a maximum? What is the maximum amount of glucose in the bloodstream?
b.When is the amount of glucose in the bloodstream 0.1 g?
c.What happens to the amount of glucose in the bloodstream after a very long time?
119. Species Biologists have long noticed a relationship between the area of a piece of land and the number of species found there. The following data shows a sample of the British Isles
a51.3. g050.08,c50.1,
g1t25 c
a1ag02c abe2at, c1t25e2t2e22t.
I1x2 510e20.3x, p1t2
p1t2 5 1.7931011 12026.872t20.99, Year Cases
1995 295
1997 166
1999 109
2001 115
2003 94
2005 107
2007 79
*These data include only those infants born in the 25 states with confiden- tial name-based HIV infection reporting.
a.Plot the data on a graphing calculator, letting corre- spond to the year 1995.
b.Using the regression feature on your calculator, find a qua- dratic, a cubic, and an exponential function that models this data.
c.Plot the three functions with the data on the same coordi- nate axes. Which function or functions best capture the behavior of the data over the years plotted?
d.Find the number of cases predicted by all three functions for 2015. Which of these are realistic? Explain.
112.Respiratory Rate Researchers have found that the 95th per- centile (the value at which 95% of the data is at or below) for respiratory rates (in breaths per minute) during the first 3 years of infancy are given by
for awake infants and
for sleeping infants, where xis the age in months. Source:
Pediatrics.
a.What is the domain for each function?
b.For each respiratory rate, is the rate decreasing or increas- ing over the first 3 years of life? (Hint:Is the graph of the quadratic in the exponent opening upward or downward?
Where is the vertex?)
c.Verify your answer to part b using a graphing calculator.
d.For a 1-year-old infant in the 95th percentile, how much higher is the waking respiratory rate than the sleeping res- piratory rate?
113.Polar Bear Mass One formula for estimating the mass (in kg) of a polar bear is given by
where gis the axillary girth in centimeters. It seems reasonable that as girth increases, so does the mass. What is the largest girth for which this formula gives a reasonable answer? What is the predicted mass of a polar bear with this girth? Source:Wildlife Management.
m1g25e0.02 10.062g20.000165g2, y5101.7285820.0139928x10.00017646x2
y5101.8241120.0125995x10.00013401x2
t50
CHAPTER 2 Review 117 and how many vascular plants are found on each. Source:
Journal of Biogeography.
122.Planets The following table contains the average distance D from the sun for the eight planets and their period Pof revolu- tion around the sun in years. Source:The Natural History of the Universe.
Isle Area (km2) Species
Ailsa 0.8 75
Fair 5.2 174
Iona 9.1 388
Man 571.6 765
N. Ronaldsay 7.3 131
Skye 1735.3 594
Stronsay 35.2 62
Wight 380.7 1008
a. One common model for this relationship is logarithmic.
Using the logarithmic regression feature on a graphing cal- culator, find a logarithmic function that best fits the data.
b. An alternative to the logarithmic model is a power func- tion of the form Using the power regression feature on a graphing calculator, find a power function that best fits the data.
c. Graph both functions from parts a and b along with the data. Give advantages and drawbacks of both models.
d. Use both functions to predict the number of species found on the isle of Shetland, with an area of 984.2 km2. Com- pare with the actual number of 421.
e. Describe one or more situations where being able to pre- dict the number of species could be useful.
P hy s i c a l S c i e n c e s
120.Oil Production The production of an oil well has decreased exponentially from 128,000 barrels per year 5 years ago to 100,000 barrels per year at present.
a.Letting represent the present time, write an exponen- tial equation for production yin terms of time tin years.
b.Find the time it will take for production to fall to 70,000 barrels per year.
121.Dating Rocks Geologists sometimes measure the age of rocks by using “atomic clocks.” By measuring the amounts of potassium-40 and argon-40 in a rock, the age tof the speci- men (in years) is found with the formula
where A and K, respectively, are the numbers of atoms of argon-40 and potassium-40 in the specimen.
a.How old is a rock in which and
b.The ratio for a sample of granite from New Hampshire is 0.212. How old is the sample?
c.Let A/K5r.What happens to tas rgets larger? Smaller?
A/K
K.0?
A50 t511.2631092ln3118.331A/K2 4
ln 2 ,
t50
S5b1Ac2.
Planet Distance (D) Period (P)
Mercury 0.39 0.24
Venus 0.72 0.62
Earth 1 1
Mars 1.52 1.89
Jupiter 5.20 11.9
Saturn 9.54 29.5
Uranus 19.2 84.0
Neptune 30.1 164.8
The distances are given in astronomical units (A.U.); 1 A.U. is the average distance from Earth to the sun. For example, since Jupiter’s distance is 5.2 A.U., its distance from the sun is 5.2 times farther than Earth’s.
a.Find functions of the form for 1.5, and 2 that fit the data at Neptune.
b.Use a graphing calculator to plot the data in the table and to graph the three functions found in part a. Which function best fits the data?
c.Use the best-fitting function from part b to predict the period of Pluto (which was removed from the list of planets in 2006), which has a distance from the sun of 39.5 A.U.
Compare your answer to the true value of 248.5 years.
d.If you have a graphing calculator or computer program with a power regression feature, use it to find a power func- tion (a function of the form that approximately fits the data. How does this answer compare with the answer to part b?
P5kDn) n51, P5kDn