Chapter 5

Solve the equation. $$\log \sqrt{x^{2}-1}=2$$

List all asymptotes of the graph of the function and the approximate coordinates of each local extremum. $$f(x)=e^{-x^{2}}$$

Do the graphs of \(f(x)=\log x^{2}\) and \(g(x)=2 \log x\) appear to be the same? How do they differ?

Rationalize the denominator and simplify your answer. $$\frac{\sqrt{x}}{\sqrt{x}-\sqrt{c}}$$

Solve the equation. $$\log \sqrt[4]{x^{2}+15 x}=2 / 5$$

List all asymptotes of the graph of the function and the approximate coordinates of each local extremum. $$g(x)=-x e^{x^{2} / 20}$$

Do the graphs of \(h(x)=\log x^{3}\) and \(k(x)=3 \log x\) appear to be the same?

Rationalize the denominator and simplify your answer. $$\frac{10}{\sqrt[3]{2}}$$

Solve the equation. $$\ln \left(x^{2}+1\right)-\ln (x-1)=1+\ln (x+1)$$

There is a colony of fruit flies in Andy's kitchen. Assume we can model the population \(t\) days from now by the function \(p(t)=100 \cdot(12)^{t / 10} \cdot\). An average fruit fly is about. 1 inches long. (a) How many fruit flies are currently in Andy's kitchen? (b) How many will there be at this time next week? In two weeks? (c) In how many days will the population reach \(2500 ?\) (d) Is it realistic to assume that this model will remain valid for a year? Justify your answer. IHint: According to the model, what will the population be in a year?

List the transformations that will change the graph of \(g(x)=\ln x\) into the graph of the given function. $$f(x)=2 \cdot \ln x$$

Rationalize the denominator and simplify your answer. $$\frac{-6}{\sqrt[3]{4}}$$

Solve the equation. $$\frac{\ln (2 x+1)}{\ln (3 x-1)}=2$$

If current rates of deforestation and fossil fuel consumption continue, then the amount of atmospheric carbon dioxide in parts per million (ppm) will be given by \(f(x)=375 e^{00609 x},\) where \(x=0\) corresponds to 2000 (a) What is the amount of carbon dioxide in \(2003 ?\) In \(2022 ?\) (b) In what year will the amount of carbon dioxide reach \(500 \mathrm{ppm} ?\)

List the transformations that will change the graph of \(g(x)=\ln x\) into the graph of the given function. $$f(x)=\ln x-7$$

Use the fact that \(x^{3}+y^{3}=\) \((x+y)\left(x^{2}-x y+y^{2}\right)\) to rationalize the denominator. $$\frac{1}{\sqrt[3]{3}+1}$$

Deal with radioactive decay and the function \(M(x)=c\left(.5^{x / h}\right) \). A sample of 300 grams of uranium decays to 200 grams in .26 billion years. Find the half-life of uranium.

The pressure of the atmosphere \(p(x)\) (in pounds per square inch) is given by $$ p(x)=k e^{-0000425 x} $$ where \(x\) is the height above sea level (in feet) and \(k\) is a constant. (a) Use the fact that the pressure at sea level is 15 pounds per square inch to find \(k\) (b) What is the pressure at 5000 feet? (c) If you were in a spaceship at an altitude of 160,000 feet, what would the pressure be?

List the transformations that will change the graph of \(g(x)=\ln x\) into the graph of the given function. $$h(x)=\ln (x-4)$$

Deal with radioactive decay and the function \(M(x)=c\left(.5^{x / h}\right) \). It takes 1000 years for a sample of 300 mg of radium- 226 to decay to 195 mg. Find the half-life of radium- 226 .

(a) The function \(g(t)=.6-e^{-0.479 t}\) gives the percentage of the United States population (expressed as a decimal) that has seen a new television show \(t\) weeks after it goes on the air. According to this model, what percentage of people have seen the show after 24 weeks? (b) The show will be renewed if over half the population has seen it at least once. Approximately when will \(50 \%\) of the people have seen the show? (c) According to this model, when will 59.9 \% of the people have seen it? When will \(60 \%\) have seen it?

List the transformations that will change the graph of \(g(x)=\ln x\) into the graph of the given function. $$k(x)=\ln (x+2)$$

Deal with radioactive decay and the function \(M(x)=c\left(.5^{x / h}\right) \). A 3 -gram sample of an isotope of sodium decays to 1 gram in 23.7 days. Find the half-life of the isotope of sodium.

List the transformations that will change the graph of \(g(x)=\ln x\) into the graph of the given function. $$h(x)=\ln (x+3)-4$$

Deal with radioactive decay and the function \(M(x)=c\left(.5^{x / h}\right) \). The half-life of cobalt- 60 is 5.3 years. How long will it take for 100 grams to decay to 33 grams?

The number of subscribers to basic cable TV (in millions) can be approximated by $$ g(x)=\frac{76.7}{1+16\left(.8444^{x}\right)^{\prime}} $$ where \(x=0\) corresponds to \(1970 .\) (a) Estimate the number of subscribers in 2005 and 2010 . (b) When does the number of subscribers reach 70 million? (c) According to this model, will the number of subscribers ever each 90 million?

List the transformations that will change the graph of \(g(x)=\ln x\) into the graph of the given function. $$k(x)=\ln (x-2)+2$$

Find the difference quotient of the given function. Then rationalize its numerator and simplify. $$f(x)=\sqrt{x+1}$$

Deal with radioactive decay and the function \(M(x)=c\left(.5^{x / h}\right) \). After six days a sample of radon- 222 decayed to \(33.6 \%\) of its original mass. Find the half-life of radon- \(222 .\)

Sketch the graph of the function. $$f(x)=\log (x-3)$$

Find the difference quotient of the given function. Then rationalize its numerator and simplify. $$g(x)=2 \sqrt{x+3}$$

The Gateway Arch (Figure \(5-15\) ) is 630 feet high and 630 feet wide at ground level. Suppose it were placed on a coordinate plane with the \(x\) -axis at ground level and the \(y\) -axis going through the center of the arch. Find a catenary function \(g(x)=A\left(e^{k x}+e^{-k x}\right)\) and a constant \(C\) such that the graph of the function \(f(x)=g(x)+C\) provides a model of the arch. [Hint: Experiment with various values of \(A, k, C\) as in the Graphing Exploration on page \(365 .\) Many correct answers are possible.]

Sketch the graph of the function. $$g(x)=2 \ln x+3$$

Find the difference quotient of the given function. Then rationalize its numerator and simplify. $$f(x)=\sqrt{x^{2}+1}$$

(a) A genetic engineer is growing cells in a fermenter. The cells multiply by splitting in half every 15 minutes. The new cells have the same DNA as the original ones. Complete the following table. \begin{array}{|c|c|} \hline \begin{array}{c} \text { Time } \\ \text { (hours) } \end{array} & \text { Number of Cells } \\ \hline 0 & 1 \\ \hline .25 & 2 \\ \hline .5 & 4 \\ \hline .75 & \\ \hline 1 & \\ \hline \end{array} (b) Write the rule of the function that gives the number of \(C\) cells at time \(t\) hours.

Sketch the graph of the function. $$h(x)=-2 \log x$$

Find the difference quotient of the given function. Then rationalize its numerator and simplify. $$g(x)=\sqrt{x^{2}-x}$$

Sketch the graph of the function. $$f(x)=\ln (-x)-3$$

Use the equation \(y=92.8935 \cdot x^{-6669}\) which gives the approximate distance \(y\) (in millions of miles) from the sun to a planet that takes \(x\) earth years to complete one orbit of the sun. Find the distance from the sun to the planet whose orbit time is given. Mercury (.24 years)

Deal with radioactive decay and the function \(M(x)=c\left(.5^{x / h}\right) \). A Native American mummy was found recently. If it has lost \(26.4 \%\) of its carbon- \(14,\) approximately how long ago did the Native American die?

A weekly census of the tree-frog population in Frog Hollow State Park produces the following results. $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Week } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Population } & 18 & 54 & 162 & 486 & 1458 & 4374 \\ \hline \end{array}$$ (a) Find a function of the form \(f(x)=P a^{x}\) that describes the frog population at time \(x\) weeks. (b) What is the growth factor in this situation (that is, by what number must this week's population be multiplied to obtain next week's population)? (c) Each tree frog requires 10 square feet of space and the park has an area of 6.2 square miles. Will the space required by the frog population exceed the size of the park in 12 weeks? In 14 weeks? [Remember: 1 square mile \(\left.=5280^{2} \text { square feet. }\right]\)

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

Use the equation \(y=92.8935 \cdot x^{-6669}\) which gives the approximate distance \(y\) (in millions of miles) from the sun to a planet that takes \(x\) earth years to complete one orbit of the sun. Find the distance from the sun to the planet whose orbit time is given. Mars ( 1.88 years)

An eccentric billionaire offers you a job for the month of September. She says that she will pay you \(2 \notin\) on the first day, \(4 \notin\) on the second day, \(8 \notin\) on the third day, and so on, doubling your pay on each successive day. (a) Let \(P(x)\) denote your salary in dollars on day \(x .\) Find the rule of the function \(P\). (b) Would you be better off financially if instead you were paid \(\$ 10,000 \text { per day? [ Hint: Consider } P(30) .]\)

Use the equation \(y=92.8935 \cdot x^{-6669}\) which gives the approximate distance \(y\) (in millions of miles) from the sun to a planet that takes \(x\) earth years to complete one orbit of the sun. Find the distance from the sun to the planet whose orbit time is given. Saturn ( 29.46 years)

Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). At what annual rate of interest should 1000 dollars be invested so that it will double in 10 years if interest is compounded quarterly?

Take an ordinary piece of typing paper and fold it in half; then the folded sheet is twice as thick as the single sheet was. Fold it in half again so that it is twice as thick as before. Keep folding it in half as long as you can. Soon the folded paper will be so thick and small that you will be unable to continue, but suppose you could keep folding the paper as many times as you wanted. Assume that the paper is .002 inches thick. (a) Make a table showing the thickness of the folded paper for the first four folds (with fold 0 being the thickness of the original unfolded paper). (b) Find a function of the form \(f(x)=P a^{x}\) that describes the thickness of the folded paper after \(x\) folds. (c) How thick would the paper be after 20 folds? (d) How many folds would it take to reach the moon (which is 243,000 miles from the earth)? [Hint: One mile is \(5280 \text { feet. }]\)

Use the equation \(y=92.8935 \cdot x^{-6669}\) which gives the approximate distance \(y\) (in millions of miles) from the sun to a planet that takes \(x\) earth years to complete one orbit of the sun. Find the distance from the sun to the planet whose orbit time is given. Pluto \((247.69\) years)

Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). How long does it take 500 dollars to triple if it is invested at \(6 \%\) compounded: (a) annually, (b) quarterly, (c) daily?

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. $$1800$$