Chapter 4

\(f(x)=\log _{x} x\)

In Exercises 7 through 14 , find all real numbers \(x\) that satisfy the given equation. 7\. \(8=2 e^{0.04 x}\)

\(4 \ln x=8\)

\(5^{x}=e^{3}\)

\(\ln (x-2)+3=\ln (x+1)\)

In Exercises 15 through 30 , find the derivative \(\frac{d y}{d x} .\) In some of these problems, you may need to use implicit differentiation or logarithmic differentiation. 15\. \(y=x^{2} e^{-x}\)

\(y=x \ln x^{2}\)

\(y=\ln \sqrt{x^{2}+4 x+1}\)

\(y=\log _{3}\left(x^{2}\right)\)

\(y=\ln \left(\frac{e^{3 x}}{1+x}\right)\)

\(y=\frac{\left(x^{2}+e^{2 x}\right)^{3} e^{-2 x}}{\left(1+x-x^{2}\right)^{2 / 3}}\)

\(y=\frac{e^{-2 x}\left(2-x^{3}\right)^{3 / 2}}{\sqrt{1+x^{2}}}\)

In Exercises 31 through 38 , determine where the given function is increasing and decreasing and where its graph is concave upward and concave downward. Sketch the graph, showing as many key features as possible (high and low points, points of inflection, asymptotes, intercepts, cusps, vertical tangents). 31\. \(f(x)=e^{x}-e^{-x}\)

\(F(u)=u^{2}+2 \ln (u+2)\)

\(g(t)=\frac{\ln (t+1)}{t+1}\)

In Exercises 39 through 42 , find the largest and small values of the given function over the prescribed closi bounded interval. 39\. \(f(x)=\ln \left(4 x-x^{2}\right) \quad\) for \(1 \leq x \leq 3\)

\(h(t)=\left(e^{-t}+e^{t}\right)^{5} \quad\) for \(-1 \leq t \leqq 1\)

In Exercises 43 through 46 , find an equation for the tangent line to the given curve at the specified point. 43\. \(y=x \ln x^{2}\) where \(x=1\)

\(y=\left(x^{2}-x\right) e^{-x}\) where \(x=0\)

\(y=x^{3} e^{2-x}\) where \(x=2\)

\(y=(x+\ln x)^{3}\) where \(x=1\)

Find \(f(9)\) if \(f(x)=e^{k x}\) and \(f(3)=2\).

RADIOACTIVE DECAY A radioactive substance decays exponentially. If 500 grams of the substance were present initially and 400 grams are prescnt 50 years later, how many grams will be present after 200 years?

COMPOUND INTEREST A sum of money is invested at a certain fixed interest rate, and the interest is compounded continuously. After 10 years, the money has doubled. How will the balance at the end of 20 years compare with the initial investment?

GROWTH OF BACTERIA The following data were compiled by a researcher during the first 10 minutes of an experiment designed to study the growth of bacteria: \begin{tabular}{l|c|c} Number of minutes & 0 & 10 \\ \hline Number of bacteria & 5,000 & 8,000 \end{tabular} Assuming that the number of bacteria grows exponentially, how many bacteria will be present after 30 minutes?

RADIOACTIVE DECAY The following data were compiled by a researcher during an experiment designed to study the decay of a radioactive substance: \begin{tabular}{l|c|c} Number of hours & 0 & 5 \\ \hline Grams of substance & 1,000 & 700 \end{tabular} Assuming that the sample of radioactive substance decays exponentially, how much is left after 20 hours?

SALES FROM ADVERTISING It is estimated that if \(x\) thousand dollars are spent on advertising. approximately \(Q(x)=50-40 e^{-0.1 x}\) thousand units of a certain commodity will be sold. a. Sketch the sales curve for \(x \geq 0\). b. How many units will be sold if no money is spent on advertising? c. How many units will be sold if \(\$ 8,000\) is spent on advertising? d. How much should be spent on advertising to generate sales of 35,000 units? e. According to this model, what is the most optimistic sales projection?

COMPOUND INTEREST How quickly will \(\$ 2,000\) grow to \(\$ 5,000\) when invested at an annual interest rate of \(8.5\) if interest is compounded: a. Quarterly b. Continuously

COMPOUND INTEREST How much should you invest now at an annual interest rate of \(6.25 \%\) so that your balance 10 years from now will be \(\$ 2,000\) if interest is compounded: a. Monthly b. Continuously

DEBT REPAYMENT You have a debt of \(\$ 10,000\), which is scheduled to be repaid at the end of 10 years. If you want to repay your debt now, how much should your creditor demand if the prevailing annual interest rate is: a. \(7 \%\) compounded monthly b. \(6 \%\) compounded continuously

EFFECTIVE RATE OF INTEREST Which investment has the greater effective interest rate: \(8.25 \%\) per year compounded quarterly or \(8.20 \%\) per year compounded continuously?

DEPRECIATION The value of a certain industrial machine decreases exponentially. If the machine was originally worth \(\$ 50,000\) and was worth \(\$ 20,000\) five years later, how much will it be worth when it is 10 years old?

POPULATION GROWTH It is estimated that \(t\) years from now the population of a certain country will be \(P\) million people, where $$ P(t)=\frac{30}{1+2 e^{-0.05 t}} $$ a. Sketch the graph of \(P(t)\). b. What is the current population? c. What will be the population in 20 years? d. What happens to the population in the long run?

BACTERIAL GROWTH The number of bacteria in a certain culture grows exponentially. If 5,000 bacteria were initially present and 8,000 were present 10 minutes later, how long will it take for the number of bacteria to double?

PROFIT A manufacturer of digital cameras estimates that when cameras are sold for \(x\) dollars apiece, consumers will buy \(8,000 e^{-0.02 x}\) cameras each week. He also determines that profit is maximized when the selling price \(x\) is \(1.4\) times the cost of producing cach unit. What price maximizes weekly profit? How many units are sold each week at this optimal price?

OPTIMAL HOLDING TIME Beth owns an asset whose value \(t\) years from now will be \(V(t)=2,000 e^{\sqrt{2 t}}\) dollars. If the prevailing interest rate remains constant at \(5 \%\) per year compounded continuously, when will it be most advantageous to sell the collection and invest the proceeds?

RULE OF 70 Investors are often interested in knowing how long it takes for a particular investment to double. A simple means for making this determination is the "rule of 70 ," which says: The doubling time of an investment with an annual interest rate \(r \%\) compounded continuously is given by \(d=\frac{70}{r}\). a. For interest rate \(r\), use the formula \(B=P e^{r t}\) to find the doubling time for \(r=4,6,9,10\), and 12. In each case, compare the value with the value obtained from the rule of 70 . b. Some people prefer a "rule of \(72^{*}\) and others use a "rule of 69 ." Test these alternative rules as in part (a), and write a paragraph on which rule you would prefer to use.

ANIMAL DEMOGRAPHY A naturalist at an animal sanctuary has determined that the function \(f(x)=\frac{4 e^{-(\ln x)^{2}}}{\sqrt{\pi} x}\) provides a good measure of the number of animals in the sanctuary that are \(x\) years old. Sketch the graph of \(f(x)\) for \(x>0\), and find the most likely age among the animals, that is, the age for which \(f(x)\) is largest.

COOLING Jayla falls into a lake where the water temperature is \(-3^{\circ} \mathrm{C}\). Her body temperature after \(t\) minutes in the water is \(T(t)=35 e^{-0.32 t}\). She will lose consciousness when her body temperature reaches \(27^{\circ} \mathrm{C}\). How long do rescuers have to save her? How fast is Jayla's body temperature dropping at the time it reaches \(27^{\circ} \mathrm{C}\) ?

FORENSIC SCIENCE The temperature \(T\) (in degrees Celsius) of the body of a murder victim found in a room where the air temperature is \(20^{\circ} \mathrm{C}\) is given by $$ T(t)=20+17 e^{-0.07 t} $$ where \(t\) is the number of hours after the victim's death. a. Graph the body temperature \(T(t)\) for \(t \geq 0\). What is the horizontal asymptote of this graph, and what does it represent? b. What is the temperature of the victim's body after 10 hours? How long does it take for the body's temperature to reach \(25^{\circ} \mathrm{C}\) ? c. Abel Baker is a clerk in the firm of Dewey, Cheatum, and Howe. He comes to work early one morning and finds the corpse of his boss, Will Cheatum, draped across his desk. He calls the police, and at 8 A.M., they determine that the temperature of the corpse is \(33^{\circ} \mathrm{C}\). Since the Last item entered on the victim's notepad was, "Fire that idiot, Baker," Abel is considered the prime suspect. Actually, Abel is bright cnough to have been reading this text in his spare time. He glances at the thermostat to confirm that the room temperature is \(20^{\circ} \mathrm{C}\). For what time will he need an alibi to establish his innocence?

CONCENTRATION OF DRUG Suppose that \(t\) hours after an antibiotic is administered orally, its concentration in the patient's bloodstream is given by a surge function of the form \(C(t)=A t e^{-k t}\), where \(A\) and \(k\) are positive constants and \(C\) is measured in micrograms per milliliter of blood. Blood samples are taken periodically, and it is determined that the maximum concentration of drug occurs 2 hours after it is administered and is 10 micrograms per milliliter. a. Use this information to determine \(A\) and \(k\). b. A new dose will be administered when the concentration falls to 1 microgram per milliliter. When does this occur?

POPULATION GROWTH According to a logistic model based on the assumption that the carth can support no more than 40 billion people, the world's population (in billions) \(t\) years after 1960 is given by a function of the form \(P(t)=\frac{40}{1+C e^{-k t}}\) where \(C\) and \(k\) are positive constants. Find the function of this form that is consistent with the fact that the world's population was approximately 3 billion in 1960 and 4 billion in 1975 . What does your model predict for the population in the year 2010 ? Check the accuracy of the model by consulting the Internet.

ACIDITY (pH) OF A SOLUTION The acidity of a solution is measured by its \(\mathrm{pH}\) value, which is defined by \(\mathrm{pH}=-\log _{10}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\), where \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) is the hydronium ion concentration (moles/liter) of the solution. On average, milk has a pH value that is three times the \(\mathrm{pH}\) value of a lime, which in tum has half the \(\mathrm{pH}\) value of an orange. If the average \(\mathrm{pH}\) of an orange is \(3.2\), what is the average hydronium ion concentration of a lime?

MORTALITY RATES It is sometimes useful for actuaries to be able to project mortality rates within a given population. A formula sometimes used for computing the mortality rate \(D(t)\) for women in the age group \(25-29\) is $$ D(t)=\left(D_{0}-0.00046\right) e^{-0.162 t}+0.00046 $$ where t is the number of years after a fixed base year and D0 is the mortality rate when t 0. a. Suppose the initial mortality rate of a particular group is 0.008 (8 deaths per 1,000 women). What is the mortality rate of this group 10 years later? What is the rate 25 years later? b. Sketch the graph of the mortality function D(t) for the group in part (a) for 0 t 25.

GROSS DOMESTIC PRODUCT The gross domestic product (GDP) of a certain country was 100 billion dollars in 1995 and 165 billion dollars in 2005 . Assuming that the GDP is growing exponentially, what will it be in the year 2015 ?

ARCHAEOLOGY "Lucy," the famous prehuman whose skeleton was discovered in Africa, has been found to be approximately \(3.8\) million years old. a. Approximately what percentage of original \({ }^{14} \mathrm{C}\) would you expect to find if you tried to apply carbon dating to Lucy? Why would this be a problem if you were actually trying to "date" Lucy? b. In practice, carbon dating works well only for relatively "recent" samples - those that are no more than approximately 50,000 years old. For older samples, such as Lucy, variations on carbon dating have been developed, such as potacsium-argon and rubidium-strontium dating. Read an article on altemative dating methods, and write a paragraph on how they are used.

A population model developed by the U.S. Census Bureau uses the formula $$ P(t)=\frac{202.31}{1+e^{3.938-0.314 s}} $$ to estimate the population of the United States (in millions) for every 10th year from the base year 1790. Thus, for instance, t 0 corresponds to 1790, t 1 to 1800, t 10 to 1890, and so on. The model excludes Alaska and Hawaii. a. Use this formula to compute the population of the contiguous United States for the years 1790, 1800, 1830, 1860, 1880, 1900, 1920, 1940, 1960, 1980, 1990, and 2000. b. Sketch the graph of P(t). When does this model predict that the population of the contiguous United States will be increasing most rapidly? c. Use the Internet to find the actual population figures for the years listed in part (a). Does the given population model seem to be accurate? (Remember to exclude Alaska and Hawaii.) Write a paragraph describing some possible reasons for any major differences between the predicted population figures and the actual census figures.

Use a graphing utility to graph \(y=2^{-x}, y=3^{-x}\), \(y=5^{-x}\), and \(y=(0.5)^{-x}\) on the same set of axes. How does a change in base affect the graph of the exponential function? (Suggestion: Use the graphing window \([-3,3] 1\) by \([-3,3] 1\).)

Use a graphing utility to draw the graphs of \(y=\sqrt{3^{x}}, y=\sqrt{3^{-x}}\), and \(y=3^{-x}\) on the same set of axes. How do these graphs differ? (Suggestion: Use the graphing window \([-4,4] 1\) by \([-2,6] 1\) )

Use a graphing utility to draw the graphs of \(y=3^{x}\) and \(y=4-\ln \sqrt{x}\) on the same axes. Then use TRACE and ZOOM to find all points of intersection of the two graphs.