Chapter 9

For Problems 1 through 9, simplify the following expressions. $$ \frac{x^{-1}+z^{-1}}{(z+x) x^{-2}} $$

The number of bacteria in a certain culture is known to triple every day. Suppose that at noon today there are 200 bacteria. (a) Construct a table of values to find a function that gives the number of bacteria after \(t\) days. (b) Approximately what was the population count at noon yesterday? At noon 4 days ago? (c) From now on, suppose the population at noon today is called \(B_{0}\) rather than being specifically 200 . Find a function that gives the number of bacteria after \(t\) days. (d) Express the number of bacteria as a function of \(w\), where \(w\) is time measured in weeks. (e) How many bacteria will be present at noon one week from today?

Let \(g(t)=3^{5 t} .\) Show that $$ \frac{g(t+h)-g(t)}{h}=g(t) \cdot \frac{g(h)-g(0)}{h} . $$

Carbon Dating: Carbon-14, with a half-life of approximately 5730 years, can be used to date organic remains on earth. (See Exercise \(9.8\) for an explanation of carbon dating.) (a) Let \(C_{0}\) denote the amount of radioactive carbon in an organism when it is alive. Find a formula for \(C(t)\), the amount of radioactive carbon in this organism \(t\) years after it has died. (b) In the summer of 1993, while in Syria, I visited the ruins of Palmyra (Tadmor in Arabic-the City of Dates). Palmyra was mentioned in tablets dating as far back as the nineteenth century B.C., but it reached its heyday in the time of Queen Zenobia around 137 A.D. While exploring the site near the Funerary Towers, I came across something that looked disturbingly like that of a human leg bone lying in the sand. A Harvard medical student I met at the site confirmed that this was indeed a human bone-really. Assuming that the relevant person died around 137 A.D., what percentage of the original \(C_{14}\) remained in the bone? (c) Rewrite the equation you got in part (a) so it is in the form \(f(t)=C_{0} a^{t} .\) From this, fill in the blank in the following statement: Each year the amount of carbon-14 in a deceased organism decreases by \(\% .\)

For Problems 1 through 9, simplify the following expressions. $$ \frac{(x y)^{-3}}{x y^{-3}} $$

As the story goes, a long time ago there was a Persian ruler who enjoyed the game of chess so much that in order to demonstrate his gratitude he offered its inventor any reward the man wanted. When the man was called in front of the ruler he requested 1 grain of rice for the first square of the chessboard, 2 for the second, 4 for the third, 8 for the fourth, and so on until rice was allocated for all 64 squares of the chessboard. At first the ruler thought that the request was quite modest-he would have been happy to give the man jewels instead of rice. Eventually he realized that the request was not modest at all and-according to some versions of this story- he ordered the man beheaded. What was all the fuss about? (This classic story is so popular that it has even been used on the TV show "I Love Lucy.") (a) How many grains of rice were allocated to the 64 th square? (b) If we assume that a grain of rice is \(0.02 \mathrm{~g}\), approximately how many grams of rice are allocated to the 64 th square? Compare this with the annual world production of rice \(\left(\approx 4 \times 10^{8}\right.\) tons in 1980\()\). Note: 1 ton is \(907.18\) kilograms. For general edification, a table of commonly used prefixes is supplied below. $$ \begin{array}{llll} {\text { Symbol }} && \text { Factor } & \\ \hline \text { G } & \text { giga- } & 1000000000 & =10^{9} \\ \text { M } & \text { mega- } & 1000000 & =10^{6} \\ \text { k } & \text { kilo- } & 1000 & =10^{3} \\ \text { m } & \text { milli- } & 0.001 & =10^{-3} \\ \text { n } & \text { nano- } & 0.000000001 & =10^{-9} \end{array} $$ (c) Challenge problem (optional): Find the total mass of the rice allocated to all 64 squares. Note: If you have to add 64 numbers without any shortcut, please skip this part of the problem!

Let \(f(t)=3^{t}\). (a) Sketch a graph of \(f\). (b) Approximate \(f^{\prime}(1)\), the slope of the tangent line to the graph of \(f(t)=3^{t}\) at \(t=1\), by computing the slope of the secant line through (1, \(f(1)\) ) and \((1.0001, f(1.0001))\). (c) Approximate \(f^{\prime}(0)\), the slope of the tangent line to the graph of \(f(t)=3^{t}\) at \(t=0\), by computing the slope of the secant line through \((0, f(0))\) and \((0.0001, f(0.0001))\). (d) Sketch a rough graph of the slope function \(f^{\prime}\).

For Problems 1 through 9, simplify the following expressions. $$ \frac{\sqrt{2 x^{3}}+\sqrt{12 y^{3} x^{4}}}{\sqrt{6 y^{2} x^{5}}} $$

Consider two strains of bacteria, one \((E .\) coli \()\) whose population doubles every 20 minutes and another, strain \(X\), whose population doubles every 15 minutes. Suppose that at present the number of \(E\). coli is 600 and the number of bacteria of strain \(X\) is \(100 .\) (a) Express the number of \(\mathrm{E}\). coli as a function of \(h\), the number of hours from now. (b) Express the number of strain \(X\) bacteria as a function of \(h\), the number of hours from now. (c) After approximately how many hours will the populations be equal in number?

The Exploratory Problems indicated that exponential functions grow at a rate proportional to themselves, i.e., if \(f(x)=a^{x}\), then \(f^{\prime}(x)=k a^{x}\), for some constant \(k\). Approximate the appropriate constant if \(f(x)=7^{x}\).

For Problems 1 through 9, simplify the following expressions. $$ \frac{x^{-1}}{z x^{-1}+z^{-1}} $$

In the 1980 s the small town of Old Bethpage, New York, made the front page of the New York Times magazine section as an illustration of what was termed a "dying suburb." In Old Bethpage schools are being converted to nursing homes as the population ages and the baby boomers move out. Suppose that the number of school-age children in 1980 was \(C_{0}\), and was decreasing at a rate of \(6 \%\) per year. Let's assume that the number of school-age children continues to drop at a rate of \(6 \%\) each year. Let \(C(t)\) be the number of school-age children in Old Bethpage \(t\) years after 1980 . (a) Find \(C(t)\). (b) Express the number of school-age children in Old Bethpage in 1994 as a percentage of the 1980 population. (c) Use your calculator to estimate the year in which the population of school-age children in Old Bethpage will be half of its size in 1980 .

In the Exploratory Problems you approximated the derivatives of \(2^{x}, 3^{x}\), and \(10^{x}\) for various values of \(x\), and, after looking at your results, you conjectured about the patterns. Now, using the definition of the derivative of \(f\) at \(x=a\), we return to this, focusing on the function \(f(x)=5^{x}\). (a) Using the definition of the derivative of \(f\) at \(x=a\), $$ f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ give an expression for \(f^{\prime}(0)\), the slope of the tangent line to the graph of at \(x=0\). (b) Show that for the function \(f(x)=5^{x}\), the difference quotient, \(\frac{f(x+h)-f(x)}{h}\), is equal to \(f(x) \cdot \frac{f(h)-f(0)}{h}\). (c) Using the definition of derivative, $$ f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} $$ conclude that the derivative of \(f(x)=5^{x}\) is $$ f^{\prime}(0) \cdot f(x) $$ Notice that you have now proven that the derivative of \(5^{x}\) is proportional to \(5^{x}\), with the proportionality constant being the slope of the tangent line to \(5^{x}\) at \(x=0\). $$ f^{\prime}(x)=f^{\prime}(0) \cdot f(x) $$ (d) Approximate the slope of the tangent line to \(5^{x}\) at \(x=0\) numerically.

For Problems 1 through 9, simplify the following expressions. $$ \frac{z^{0} x^{-1} y^{-2}}{z^{-2} x^{-1} y^{2}} $$

In the middle of the \(1994-95\) academic year, in the middle of the week, in the middle of the day, there was a bank robbery and subsequent shootout in the middle of Harvard Square. Throughout the afternoon the news spread by word-of- mouth. Suppose that at the time of the occurrence 30 people know the story. Every 15 minutes each person who knows the news passes it along to one other person. Let \(N(t)\) be the number of people who know at time \(t\). (a) Make a table with time in one column and \(N(t)\) in the other. Identify the pattern and write \(N\) as a function of \(t\). (b) If you've written your equation for \(N(t)\) with \(t\) in minutes, convert to hours. If you've done it in hours, convert to minutes. Make a table to check your answers. (It's easy to make a mistake the first time you do this.)

For Problems 5 through 9 , differentiate the function given. $$ f(x)=x^{3} e^{x} $$

Let \(D(t), H(t)\), and \(J(t)\) represent the annual salaries (in dollars) of David, Henry, and Jennifer, and suppose that these functions are given by the following formulas, where \(t\) is in years. \(t=0\) corresponds to this year's salary, \(t=1\) to the salary one year from now, and so on. The domain of each function is \(t=0,1,2, \ldots\) up to retirement. $$ \begin{array}{c} D(t)=40,000+2500 t \\ H(t)=50,000(0.97)^{t} \\ J(t)=40,000(1.05)^{t} \end{array} $$ (a) Describe in words how each employee's salary is changing. (b) Suppose you are just four years away from retirement-you'll collect a salary for four years, including the present year. Which person's situation would you prefer to be your own? (c) If you are in your early twenties and looking forward to a long future with the company, which would you prefer?

For Problems 1 through 9, simplify the following expressions. $$ \frac{x^{n+1} y^{2 n}}{\left(\frac{x}{y}\right)^{n}} $$

Differentiate the function given. $$ f(x)=\frac{e^{2 x}}{x} $$

In Anton Chekov's play "Three Sisters," Lieutenant-Colonel Vershinin says the following in reply to Masha's complaint that much of her knowledge is unnecessary. "I don't think there can be a town so dull and dismal that intelligent and educated people are unnecessary in it. Let us suppose that of the hundred thousand people living in this town, which is, of course, uncultured and behind the times, there are only three of your sort. ... Life will get the better of you, but you will not disappear without a trace. After you there may appear perhaps six like you, then twelve and so on until such as you form a majority. In two or three hundred years life on earth will be unimaginably beautiful, marvelous. Man needs such a life and, though he hasn't it yet, he must have a presentiment of it, expect it, dream it, prepare for it; for that he must know more than his father and grandfather. And you complain about knowing a great deal that is unnecessary." Let us assume that Vershinin means that this doubling occurs every generation and take a generation to be 25 years. Suppose that the total population of the town remains unchanged. (a) In approximately how many years will the people "such as [Masha] form a majority"? (b) What percentage of the town will be "intelligent and educated" in the 200 years that Vershinin mentions? (c) Now assume that the total population grows at a rate of \(2 \%\) per year. Answer questions (a) and (b) with this new assumption.

For Problems 1 through 9, simplify the following expressions. $$ \frac{\left(a b^{x}\right)^{-2}}{(a b)^{-x}} $$

Differentiate the function given. $$ f(x)=3 e^{-x} $$

Many trainers recommend that at the start of the season, a cyclist should increase his or her weekly mileage by not more than \(15 \%\) each week. (a) If a cyclist maintains a "base" of 50 miles per week during the winter, what is his or her maximum recommended weekly mileage for the fifth week of the season? (b) Find a formula for \(M(w)\), the maximum weekly recommended mileage \(w\) weeks into the season. Assume that initially the cyclist has a base of \(A\) miles per week.

For Problems 1 through 9, simplify the following expressions. $$ \frac{(a b)^{-x}}{a^{-x}+b^{-y}} $$

For Problems 1 through 9, simplify the following expressions. $$ \frac{\left(a^{-x+1} b\right)^{3}}{\left(a^{2} b^{3}\right)^{x}} $$

Differentiate the function given. $$ f(x)=e^{2 x}\left(x^{2}+2 x+2\right) $$

For Problems 10 through 15, factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ b^{x+y}+b^{x} $$

Use the tangent line approximation of \(e^{x}\) at \(x=0\) to approximate \(e^{-1} .\) Is your answer larger than \(e^{-1}\) or smaller?

Factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ b^{2 x}+b^{x+1} $$

Find the equation of the line tangent to \(f(x)=e^{x}\) at \(x=1\).

(a) If rabbits grow according to \(R(t)=1010(2)^{t / 3}, t\) in years, after how many years does the rabbit population double? What is the percent increase in growth each year? (b) If the sheep population in Otrahonga, New Zealand, is growing according to \(S(t)=3162(1.065)^{t}, t\) in years, after approximately how many years does the sheep population double? What is the percent increase in growth each year?

Factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ 3 b^{2 x+1}-4 b^{2 x-1} $$

Differentiate the following. (a) \(f(x)=\frac{x^{2} e^{x}}{3}\) (b) \(f(x)=\frac{5 x^{2}}{3 e^{2}}\) (c) \(f(x)=\frac{1}{x e^{5 x}}\)

Exploratory: Which grows faster, \(2^{x}\) or \(x^{2}\) ? (a) Using what you know about these two functions and experimenting numerically and graphically, guess the following limits: i. \(\lim _{x \rightarrow \infty} x 2^{-x}\) ii. \(\lim _{x \rightarrow-\infty} \frac{x^{2}}{2^{x}} \quad\) iii. \(\lim _{x \rightarrow \infty} \frac{x^{2}}{2^{x}} \quad\) iv. \(\lim _{x \rightarrow \infty} \frac{2^{x}}{x^{2}}\) (b) For \(|x|\) large, which function is dominant, \(2^{x}\) or \(x^{2} ?\) Would you have answered differently if we looked at \(3^{x}\) and \(x^{3}\) instead?

Factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ \left(a b^{2}\right)^{x}+\left(\frac{a}{b}\right)^{-x} $$

Devaluation: Due to inflation, a dollar loses its purchasing power with time. Suppose that the dollar loses its purchasing power at a rate of \(2 \%\) per year. (a) Find a formula that gives us the purchasing power of \(\$ 1 t\) years from now. (b) Use your calculator to approximate the number of years it will take for the purchasing power of the dollar to be cut in half.

Factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ b^{3 x}-(2 b)^{-1+2 x} $$

A population of beavers is growing exponentially. In June 1993 (our benchmark year when \(t=0\) ) there were 100 beavers. In June \(1994(t=1)\) there were 130 beavers. (a) Write a function \(B(t)\) that gives the number of beavers at time \(t\). (b) What is the percent increase in the beaver population from one year to the next?

15\. Consider the function $$ f(x)=\frac{x^{2}}{e^{x}} $$ (a) Compute \(f^{\prime}(x)\). (The Quotient Rule is unnecessary.) (b) For what values of \(x\) is \(f^{\prime}(x)\) positive? For what values of \(x\) is \(f^{\prime}(x)\) negative? (c) For what values of \(x\) is \(f(x)\) increasing? For which is it decreasing? Give exact answers. (d) What is the smallest value ever taken on by \(f(x)\) ? Explain your reasoning.

For Problems 16 through 21, if the statement is always true, write "True;" if the statement is not always true, produce a counterexample. In these problems, \(a\) and \(b\) are positive constants. $$ \sqrt{a^{2 x}}=a^{x} $$

According to a report from the General Accounting Office, during the 14 -year period between the school year \(1980-1981\) and the school year \(1994-1995\), the average tuition at four-year public colleges increased by \(234 \%\). During the same period, average household income increased by \(82 \%\), and the Labor Department's Consumer Price Index (CPI) increased by 74\%. (Boston Globe, August 16, 1996.) (a) Assuming exponential growth, determine the annual percentage increase for each of these three measures. (b) The average cost of tuition in \(1994-1995\) was \(\$ 2865\) for in-state students. What was it in \(1980-1981\) ? (c) Starting with an initial value of one unit for each of the three quantities, average tuition at four-year public colleges, average household income, and the Consumer Price Index, sketch on a single set of axes the graphs of the three functions over this 14 -year period. (d) Suppose that a family has two children born 14 years apart. In \(1980-1981\), the tuition cost of sending the elder child to college represented \(15 \%\) of the family's total income. Assuming that their income increased at the same pace as the average household, what percent of their income was needed to send the younger child to college in \(1994-1995 ?\)

Suppose that in a certain scratch-ticket lottery game, the probability of winning with the purchase of one card is 1 in 500 , or \(0.2 \%\); hence, the probability of losing is \(100 \%-0.2 \%=99.8 \%\). But what if you buy more than one ticket? One way to calculate the probability that you will win at least once if you buy \(n\) tickets is to subtract from \(100 \%\) the probability that you will lose on all \(n\) cards. This is an easy calculation; the probability that you will lose two times in a row is \((99.8 \%)(99.8 \%)=\) (a) What is the probability that you will win at least once if you play three times? (b) Find a formula for \(P(n)\), the percentage chance of winning at least once if you play the game \(n\) times. (c) How many tickets must you buy in order to have a \(25 \%\) chance of winning? A \(50 \%\) chance? (d) Does doubling the number of tickets you buy also double your chances of winning? (e) Sketch a graph of \(P(n) .\) Use \([0,100]\) as the range of the graph. Explain the practical significance of any asymptotes.

If the statement is always true, write "True;" if the statement is not always true, produce a counterexample. In these problems, \(a\) and \(b\) are positive constants. $$ \sqrt{\frac{a^{2 x}}{b^{-2 x}}}=(a b)^{x} $$

For Problems 22 through 24, simplify as much as possible. $$ \frac{a^{x+y}-a^{2 x}}{a^{x}} $$

Simplify as much as possible. $$ \frac{a^{2 x}-b^{4 x}}{a^{x}+b^{x}} $$

Simplify as much as possible. $$ \frac{A^{4+p}-A^{5 p}}{A^{2+p}-A^{3 p}} $$

True or False: If the statement is always true, write "True;" if the statement is not always true, produce a counterexample. (a) \(\left(a^{2}+b^{2}\right)^{1 / 2}=a+b\) (b) \((a+b)^{-1}=\frac{1}{a+b}, \quad a, b \neq 0\) (c) \((a+b)^{-1}=\frac{1}{a}+\frac{1}{b}, \quad a, b \neq 0\) (d) \(R^{-1 / 2}=-\frac{1}{\sqrt{R}}, \quad R>0\) (e) \(x^{z}+x^{z}=2 x^{z}\) (f) \(x^{z} x^{z}=x^{\left(z^{2}\right)}\) (g) \(x^{z} x^{z}=x^{2 z}\)

Simplify as much as possible: (a) \(\frac{x^{2 y}+x^{y+2}}{x^{y}}\) (b) \(\frac{\frac{\sqrt{x}}{x^{-1 / 2} y}-1}{y-\frac{x^{2}}{y}}\) (c) \(\frac{A^{B+4}-A^{3 B}}{A^{B}\left(A^{2}-A^{B}\right)}\) (d) \(\frac{y^{3 w}-y^{w+4}}{y^{w}\left(y^{w}+y^{2}\right)}\)

In Problems 34 through 37, evaluate the limits. $$ \text { (a) } \lim _{x \rightarrow \infty}-2(1.1)^{-x} $$

Evaluate the limits. (a) \(\lim _{x \rightarrow \infty}\left(0.89^{x}-1\right)\) (b) \(\lim _{x \rightarrow-\infty}\left(0.89^{x}-1\right)\)