Chapter 5

Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). (a) How long will it take to triple your money if you invest 500 dollars at a rate of \(5 \%\) per year compounded annually? (b) How long will it take at \(5 \%\) compounded quarterly?

Suppose you invest \(\$ 1200\) in an account that pays \(4 \%\) interest, compounded annually and paid from date of deposit to date of withdrawal. (a) Find the rule of the function \(f\) that gives the amount you would receive if you closed the account after \(x\) years. (b) How much would you receive after 3 years? After 5 years and 9 months? (c) When should you close the account to receive \(\$ 1850 ?\)

Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). At what rate of interest (compounded annually) should you invest 500 dollars if you want to have 1500 dollars in 12 years?

Anne now has a balance of \(\$ 800\) on her credit card, on which \(1.5 \%\) interest per month is charged. Assume that she makes no further purchases or payments (and that the credit card company doesn't turn her account over to a bill collector (a) Find the rule of the function \(g\) that gives Anne's total credit card debt after \(x\) months. (b) How much will Anne owe after one year? After two years? (c) When will she owe twice the amount she owes now?

Find a viewing window (or windows) that shows a complete graph of the function. $$f(x)=\frac{\log x}{x}$$

The population of Mexico was 100.4 million in 2000 and is expected to grow at the rate of \(1.4 \%\) per year. (a) Find the rule of the function \(f\) that gives Mexico's population (in millions) in year \(x,\) with \(x=0\) corresponding to 2000. (b) Estimate Mexico's population in 2010 . (c) When will the population reach 125 million people?

Find a viewing window (or windows) that shows a complete graph of the function. $$l(x)=e^{e^{x}}$$

Between 1790 and \(1860,\) the population y of the United States (in millions) in year x was given by \(y=3.9572\left(1.0299^{\circ}\right),\) where \(x=0\) corresponds to \(1790 .\)F ind the U.S. population in the given year. $$1859$$

Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). Find a formula that gives the time needed for an investment of \(P\) dollars to double, if the interest rate is \(r \%\) compounded annually.

The number of digital devices (such as MP3 players, handheld computers, cell phones, and PCs) in the world was approximately .94 billion in 1999 and is growing at a rate of \(28.3 \%\) a year. (a) Find the rule of a function that gives the number of digital devices (in billions) in year \(x,\) with \(x=0\) corresponding to 1999 (b) Approximately how many digital devices will be in use in \(2010 ?\) (c) If this model remains accurate, when will the number of digital devices reach 6 billion?

Find a viewing window (or windows) that shows a complete graph of the function. $$r(x)=\ln \left(e^{x}\right)$$

Here are some of the reasons why restrictions are necessary when defining fractional powers of a negative number. (a) Explain why the equations \(x^{2}=-4, x^{4}=-4\) \(x^{6}=-4,\) etc. \(,\) have no real solutions. Hence, we cannot define \(c^{1 / 2}, c^{1 / 4}, c^{1 / 6}\) when \(c=-4\) (b) since \(1 / 3\) is the same as \(2 / 6,\) it should be true that \(c^{1 / 3}=c^{2 / 6},\) that is, that \(\sqrt[3]{c}=\sqrt[6]{c^{2}} .\) Show that this is false when \(c=-8\)

Deal with functions of the form \(f(x)=P e^{k x}\) where \(k\) is the continuous exponential growth rate (see Example 6 ). The present concentration of carbon dioxide in the atmosphere is 364 parts per million (ppm) and is increasing exponentially at a continuous yearly rate of \(.4 \%\) (that is, \(k=.004) .\) How many years will it take for the concentration to reach 500 ppm?

Find the average rate of change of the function. \(f(x)=\ln (x-2),\) as \(x\) goes from 3 to 5

Use a calculator to find \((3141)^{-3141}\). Explain why your answer cannot possibly be the number \((3141)^{-3141} .\) Why does your calculator behave the way that it does?

Deal with functions of the form \(f(x)=P e^{k x}\) where \(k\) is the continuous exponential growth rate (see Example 6 ). The amount \(P\) of ozone in the atmosphere is currently decaying exponentially each year at a continuous rate of \(\frac{1}{4} \%\) (that is, \(k=-.0025\) ). How long will it take for half the ozone to disappear (that is, when will the amount be \(P / 2\) )? [Your answer is the half-life of ozone.]

The U.S. Department of Commerce estimated that there were 54 million Internet users in the United States in 1999 and 85 million in 2002 . (a) Find an exponential function that models the number of Internet users in year \(x,\) with \(x=0\) corresponding to 1999 (b) For how long is this model likely to remain accurate? [Hint: The current U.S. population is about 305 million.]

Find the average rate of change of the function. $$g(x)=x-\ln x, \text { as } x \text { goes from } .5 \text { to } 1$$

(a) Graph \(f(x)=x^{5}\) and explain why this function has an inverse function. (b) Show algebraically that the inverse function is \(g(x)=x^{1 / 5}\) (c) Does \(f(x)=x^{6}\) have an inverse function? Why or why not?

At the beginning of an experiment, a culture contains 200 H. pylori bacteria. An hour later there are 205 bacteria. Assuming that the \(H\). pylori bacteria grow exponentially, how many will there be after 10 hours? After 2 days?

Find the average rate of change of the function. \(g(x)=\log \left(x^{2}+x+1\right),\) as \(x\) goes from -5 to -3

If \(n\) is an odd positive integer, show that \(f(x)=x^{n}\) has an inverse function and find the rule of the inverse function. [Hint: Exercise \(71 \text { is the case when } n=5 .]\)

Deal with functions of the form \(f(x)=P e^{k x}\) where \(k\) is the continuous exponential growth rate (see Example 6 ). Between 1996 and \(2004,\) the number of United States subscribers to cell-phone plans has grown nearly exponentially. In 1996 there were 44,043,000 subscribers and in 2004 there were \(182,140,000^{\dagger}\) (a) What is the continuous growth rate of the number of cell-phone subscribers? (b) In what year were there 60,000,000 cell-phone subscribers? (c) Assuming that this rate continuous, in what year will there be 350,000,000 subscribers? (d) In 2007 the United States population was approximately 300 million. Is your answer to part (c) realistic? If not, what could have gone wrong?

The population of India was approximately 1030 million in 2001 and was 967 million in \(1997 .\) If the population continues to grow exponentially at the same rate, what will it be in \(2010 ?\)

Find the average rate of change of the function. \(f(x)=x \log |x|,\) as \(x\) goes from 1 to 4

Use the catalog of basic functions (page 170 ) and Section 3.4 to describe the graph of the given function. $$g(x)=\sqrt{x+3}$$

Deal with functions of the form \(f(x)=P e^{k x}\) where \(k\) is the continuous exponential growth rate (see Example 6 ). The probability \(P\) percent of having an accident while driving a car is related to the alcohol level of the driver's blood by the formula \(P=e^{k t},\) where \(k\) is a constant. Accident statistics show that the probability of an accident is \(25 \%\) when the blood alcohol level is \(t=.15\). (a) Find \(k .\) IUse \(P=25,\) not .25 .1 (b) At what blood alcohol level is the probability of having an accident \(50 \% ?\)

Kerosene is passed through a pipe filled with clay to remove various pollutants. Each foot of pipe removes \(25 \%\) of the pollutants. (a) Write the rule of a function that gives the percentage of pollutants remaining in the kerosene after it has passed through \(x\) feet of pipe. [See Example 7.] (b) How many feet of pipe are needed to ensure that \(90 \%\) of the pollutants have been removed from the kerosene?

Use the catalog of basic functions (page 170 ) and Section 3.4 to describe the graph of the given function. $$h(x)=\sqrt{x}-2$$

Deal with functions of the form \(f(x)=P e^{k x}\) where \(k\) is the continuous exponential growth rate (see Example 6 ). Under normal conditions, the atmospheric pressure (in millibars) at height \(h\) feet above sea level is given by \(P(h)=\) \(1015 e^{-k t},\) where \(k\) is a positive constant. (a) If the pressure at 18,000 feet is half the pressure at sea level, find \(k\). (b) Using the information from part (a), find the atmospheric pressure at 1000 feet, 5000 feet, and 15,000 feet.

If inflation runs at a steady \(3 \%\) per year, then the amount a dollar is worth decreases by \(3 \%\) each year. (a) Write the rule of a function that gives the value of a dollar in year \(x .\) (b) How much will the dollar be worth in 5 years? In 10 years? (c) How many years will it take before today's dollar is worth only a dime?

(a) Find the average rate of change of \(f(x)=\ln x^{2},\) as \(x\) goes from .5 to 2 (b) Find the average rate of change of \(g(x)=\ln (x-3)^{2},\) as \(x\) goes from 3.5 to 5 (c) What is the relationship between your answers in parts (a) and (b) and why is this so?

Use the catalog of basic functions (page 170 ) and Section 3.4 to describe the graph of the given function. $$k(x)=\sqrt{x+4}-4$$

Deal with functions of the form \(f(x)=P e^{k x}\) where \(k\) is the continuous exponential growth rate (see Example 6 ). One hour after an experiment begins, the number of bacteria in a culture is \(100 .\) An hour later, there are 500 . (a) Find the number of bacteria at the beginning of the experiment and the number three hours later. (b) How long does it take the number of bacteria at any given time to double?

You have 5 grams of carbon- \(14,\) whose half-life is 5730 years. (a) Write the rule of the function that gives the amount of carbon- 14 remaining after \(x\) years. [See the box preceding Example 8.1 (b) How much carbon-14 will be left after 4000 years? After 8000 years? (c) When will there be just 1 gram left?

Show that \(g(x)=\ln \left(\frac{x}{1-x}\right)\) is the inverse function of \(f(x)=\frac{1}{1+e^{-x}} .(\text { See Section } 3.7 .)\)

(a) Suppose \(r\) is a solution of the equation \(x^{n}=c\) and \(s\) is a solution of \(x^{n}=d .\) Verify that \(r s\) is a solution of \(x^{n}=c d\) (b) Explain why part (a) shows that \(\sqrt[n]{c d}=\sqrt[n]{c} \sqrt[n]{d}\)

(a) The half-life of radium is 1620 years. If you start with 100 milligrams of radium, what is the rule of the function that gives the amount remaining after \(t\) years? (b) How much radium is left after 800 years? After 1600 years? After 3200 years?

The doubling function \(D(x)=\frac{\ln 2}{\ln (1+x)}\) gives the years required to double your money when it is invested at interest rate \(x\) (expressed as a decimal), compounded annually. (a) Find the time it takes to double your money at each of these interest rates: \(4 \%, 6 \%, 8 \%, 12 \%, 18 \%, 24 \%\) \(36 \%\) (b) Round the answers in part (a) to the nearest year and compare them with these numbers: \(72 / 4,72 / 6,72 / 8\) \(72 / 12,72 / 18,72 / 24,72 / 36 .\) Use this evidence to state a rule of thumb for determining approximate doubling time, without using the function \(D .\) This rule of thumb, which has long been used by bankers, is called the rule of 72 .

The output \(Q\) of an industry depends on labor \(L\) and capital \(C\) according to the equation $$Q=L^{1 / 4} C^{3 / 4} $$ (a) Use a calculator to determine the output for the following resource combinations. $$\begin{array}{|c|c|c|}\hline L & C & Q=L^{1 / 4} C^{3 / 4} \\\\\hline 10 & 7 & \\ \hline 20 & 14 & \\\\\hline 30 & 21 & \\\\\hline 40 & 28 & \\\\\hline 60 & 42 & \\ \hline\end{array}$$ (b) When you double both labor and capital, what happens to the output? When you triple both labor and capital, what happens to the output?

The spread of a flu virus in a community of 45,000 people is given by the function $$f(t)=\frac{45,000}{1+224 e^{-.899 t}}$$ where \(f(t)\) is the number of people infected in week \(t\). (a) How many people had the flu at the outbreak of the epidemic? After three weeks? (b) When will half the town be infected?

Find a function \(f(x)\) with the property \(f(r+s)=f(r) f(s)\) for all real numbers \(r\) and \(s\).

Suppose \(f(x)=A \ln x+B,\) where \(A\) and \(B\) are constants. If \(f(1)=10\) and \(f(e)=1,\) what are \(A\) and \(B ?\)

Do Exercise 77 when the equation relating output to resources is \(Q=L^{1 / 4} C^{1 / 2}\)

The beaver population near a certain lake in year \(t\) is approximately $$p(t)=\frac{2000}{1+199 e^{-.5544 t}}$$ (a) When will the beaver population reach \(1000 ?\) (b) Will the population ever reach \(2000 ?\) Why?

If \(f(x)=A \ln x+B\) and \(f(e)=5\) and \(f\left(e^{2}\right)=8,\) what are \(A\) and \(B ?\)

Do Exercise 77 when the equation relating output to resources is \(Q=L^{1 / 2} C^{3 / 4}\)

Assume that you watched 1000 hours of television this year, and will watch 750 hours next year, and will continue to watch \(75 \%\) as much every year thereafter. (a) In what year will you be down to ten hours per year? (b) In what year would you be down to one hour per year?

(a) Using the viewing window with \(-4 \leq x \leq 4\) and \(-1 \leq y \leq 8, \operatorname{graph} f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=2^{x}\) on the same screen. If you think of the \(y\) -axis as a mirror, how would you describe the relationship between the two graphs? (b) Without graphing, explain how the graphs of \(g(x)=2^{x}\) and \(k(x)=2^{-x}\) are related.

The height \(h\) above sea level (in meters) is related to air temperature \(t\) (in degrees Celsius), the atmospheric pressure \(p\) (in centimeters of mercury at height \(h\) ), and the atmospheric pressure \(c\) at sea level by $$ h=(30 t+8000) \ln (c / p) $$ If the pressure at the top of Mount Rainier is 44 centimeters on a day when sea level pressure is 75.126 centimeters and the temperature is \(7^{\circ} \mathrm{C},\) what is the height of Mount Rainier?